On the Total Variation of the Jacobian

Irene Fonseca Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA (fonseca@andrew.cmu.edu) Nicola Fusco Dipartimento di Matematica “R. Caccioppoli”, Università di Napoli, Via Cintia, 80126 Napoli, Italy (n.fusco@unina.it) Paolo Marcellini Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67A, 50134 Firenze, Italy (marcell@math.unifi.it)

Contents

1 Introduction 2 2 Statement of the main results 4 3 Geometrical interpretation 11 4 Det Du versus det Du 13 5 The general 2−d case 17 6 The “eight” curve 31 7 The n−dimensional case 37 8 Relaxation in the general polyconvex case 50 9 A relevant n−dimensional class of maps 52 10 Some 2− and 3−dimensional examples 54

Keywords: Jacobian, total variation, relaxation, singular measures, topological degree. 2000 Mathematics Subject Classification: 49J45, 49K20, 35F30, 74B20, 74G70

1 Introduction

Well established theories in the Calculus of Variations and in Partial Differential Equations have been challenged in recent years by new phenomena in solid physics and in materials sciences which demand innovative approaches and new ideas. In this paper we address the study of the Jacobian determinant det Du of fields u : Ω → ℝ^{n} outside the traditional regularity space W^{1,n} (Ω; ℝ^{n}).

The primary motivation that led to this subject is threefold:

– applications of high-temperature superconducting magnetic materials have had a tremendous impact in the development of a whole mathematical theory based on Ginzburg-Landau model, and where vorticity plays a very important role (see [15]). As pointed out by Jerrard and Soner in [40], the formation of vortices is accompanied by highly localized defectiveness at points or along rays, and the ability to extend and interpret the mechanism of change of volume dictated by the Jacobian to the range p ∈ (n − 1, n) may shed some light into this theory;

– the formation of (radially symmetric) holes in rubber-like (nonlinear) elastic materials is studied in the theory of cavitation, and its advance is heavily hinged on the characterization of the distributional Jacobian determinant (see (1); see also (39) below) for certain ranges of p < n. This issue has attracted the attention of several mathematical researchers for the past twenty years, and although some progress has been made, pioneered by John Ball [4], [5], and followed by James and Spector [39], Müller and Spector [53], Sivaloganathan [57], by Marcellini [46], using an alternative (and closer to the point of view of the present paper) approach, and many others, we believe that we have only scratched the surface of a very rich field in the Calculus of Variations virtually unexplored until recently;

– the relevance of the distributional Jacobian determinant in the study of harmonic mappings with singularities (see [9]), and in the study of density results of smooth functions in H^{1}(B(0, 1); S^{2}), where B(0, 1) ⊂ ℝ^{3}. Bethuel [6] showed that this density result holds for u ∈ H^{1}(B(0, 1); S^{2}) if det Du = 0.

To fix the notations, we consider a vector-valued map u : Ω ⊂ ℝ^{n} → ℝ^{n}, defined on an open set Ω of ℝ^{n}, for some n ≥ 2. We denote by Du = Du (x) the gradient of u at x ≡ (x_{1}, x_{2}, …, x_{n}) ∈ Ω, i.e., the n × n matrix (Jacobian matrix) of the partial derivatives of u ≡ (u^{1}, u^{2}, …, u^{n}) and by

det Du (x) := \frac{∂ (u^{1}, u^{2}, …, u^{n})}{∂ (x_{1}, x_{2}, …, x_{n})}

its determinant (Jacobian determinant).

If u ∈ W^{1,n} (Ω; ℝ^{n}), since |det Du (x)| ≤ n^{−n/2} |Du (x)|^{n}, then the Jacobian determinant det Du is a function of class L^{1} (Ω; ℝ^{n}). In this case the set function

E ⊂ Ω ⟶ m (E) := ∫_{E} det Du (x) dx

is a measure in Ω, whose total variation |m| in Ω is given by

|m| (Ω) := ∫_{Ω} |det Du (x)| dx .

When u ∉ W^{1,n} (Ω; ℝ^{n}) it may still be possible to consider the distributional Jacobian determinant

Det Du := ∑_{i=1}^{n} (−1)^{i+1} \frac{∂}{∂x_{i}} (u^{1} \frac{∂ (u^{2}, …, u^{n})}{∂ (x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n})})  (1)

(or any other permutation in the set {u^{1}, u^{2}, …, u^{n}}, with the sign of the permutation), which coincides almost everywhere with the pointwise Jacobian determinant det Du if u ∈ W^{1,n} (Ω; ℝ^{n}), but which may be different otherwise. The definition of the distributional Jacobian determinant Det Du is based on integration by parts of the formal expression in (1), after multiplication by a test function.

To render the definition mathematically precise it is then necessary to make some assumptions on u. We may assume that u^{1} (or, for symmetry reasons, also the full vector u) is bounded and the gradient Du is of class L^{n−1} (or, more generally, the (n − 1) × n matrix (Du^{2}, …, Du^{n}) is of class L^{n−1}), i.e., u ∈ L^{∞} (Ω; ℝ^{n}) ∩ W^{1,n−1} (Ω; ℝ^{n}). Another possibility is to require that u ∈ W^{1,p} (Ω; ℝ^{n}) for some p > n^{2}/ (n + 1) (the strict inequality is useful for compactness reasons); in fact, in this case by the Sobolev Imbedding Theorem u ∈ L^{n^{2}} (Ω; ℝ^{n}) and the products in (1) are well defined in L^{1} because 1/n^{2} + (n − 1) · (n + 1) /n^{2} = 1. Local summability assumptions are also allowed. In this paper we assume that u ∈ L_{loc}^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) for some p > n − 1.

Since the fundamental work of Morrey [48], who treated weak continuity properties of Det Du in (1) (see also Reshetnyak [56]), Det Du has played a pivotal role in the calculus of variations, in addition to its well established one in geometry. Ball pointed out in [4] some relevant applications of the Jacobian determinant to nonlinear elasticity, and sharp weak continuity properties of the Jacobian has been investigated in a series of papers by Müller, starting with [49]. More detailed description of the state of the art in this subject may be found in Section 4.

In recent years, several attempts have been made to establish relations between the distribution Det Du and the “total variation” of the Jacobian determinant det Du (x). One possible definition is based on the following limit formula. Given u ∈ L_{loc}^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) for some p > n − 1, the total variation T V (u, Ω) of the Jacobian determinant is defined by

T V (u, Ω) = inf {lim inf_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx :  (2)
u_{h} ⇀ u weakly in W^{1,p} (Ω; ℝ^{n}) , u_{h} ∈ W^{1,n} (Ω; ℝ^{n})} .

Note that, a priori, definition (2) may depend on p and, more precisely, we should use the notation T V_{p} (u, Ω) instead of T V (u, Ω). However, the representation formulas for T V (u, Ω) given in this paper turn out to be independent of p, and, surprisingly, it can be shown that, for certain classes of functions u, weak convergence in W^{1,p} (Ω; ℝ^{n}) may be equivalently replaced by strong convergence (see (26)). Similar definitions may be proposed under other summability assumptions on u.

This approach has been considered by Marcellini [45], Giaquinta, Modica and Souček [35], [36], Fonseca and Marcellini [27], Bouchitté, Fonseca and Malý [8]. In particular, Marcellini [45] and Fonseca and Marcellini [27] noticed that the total variation of the Jacobian determinant may have a nonzero singular part, while Bouchitté, Fonseca and Malý [8] proved that this singular part is a measure. Giaquinta, Modica and Souček [35], [36] found that the lower limit in (2) can be different from the total variation of the measure Det Du. More comments and references are given in Section 4.

Notice that it has been first noted by Malý [41] and by Giaquinta, Modica and Souček [35] (see also Jerrard and Soner [40]) that, for some maps u ∈ L^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) with p ∈ (n − 1, n), it may happen that the distribution Det Du is identically equal to zero while the total variation of the Jacobian determinant is different from zero. Also, when Det Du is a measure, it turns out that, in general, the total variation of the Jacobian determinant det Du (x) is not the total variation of the measure Det Du. Some precise (from a quantitative point of view) examples illustrating this phenomenon are proposed in Section 10. In Section 9 we compute the total variation of a class of singular maps u : Ω → S^{n−1} ⊂ ℝ^{n}, playing a central role in the analysis of Jerrard and Soner [40], defined by

u (x) := \frac{w (x) − w (0)}{|w (x) − w (0)|} ,  (3)

where w : Ω → ℝ^{n} is a locally Lipschitz-continuous map, classically differentiable at x = 0 and such that det Dw (0) ≠ 0. We find that the total variation of the Jacobian determinant of u in Ω (an open set of ℝ^{n} containing the origin) is equal to the measure ω_{n} of the unit ball.

The aim of this paper is to give an explicit characterization of the total variation T V (u, Ω) of the Jacobian determinant det Du (x), defined in (2), for some classes of functions u ∈ L_{loc}^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) with p > n − 1, in particular for those u locally Lipschitz-continuous away from a given point x_{0} ∈ Ω (and thus with the Jacobian determinant det Du possibly singular only at x_{0}).

Statements of the main results are given in the following Section 2. In Section 3 we relate the notion of total variation of the Jacobian determinant to the topological degree. A relevant geometrical interpretation is given by Corollary 16 of Section 3. In particular, denoting by B_{1} the unit ball of ℝ^{n} and by S^{n−1} := ∂B_{1} its boundary, we prove that, if v : S^{n−1} → S^{n−1} is a map of class C^{1} onto S^{n−1}, locally invertible with local inverse of class C^{1} at any point of S^{n−1}, and if u : B_{1}\ {0} → S^{n−1} is defined by u (x) := v (\frac{x}{|x|}), then the total variation T V (u, B_{1}) of the Jacobian determinant of u may be expressed in terms of the topological degree of the maps v and ṽ, where ṽ : B_{1} → B_{1} is any Lipschitz-continuous extension of v to the unit ball B_{1}. Precisely,

T V (u, B_{1}) = ω_{n} |deg v| = ω_{n} |deg ṽ| .  (4)

Note that formula (4) does not hold, in general, if the map v : S^{n−1} → ℝ^{n} takes values on a set v (S^{n−1}) not diffeomorphic to S^{n−1} (see Theorem 4 and the examples of Section 10).

Section 4 is dedicated to explaining how the study of the total variation T V (u, Ω) fits squarely within the framework of relaxation problems with nonstandard growth conditions. In Section 5 we present a thorough study of the 2−d case, which plays a very special role. In fact, in two dimensions we are able to perform a deeper analysis and to find more general assumptions which allow us to characterize fully the total variation T V (u, Ω). In particular, it is possible to identify T V (u, Ω) of maps u : B_{1} ⊂ ℝ^{2} → Γ, with values on a set Γ which is the boundary of a simply connected domain D ⊂ ℝ^{2}, starshaped with respect to a point ξ in the interior of D (for example, when Γ = S^{1} is the boundary of the unit ball B_{1}). We emphasize Lemma 23, which we call “the umbrella lemma”, and which plays a crucial role in our argument, as explained in Section 5.

In Section 7 we move on to the general n−dimensional framework, and in Section 8 we apply the results thus obtained to the study of relaxation of polyconvex functionals. Indeed, we provide an explicit representation formula for the related energy associated to the polyconvex integral functional

F (u, Ω) := ∫_{Ω} g (M (Du)) dx ,  (5)

where g : ℝ^{N} → [0, +∞) is a convex function, M (Du) is the map with values in ℝ^{N}, N = ∑_{j=1}^{n} \binom{n}{j}^{2}, defined by

M (Du) := (Du, adj_{2}Du, …, adj_{n−1}Du, det Du) ,

and where adj_{j}Du denotes, for every j = 2, …, n − 1, the matrix of all minors j × j of Du.

Finally, in Section 9 we study in detail T V (u, Ω) when u is as in (3). Additional 2−dimensional and 3−dimensional examples are proposed in Section 10. The special, but representative, case analyzed in Section 6 concerns maps u : B_{1} ⊂ ℝ^{2} → γ = γ^{+} ∪ γ^{−}, were γ is the “eight” curve, i.e., the union of the two tangent circles γ^{±} = {(x_{1}, x_{2}) ∈ ℝ^{2} : (x_{1} ∓ 1)^{2} + (x_{2})^{2} = 1} in ℝ^{2}. In particular, we show (see Theorem 4 and Section 10) that in general formula (4), which relates the total variation T V (u, Ω) of the Jacobian determinant with the topological degree, does not hold if the map u : B_{1} ⊂ ℝ^{2} → γ takes values on the “eight” curve γ.

2 Statement of the main results

In this section we state several representation formulas for the total variation T V (u, Ω) of the Jacobian determinant, defined in (2). We consider first the 2−d case in detail, and in the second part of this section we describe the general n−dimensional case.

The reason why we focus specifically on the case n = 2 is twofold: the assumptions needed are more general than in the case n ≥ 2; we believe that it is easier to follow the main ideas of this paper, by dissociating the technical aspects from the methods.

In order to fix the notations, here we consider x_{0} = 0 and Ω ⊂ ℝ^{2} is an open set containing the origin. With an obvious abuse of notation, we write u (x) = u (x_{1}, x_{2}) = u (ϱ, ϑ), where (ρ, ϑ), ϱ ≥ 0, 0 ≤ ϑ ≤ 2π, are the polar coordinates in ℝ^{2}. We also denote by D_{τ} u the tangential derivative of u (in the τ = (− sin ϑ, cos ϑ) direction), which is related to the (vector-valued) derivative u_{ϑ} by the formula

u_{ϑ} =: \frac{∂u (ϱ cos ϑ, ϱ sin ϑ)}{∂ϑ} = ϱ [−u_{x_{1}} sin ϑ + u_{x_{2}} cos ϑ] = ϱD_{τ} u .

We denote by v : [0, 2π] → Γ ⊂ ℝ^{2} a Lipschitz-continuous map, with v (0) = v (2π), with components v (ϑ) = (v^{1} (ϑ) , v^{2} (ϑ)), and with values on a curve Γ ⊇ v ([0, 2π]). We assume that Γ can be parametrized in the following way

Γ = {ξ + r (ϑ) (cos ϑ, sin ϑ) : ϑ ∈ [0, 2π]} ,  (6)

where r (ϑ) is a piecewise C^{1}-function such that r (0) = r (2π), and r (ϑ) ≥ r_{0} for every ϑ ∈ [0, 2π] and for some r_{0} > 0. Condition (6) reduces to saying that Γ is the boundary of a domain

D := {ξ + ϱ (cos ϑ, sin ϑ) : ϑ ∈ [0, 2π] , 0 ≤ ϱ ≤ r (ϑ)} ,  (7)

starshaped with respect to a point ξ in the interior of D. In Section 5 we prove the following result.

Theorem 1 (General result in 2−d) Let u be a function of class W^{1,p} (Ω; ℝ^{2})∩W_{loc}^{1,∞} (Ω\ {0}; ℝ^{2}) for some p ∈ (1, 2). Let v : [0, 2π] → Γ, v (ϑ) = (v^{1} (ϑ) , v^{2} (ϑ)), ϑ ∈ [0, 2π], be a Lipschitz-continuous map, with v (0) = v (2π) and Γ as in (6), and such that

lim_{ϱ→0} ∥u (ϱ, ·) − v (·)∥_{L^{∞}((0,2π);ℝ^{2})} = 0 .  (8)

If the tangential derivative D_{τ} u of u satisfies the bound

sup_{ϱ>0} \frac{1}{ϱ^{2−p}} ∫_{B_{ϱ}} |D_{τ} u|^{p} dx = sup_{ϱ>0} \frac{1}{ϱ^{2−p}} ∫_{0}^{ϱ} r^{1−p} dr ∫_{0}^{2π} |u_{ϑ} (r, ϑ)|^{p} dϑ ≤ M_{0}  (9)

for a positive constant M_{0}, then the total variation of u is given by

T V (u, Ω) = ∫_{Ω} |det Du (x)| dx + \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .

Note that, by (8), there exists r > 0 such that B_{r} ⊂ Ω and u ∈ L^{∞} (B_{r}; ℝ^{2}). Therefore in the statement of Theorem 1 we have in fact u ∈ L_{loc}^{∞} (Ω; ℝ^{2}) ∩ W^{1,p} (Ω; ℝ^{2}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{2}) for some p ∈ (1, 2). Moreover, the assumption of Lipschitz-continuous of v can be replaced by the weaker assumption that v ∈ W^{1,p} ((0, 2π) ; ℝ^{2}); however we shall not discuss this latter case in details.

Consider the particular case in which the map u = u (ϱ, ϑ) does not depend on ϱ, that is u = u (ϑ). Then, as a function of ϑ, u = u (ϑ) : [0, 2π] → ℝ^{2} is a Lipschitz-continuous map and u (0) = u (2π). Considered as a function of two variables, i.e., u : Ω = B_{1} → ℝ^{2} constant with respect to ϱ ∈ (0, 1], it turns out that u ∈ L^{∞} (Ω; ℝ^{2}) ∩ W^{1,p} (Ω; ℝ^{2}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{2}) for every p ∈ [1, 2), but u ∉ W^{1,2} (Ω; ℝ^{2}) unless u (ϑ) is constant.

From the previous result, with u = v, we immediately obtain the following consequence.

Corollary 2 (Radially independent maps in 2−d) Let Γ be as in (6), and let u = v : [0, 2π] → Γ be a Lipschitz-continuous map such that v (0) = v (2π). Then det Du (x) = 0 for almost every x ∈ ℝ^{2} and the total variation of the Jacobian determinant is given by

T V (u, Ω) = \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .  (10)

We observe that formula (10) has a relevant geometrical meaning; in fact the right hand side represents the “winding number” of the curve v = (v^{1}, v^{2}). See Section 3 for a further discussion on the geometric interpretation of (10).

With the aim to compare the previous result with the n−dimensional results given below, we present the following equivalent formulation of Corollary 2.

Corollary 3 (Analytic interpretation in 2−d) Let Γ be as in (6), and let v : [0, 2π] → Γ be a Lipschitz-continuous map such that v (0) = v (2π). Then the total variation T V (u, Ω) is given by

T V (u, Ω) = |∫_{B_{1}} det Dũ (x) dx| ,  (11)

where ũ : B_{1} → ℝ^{2} is any Lipschitz-continuous extension of v to B_{1}.

Note the surprising fact that the integral in the right hand side of (11) (and in (10) as well) appears with the absolute value outside the integral sign, and not inside!

Another relevant 2−dimensional result is related to the “eight” curve in ℝ^{2}, i.e., to the union γ of the two circles γ^{+}, γ^{−}, of radius 1 with centers at (1, 0) and at (−1, 0) respectively. Some explicit examples related to the “eight” curve are given in Section 10. Below we present two estimates, an upper bound and a lower bound, which will allow us to study the examples in Section 10. However, we stress the fact that some cases related to the “eight” curve remain unsolved (see Remark 5 below). The following Theorem is proved in Section 6.

Theorem 4 (The “eight” curve) Let γ = γ^{+} ∪ γ^{−} ⊂ ℝ^{2} be the union of the two circles of radius 1 with centers at (1, 0) and at (−1, 0). Let v : [0, 2π] → γ be a Lipschitz-continuous curve, with parametric representation v (ϑ) = (v^{1} (ϑ) , v^{2} (ϑ)), ϑ ∈ [0, 2π], such that v (0) = v (2π). Let (I_{j})_{j∈ℕ} be a sequence of disjoint open intervals (possibly empty) of [0, 2π] such that the image v (I_{j}) is contained either in γ^{+} or in γ^{−}, and v (ϑ) = (0, 0) when ϑ ∉ ∪_{j∈ℕ}I_{j}. Then, with u (x) := v (x/ |x|), the following upper estimate holds

T V (u, B_{1}) ≤ \frac{1}{2} ∑_{j∈ℕ} |∫_{I_{j}} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ | .  (12)

For the lower estimate, we denote by I_{j}^{+}, with the + sign, any previous interval I_{j} such that v (I_{j}) ⊂ γ^{+}, and by I_{k}^{−} any previous interval I_{k} such that v (I_{k}) ⊂ γ^{−}. Then we also have

T V (u, B_{1}) ≥ \frac{1}{2} \{|∑_{j∈ℕ}∫_{I_{j}^{+}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| + |∑_{k∈ℕ}∫_{I_{k}^{−}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ|\} .  (13)

Remark 5 If the curve v : [0, 2π] → γ = γ^{+} ∪ γ^{−} admits only two intervals I_{1}^{+} and I_{2}^{−} where respectively v (I_{1}^{+}) ⊂ γ^{+}, v (I_{2}^{−}) ⊂ γ^{−}, then the above estimates for T V (u, B_{1}) are in fact equalities. The same happens if the intervals are three, say I_{1}^{+}, I_{2}^{−} and I_{3}^{+}; in fact this case can be reduced to the previous one by periodicity. If the intervals are four, say I_{1}^{+}, I_{2}^{−}, I_{3}^{+} and I_{4}^{−}, then we may have a gap between the lower bound and the upper bound stated in Theorem 4, unless the integral of v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1} has the same sign, respectively in I_{1}^{+}, I_{3}^{+} and in I_{2}^{−}, I_{4}^{−}. These considerations are utilized to study some of the examples of Section 10.

Moving to the n−dimensional case, we first establish by Theorem 6 a general inequality between the total variation of the distributional determinant Det Du (see (1)), that we denote by | Det Du|(Ω), and the total variation T V (u, Ω) if the Jacobian, defined in (2). Note that, in the first half of the statement of the next theorem, we do not assume that u ∈ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}), while in the second half, we require that u ∈ W_{loc}^{1,n} (Ω\ {0}; ℝ^{n}).

Theorem 6 (Comparison between | Det Du|(Ω) and T V (u, Ω)) Let p > n−1 and let u ∈ L^{∞} (Ω; ℝ^{n})∩ W^{1,p} (Ω; ℝ^{n}). If T V (u, Ω) < +∞, then T V (u, ·) and Det Du are finite Radon measures, det Du ∈ L^{1} (Ω), and

T V (u, A) = ∫_{A} |det Du (x)| dx + λ_{s} (A) ,  (14)
Det Du (A) = ∫_{A} det Du (x) dx + μ_{s} (A) ,  (15)

for every open set A ⊂ Ω, where λ_{s}, μ_{s}, are finite Radon measures, singular with respect to the Lebesgue measure L^{n}, and |μ_{s}| ≤ λ_{s}, i.e., for every open set A ⊂ Ω,

| Det Du| (A) ≤ T V (u, A) .  (16)

If, in addition, u ∈ W_{loc}^{1,n} (Ω\ {0}; ℝ^{n}), then λ_{s} = λδ_{0}, μ_{s} = μδ_{0}, for some constants λ ≥ 0, μ ∈ ℝ, with |μ| ≤ λ, where δ_{0} is the Dirac mass at the origin.

The proof of Theorem 6 is presented at the end of Section 4. We note that one of the main contributions of this paper is the identification of the defect constants λ ≥ 0, μ ∈ ℝ. However, the arguments used in this work are self contained and independent of the tools and thecniques needed to establish Theorem 6.

Let us denote by B_{r} the ball in ℝ^{n}, n ≥ 2, with center in 0 and radius r > 0. In particular, B_{1} is the ball of radius r = 1 and ∂B_{1} = S^{n−1} is its boundary.

We call the attention of the reader to the fact that, in dealing with the general n−dimensional case, we denote by v a map from S^{n−1} into ℝ^{n}, while in 2−d v = v (ϑ) does not denote a map from S^{1} into ℝ^{2}, but instead a periodic function from [0, 2π] into ℝ^{2}. Therefore, if \overline{v} is the corresponding map from S^{1} into ℝ^{2}, then we have v (ϑ) = \overline{v} (cos ϑ, sin ϑ).

Let ω_{0} ∈ S^{n−1} be fixed. For every j ∈ {1, 2, …, n − 1} let τ_{j} : S^{n−1}\ {ω_{0}} → ∂B_{1} by a vector field of class C^{1} such that, for every x ∈ S^{n−1}\ {ω_{0}}, the set of vectors {τ_{1} (ω) , τ_{2} (ω) , …, τ_{n−1} (ω)} is an orthonormal basis for the tangent plane to the surface ∂B_{1} at the point ω.

The following theorem provides a general representation formula for the total variation of the distributional determinant |Det Du| (Ω). Note that, under the assumption u ∈ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}), by formula (19) we give a representation of the total variation of the singular measure μ_{s} in (14).

Theorem 7 (Total variation of the distributional determinant) Let n ≥ 2 and let Ω be an open set containing the origin. Let u ∈ W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for some p ∈ (n − 1, n).

Let v : ∂B_{1} = S^{n−1} → ℝ^{n}, v ∈ W^{1,∞} (S^{n−1}; ℝ^{n}), v = (v^{1}, v^{2}, …, v^{n}), be a Lipschitz-continuous map such that

lim_{ϱ→0^{+}} max {|u (ϱω) − v (ω)| : ω ∈ S^{n−1}} = 0 .  (17)

Let us assume that

sup_{ϱ>0} \frac{1}{ϱ^{n−p}} ∫_{B_{ϱ}} |D_{τ} u|^{p} dx ≤ M_{0}  (18)

for a positive constant M_{0}. If det Du ∈ L^{1} (Ω) then Det Du is a Radon measure and its the total variation |Det Du| is given by

|Det Du| (Ω) = ∫_{Ω} |det Du (x)| dx  (19)

+\frac{1}{n} |∫_{∂B_{1}} ∑_{i=1}^{n} (−1)^{i+1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) dH^{n−1}| .

Moreover, if we denote by ũ : B_{1} → ℝ^{n} any Lipschitz-continuous extension of v to B_{1}, then

|Det Du| (Ω) = ∫_{Ω} |det Du (x)| dx + |∫_{B_{1}} det Dũ (x) dx| .  (20)

By assumption (17) there exists r > 0 such that u ∈ L^{∞} (B_{r}; ℝ^{2}). Thus, in the statement of Theorem 7 (and in Theorem 10 below), we actually have that u is a function of class L_{loc}^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for some p ∈ (n − 1, n).

Remark 8 A simple calculation shows that in 2−d the last term on the right hand side of (19) reduces to

\frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| ,

where v : [0, 2π] → ℝ^{2} is the asymptotic limit map in (8). Indeed, denoting by \overline{v} : S^{1} → ℝ^{2} the map related to v through the condition v (ϑ) := \overline{v} (cos ϑ, sin ϑ), we have

\frac{dv^{i}}{dϑ} = \frac{∂\overline{v}^{i}}{∂x_{1}} (− sin ϑ) + \frac{∂\overline{v}^{i}}{∂x_{2}} cos ϑ , i = 1, 2,

and, since the unit tangent vector τ : [0, 2π] → ∂B_{1} can be represented by τ (θ) = (− sin ϑ, cos ϑ), we obtain

\frac{dv^{i}}{dϑ} = \frac{∂\overline{v}^{i}}{∂τ} , i = 1, 2.

With the notation ω = (\frac{x_{1}}{|x|}, \frac{x_{2}}{|x|}) = (cos ϑ, sin ϑ) ∈ ∂B_{1} = S^{1}, we finally have

∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ = ∫_{0}^{2π} \{\overline{v}^{1} \frac{∂\overline{v}^{2}}{∂τ} − \overline{v}^{2} \frac{∂\overline{v}^{1}}{∂τ} \} dϑ

= ∫_{∂B_{1}} ∑_{i=1}^{2} (−1)^{i+1} \overline{v}^{i} (ω) \frac{d\overline{v}^{i}}{dτ} (ω) dH^{1} .

Therefore (19) in 2−d becomes

|Det Du| (Ω) = ∫_{Ω} |det Du (x)| dx + \frac{1}{2} |∫_{0}^{2π} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| ,

and the conclusion of Theorem 1 now can be restated in the form

T V (u, Ω) = |Det Du| (Ω) .

Remark 9 In the case of the “eight” curve studied by Theorem 4, with v : [0, 2π] → γ = γ^{+} ∪γ^{−} ⊂ ℝ^{2} and u (x) = v (x/ |x|), we have

T V (u, B_{1}) ≥ \frac{1}{2} \{|∑_{j∈ℕ}∫_{I_{j}^{+}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| + |∑_{k∈ℕ}∫_{I_{k}^{−}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ|\}

≥ \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| = |Det Du| (B_{1}) .  (21)

Therefore, as in the general case (see Theorem 6 and (16) in particular), T V (u, B_{1}) ≥ |Det Du| (B_{1}). Moreover, in view of the inequalities in (21), we can easily find an example such that the strict inequality T V (u, B_{1}) > |Det Du| (B_{1}) holds. See Section 10.

Next we state the main result for the n−d case, analogous to Theorem 1. The proof of the theorem may be found in Section 7.

Theorem 10 (General result in n−d) Let n ≥ 2 and let Ω be an open set containing the origin. Let u ∈ W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for some p ∈ (n − 1, n) and let v ∈ W^{1,∞} (S^{n−1}; ℝ^{n}) satisfying (17) and (18). If det Du ∉ L^{1} (Ω) then T V (u, Ω) = +∞. If det Du ∈ L^{1} (Ω), then the total variation of the distributional determinant |Det Du| (Ω) is given by (19) and T V (u, Ω) ≥ |Det Du| (Ω). Moreover, if the quantity

∑_{i=1}^{n} (−1)^{i+1} v^{i} \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})}  (22)

has constant sign H^{n−1}−almost everywhere on ∂B_{1}, then

T V (u, Ω) = |Det Du| (Ω) .  (23)

In Section 9 we apply Theorem 10 to calculate explicitly the total variation of the singular map u : Ω\ \{0\} → ℝ^{n}, u (x) = \frac{w(x)−w(0)}{|w(x)−w(0)|}, where w is a map differentiable at x = 0, with det Dw (0) ≠ 0, to obtain T V (u, Ω) = |B_{1}| = ω_{n}.

Remark 11 We conjecture that formula (23) holds independently of the sign condition (22) for a certain of subclass of mappings u with asymptotic limit v at x = 0, in particular if v : S^{n−1} → ℝ^{n} takes values in S^{n−1}. Theorem 1 above asserts that this conjecture is true in the 2−dimensional case, and when v (S^{1}) is the set Γ in (6), boundary of a starshaped set. With the Example 48 we propose a 3−d case where the conjecture is also true. However, if v (S^{n−1}) is not diffeomorphic to S^{n−1}, as in the case of the “eight” curve considered in Theorem 4 (see also the examples of Section 10), then the representation formula for T V (v, B_{1}) should take into account the topology of v (S^{n−1}).

As further applications of Theorem 10, now we consider radially independent maps u : Ω → ℝ^{n}, defined through a Lipschitz-continuous map v : S^{n−1} → ℝ^{n} by the position

u (x) := v (\frac{x}{|x|}) , ∀ x ∈ B_{1}\ \{0\} .

Clearly u ∈ W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for every p ∈ [1, n), but u ∉ W^{1,n} (Ω; ℝ^{n}) unless v is a constant function. We obtain immediately from Theorem 10 the following result.

Corollary 12 (Radially independent maps) Let v : ∂B_{1} = S^{n−1} → ℝ^{n}, v = (v^{1}, v^{2}, …, v^{n}), be a Lipschitz-continuous map. For every open set Ω containing the origin we consider the map u : Ω → ℝ^{n}, defined by u (x) := v (x/ |x|) for x ∈ Ω\ {0}. For every p ∈ (n − 1, n) the total variation of the Jacobian of u is given by

T V (u, Ω) = \frac{1}{n} |∫_{∂B_{1}} ∑_{i=1}^{n} (−1)^{i+1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) dH^{n−1}| ,  (24)

provided the quantity (22) has constant sign H^{n−1}−almost everywhere on ∂B_{1}.

The following result is similarly to Corollary 3, valid in the 2−d case.

Corollary 13 (Analytic interpretation in n−d) Let v : S^{n−1} → ℝ^{n} be a Lipschitz-continuous map, let Ω be an open set containing the origin, and let u : Ω → ℝ^{n} be defined by u (x) := v (x/ |x|) for x ∈ Ω\ {0}. Denote by ũ : B_{1} → ℝ^{n} the Lipschitz-continuous extension of v to B_{1} given by ũ (0) = 0 and

ũ (x) := |x| · v (\frac{x}{|x|}) , ∀ x ∈ B_{1}\ \{0\} .

If the Jacobian det Dũ (x) has constant sign H^{n−1}−almost everywhere on B_{1}, then

T V (u, Ω) = |∫_{B_{1}} det Dũ (x) dx| .  (25)

Remark 14 Let us assume that v : S^{n−1} → S^{n−1} is a map of class C^{1} onto S^{n−1}, locally invertible with C^{1} local inverse at any point of S^{n−1}. If ũ is defined as before by ũ (x) = |x| · v (x/ |x|), then also ũ : B_{1} → B_{1} is a map of class C^{1} and it is locally invertible with C^{1} local inverse at any point of B_{1}\ {0}. Then the assumption of Corollary 13 is satisfied. Indeed, det Dũ (x) ≠ 0 for every x ∈ B_{1}\ {0} and, by continuity, det Dũ (x) has constant sign in B_{1}\ {0}. We also notice that, by (101) of Lemma 39, when η (t) = t we have

det Dũ (x) = ∑_{i=1}^{n} (−1)^{i+1} v^{i} (\frac{x}{|x|}) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (\frac{x}{|x|}) ,

therefore the sign assumption in Corollary 13 is equivalent to the sign assumption of Theorem 10.

A final remark about the definition (2) of the total variation T V (u, Ω) of the Jacobian determinant det Du (x). As before, consider u ∈ L^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) for some p > n − 1. The definition in (2) of T V (u, Ω) is based on the convergence of a generic sequence {u_{h}}_{h∈ℕ} ⊂ W^{1,n} (Ω; ℝ^{n}) to u in the weak topology of W^{1,p} (Ω; ℝ^{n}). Instead, we could consider the strong norm topology and give the following definition of T V^{s} (u, Ω):

T V^{s} (u, Ω) = inf {lim inf_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx :  (26)
u_{h} → u strongly in W^{1,p} (Ω; ℝ^{n}) , u_{h} ∈ W^{1,n} (Ω; ℝ^{n})} .

Clearly we have

T V (u, Ω) ≤ T V^{s} (u, Ω) , ∀ u ∈ L^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) .  (27)

However it is interesting, and somewhat surprising, to observe that Theorems 1, 4 and 10 (as well as Corollaries 2 and 12) still hold if we replace T V (u, Ω) by T V^{s} (u, Ω). In particular, under the assumptions of Theorems 1 and 10 we have indeed

T V (u, Ω) = T V^{s} (u, Ω) ,  (28)

for every open set Ω ⊂ ℝ^{n}, and for every u ∈ L^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) with p > n − 1.

3 Geometrical interpretation

In this section we give a geometrical interpretation of the results stated in Section 2, by means of the notion of topological degree of maps between manifolds.

We recall that if w : Ω → ℝ^{n} is a Lipschitz-continuous map, then the topological degree of the map w at a point y ∈ ℝ^{n} is

deg (w, Ω, y) := ∑_{x∈w^{−1}(y)∩A(w)} sign (det Dw (x)) ,

where A (w) = {x ∈ Ω : w is differentiable at x}. The degree of the map w in the set Ω, denoted by deg w, is

deg w := \frac{1}{|w (Ω)|} ∫_{w(Ω)} ∑_{x∈w^{−1}(y)∩A(w)} sign (det Dw (x)) dy ,  (29)

= \frac{1}{|w (Ω)|} ∫_{w(Ω)} ∑_{x∈w^{−1}(y)} sign (det Dw (x)) dy

(above we used the fact that, since w is a Lipschitz-continuous map, then the measure of the sets Ω\A (w) and of its image w (Ω\A (w)) are equal to zero). See the books by Giaquinta, Modica and Souček [36] and by Fonseca and Gangbo [24] for more details.

It is well known that

∫_{Ω} det Dw (x) dx = ∫_{w(Ω)} deg (w, Ω, y) dy ,

and thus

deg w = \frac{1}{|w (Ω)|} ∫_{Ω} det Dw (x) dx .  (30)

Using of the symbol # to denote the cardinality of the set, we have

∫_{Ω} |det Dw (x)| dx = ∫_{w(Ω)} # {x ∈ Ω : w (x) = y} dy .  (31)

For our purposes it is also useful to recall the definition of degree of a map v : S^{n−1} → S^{n−1}, v onto S^{n−1}. To this aim let us denote by T_{ω} the tangential plane to S^{n−1} at the point ω ∈ S^{n−1}. If v is Lipschitz-continuous, then for H^{n−1}−a.e. ω ∈ S^{n−1} the differential dv_{ω} : T_{ω} → T_{v(ω)} exists. Similarly to the Euclidian case (29), the degree of v is defined by (see Chapter 5 of the book by Milnor [47])

deg v := \frac{1}{nω_{n}} ∫_{S^{n−1}} ∑_{ω∈v^{−1}(σ)} sign (det dv_{ω}) dH_{σ}^{n−1} ,  (32)

where, with an obvious abuse of notation, we denote by dv_{ω} also the (n − 1) × (n − 1) matrix representing the differential with respect to two fixed bases in T_{ω} and T_{v(ω)}. Using again the area formula for maps between manifolds, as in (30) we get

deg v = \frac{1}{nω_{n}} ∫_{S^{n−1}} det dv_{ω} dH_{ω}^{n−1} .  (33)

Fix ω_{0} ∈ ∂B_{1} and denote by τ_{j} : S^{n−1}\ {ω_{0}} → ℝ^{n}, for j ∈ {1, 2, …, n − 1}, a vector field of class C^{1} such that, for every x ∈ S^{n−1}\ {ω_{0}}, the set of vectors {τ_{1} (x) , τ_{2} (x) , …, τ_{n−1} (x)} is an orthonormal basis for the tangent plane to the surface S^{n−1} at the point x. The following representation formula (34) for deg v holds.

Theorem 15 Let v : S^{n−1} → S^{n−1} be a Lipschitz-continuous map onto S^{n−1}. Then, for H^{n−1}−a.e. ω ∈ S^{n−1}, we have

det dv_{ω} = ∑_{i=1}^{n} (−1)^{i+1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) .  (34)

Theorem 15 is proved below in this section. We deduce from Theorem 15 and Corollary 12 the following consequence.

Corollary 16 (Geometric interpretation) Let v : S^{n−1} → S^{n−1} be a map of class C^{1} and onto, and let u : B_{1}\ {0} → S^{n−1} be defined by u (x) := v (x/ |x|). If dv_{ω} is not singular at any ω ∈ S^{n−1}, i.e., if v is locally invertible with C^{1} local inverse at any point of S^{n−1}, then

T V (u, B_{1}) = ω_{n} |deg v| = ω_{n} |deg ṽ| ,  (35)

where ṽ: B_{1} → ℝ^{n} is any Lipschitz-continuous extension of v to B_{1}.

Remark 17 In two dimensions the total variation T V (u, B_{1}) can be expressed in terms of the degree as in (35) under the sole assumption that v maps S^{1} into a simple curve enclosing a starshaped domain (see Corollary 2). However, as shown in Section 10, this is not true anymore if v maps S^{1} into a non-simple curve, such as the “eight” curve.

Proof of Theorem 15. Fix ω ∈ S^{n−1} and denote by {τ_{1}, τ_{2}, …, τ_{n−1}}, {σ_{1}, σ_{2}, …, σ_{n−1}}, two orthonormal bases for the tangent planes to T_{ω} and T_{v(ω)}, respectively. With respect to these two bases the linear map dv_{ω} : T_{ω} → T_{v(ω)} is represented by the (n − 1) × (n − 1) matrix with coefficients

(dv_{ω})_{ij} = ⟨σ_{j}, \frac{∂v}{∂τ_{i}} (ω)⟩ .

Therefore the matrix dv_{ω} is the product of A and B, where A is the (n − 1) × n matrix whose rows are \frac{∂v}{∂τ_{i}} and B is the n × (n − 1) matrix whose columns are σ_{j}. By (iii) of Lemma 41 we have

det dv_{ω} = ∑_{i=1}^{n} det X_{,i} (A) · det X_{i,} (B) ,

where

det X_{,i} (A) = \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) ,

and in view of (114)

det X_{i,} (B) = (−1)^{i+1} ν_{i} (v (ω)) ,

where ν_{i} (v (ω)) is the i-th component of the outward unit normal S^{n−1} at v (ω), i.e., ν_{i} (v (ω)) = v^{i} (ω). This concludes the proof of (34).

Proof of Corollary 16. From the previous theorem we deduce that, if dv_{ω} is not singular at ω ∈ S^{n−1}, then det dv_{ω} ≠ 0. Therefore, the right hand side of (34) is different from zero and has constant sign, since by assumption the map v is of class C^{1}. The first equality in (35) follows from Theorem 15 and (24). The second equality is consequence of (25) and of (30).

We conclude by giving a geometrical interpretation of some of the estimates given in this paper. In the following statement we use again of the symbol # to denote the cardinality of a set.

Theorem 18 Let v : S^{n−1} → S^{n−1} be a Lipschitz-continuous map and let u : B_{1}\ {0} → S^{n−1} be defined by u (x) := v (x/ |x|). The total variation T V (u, B_{1}) of the Jacobian of u can be estimated by

T V (u, B_{1}) ≥ ω_{n} |deg v| ,  (36)
T V (u, B_{1}) ≤ \frac{1}{n} ∫_{∂B_{1}} # \{x ∈ S^{n−1} : v (x) = ω\} dH_{ω}^{n−1} .  (37)

Proof. Inequality (36) follows from inequality T V (u, B_{1}) ≥ |Det Du| (B_{1}), equality (19) of Theorem 7 on the representation of |Det Du| (B_{1}), and formula (34) of Theorem 15.

To prove (37), we apply the estimate (108) and formula (31). Precisely, we denote by ṽ: B_{1} → ℝ^{n} the extension of v defined by ṽ(0) = 0 and

ṽ(x) := |x| · v (\frac{x}{|x|}) , ∀ x ∈ B_{1}\ \{0\} .

Let ϱ_{h} → 0^{+} and define

u_{h} (x) := \{ \begin{matrix}\frac{1}{ϱ_{h}} ṽ(x) , if x ∈ B_{ϱ_{h}} \\ u (x) := v (x/ |x|) , if x ∈ B_{1}\B_{ϱ_{h}} \end{matrix} .

Clearly u_{h} ⇀ u in W^{1,p} (Ω; ℝ^{n}) and, by (31),

T V (u, B_{1}) ≤ lim inf_{h→+∞} ∫_{B_{1}} |det Du_{h} (x)| dx = lim inf_{h→+∞} ∫_{B_{ϱ_{h}}} |det D \frac{1}{ϱ_{h}} ṽ(x)| dx

= ∫_{B_{1}} |det Dṽ(x)| dx = ∫_{ṽ(B_{1})} # {x ∈ B_{1} : ṽ(x) = y} dy ,

and, since ṽ(B_{1}) ⊆ B_{1},

T V (u, B_{1}) ≤ ∫_{B_{1}} # \{x ∈ S^{n−1} : v (x) = \frac{y}{|y|}\} dy

= ∫_{0}^{1} ϱ^{n−1} dϱ ∫_{∂B_{1}} # \{x ∈ S^{n−1} : v (x) = ω\} dH_{ω}^{n−1} = \frac{1}{n} ∫_{∂B_{1}} # \{x ∈ S^{n−1} : v (x) = ω\} dH_{ω}^{n−1} .

4 Det Du versus det Du

In this section we give a brief overview of relations between Det Du, det Du and T V (u, Ω). We recall that the Jacobian det Du is given by

det Du (x) := \frac{∂ (u^{1}, u^{2}, …, u^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} = ∑_{i=1}^{n} \frac{∂u^{1}}{∂x_{i}} (adj Du)_{1}^{i} ,  (38)

where adj Du stands for the adjugate of Du, i.e., the transpose of the matrix of cofactors of Du. It is clear that when u ∈ W_{loc}^{1,n} (Ω; ℝ^{n}) then det Du ∈ L_{loc}^{1} (Ω). However, it is well known that, within some ranges of lower regularity for u, it is still possible to introduce a new concept of determinant which agrees with det Du when u ∈ W_{loc}^{1,n} (Ω; ℝ^{n}).

Consider the distributional Jacobian determinant, which, as usual, is denoted by Det Du capitalized, and is given by

Det Du := ∑_{i=1}^{n} (−1)^{i+1} \frac{∂}{∂x_{i}} (u^{1} \frac{∂ (u^{2}, …, u^{n})}{∂ (x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n})}) = ∑_{i=1}^{n} \frac{∂}{∂x_{i}} (u^{1} (adj Du)_{1}^{i}) . (39)

Note that Det Du is a distribution when u ∈ W^{1,p} (Ω; ℝ^{n}), adj Du ∈ L^{q}(Ω; ℝ^{n×n}), with 1/p + 1/q ≤ 1 + 1/n (in particular, when u ∈ W_{loc}^{1,p} (Ω; ℝ^{n}) for some p > n^{2}/(n + 1)), or when u ∈ L_{loc}^{∞} (Ω; ℝ^{n}) ∩ W_{loc}^{1,n−1} (Ω; ℝ^{n}) (actually, it suffices to require that u^{1} ∈ L_{loc}^{∞} (Ω; ℝ^{n}), and that the vector field of derivatives (Du^{2}, Du^{3}, …, Du^{n}) ∈ L_{loc}^{n−1} (Ω; ℝ^{(n−1)×n})). In the latter case, it is clear that the products in (39) are in L_{loc}^{1}(Ω). Also, if u ∈ W^{1,p} (Ω; ℝ^{n}) and adj Du ∈ L^{q}(Ω; ℝ^{n×n}) with 1/p + 1/q ≤ 1 + 1/n, then this integrability property still holds by virtue of Hölder’s inequality together with the fact that 1/q + 1/(p^{∗}) ≤ 1 and, due to the Sobolev Embedding Theorem, u ∈ L_{loc}^{p^{∗}} (Ω; ℝ^{n}).

For smooth functions the Jacobian determinant det Du (x) and the distributional Jacobian determinant Det Du coincide. In fact, if u ∈ W^{1,n} (Ω; ℝ^{n}) then using the fact that the adjugate is divergence free, it is easy to see that (38) reduces to (39). Also, Müller, Tang and Yan proved in [54] that if u ∈ W^{1,n−1} (Ω; ℝ^{n}) and if adj Du ∈ L^{n/(n−1)}(Ω; ℝ^{n×n}) then Det Du = det Du and it belongs to L^{1}(Ω). This relation may fail if u is not sufficiently regular. As an example, consider (see [36])

u (x) := \sqrt[n]{a^{n} + |x|^{n}} \frac{x}{|x|} , Ω := B_{1} ,

where B_{1}, as in the previous sections, stands for the open ball in ℝ^{n} centered at zero and with radius one. Then u ∈ W^{1,p}(B_{1}; ℝ^{n}) for all p < n, det Du = 1 a.e. in B_{1}, but

Det Du = L^{n}⌊B_{1} + ω_{n} a^{n} δ_{0},

where L^{n} denotes the Lebesgue measure in ℝ^{n} and ω_{n} is the volume of the unit ball B_{1}. Similarly, as shown in [27], if u(x) := x/|x| then det Du = 0 a.e. in B_{1} and Det Du = ω_{n} δ_{0}.

These examples suggest that, at least for some ranges of p, when Det Du is a Radon measure then its absolutely continuous part with respect to the n−dimensional Lebesgue measure reduces to det Du. Indeed, this holds when u ∈ W^{1,p} (Ω; ℝ^{n}) and adj Du ∈ L^{q}(Ω; ℝ^{n×n}) with 1/p + 1/q ≤ 1 + 1/n (see [50]); see also Theorem 6.

The presence of singular measures in Det Du is in perfect agreement with recent experiments, which suggest that, in addition to bulk energy, surface contributions and singular measures may also be energetically relevant, thus disfavoring the creation of extremely small cavities (see [16], [30] and [31]). These considerations have motivated the search for a characterization of the singular measures which may appear in the description of the distributional Jacobian determinant. If we do not impose any geometrical or analytical restrictions on the function u, then it is possible to attain Radon measures with support of arbitrary Hausdorff dimension. Precisely, it was proven by Müller [52] (see also [50]) that, given α ∈ (0, n), there exists a compact set K ⊂ B_{1} with Hausdorff dimension α, and there exists u ∈ W^{1,p}(B_{1}; ℝ^{n}) ∩ C^{0}(\overline{B_{1}}) for all p < n, such that

Det Du = det Du L^{n}⌊B_{1} + μ_{s},  (40)

where μ_{s} is a positive Radon measure, singular with respect to L^{n}, and such that supp μ_{s} = K. The situation is dramatically different if u ∈ W^{n−1}(Ω, S^{n−1}), as it can be shown that if Det Du is a finite, signed, Radon measure then Det Du is a finite integer combination of Dirac masses (see Brezis and Nirenberg [11], [12]). The use of BMO and Hardy spaces allows one to obtain higher integrability results along the lines of Müller [49], [51], and Coifman, Lions, Meyers and Semmes [17]. As an example, it can be shown that if u ∈ W^{1,n} (Ω; ℝ^{n}) is such that det Du ≥ 0, then (see also Brezis, Fusco and Sbordone [10] and Iwaniec and Sbordone [38]) det Du log(2 + det Du) ∈ L_{loc}^{1}(Ω).

As mentioned before, in this paper we assume that u is a function of class

W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n})

for some p ∈ (n − 1, n) and for an open set Ω ⊂ ℝ^{n} containing the origin. The definition of the total variation T V (u, Ω) introduced in (2) follows the approach commonly used for variational problems with non-standard growth and coercivity conditions (see [1], [2], [8], [14], [25], [27], [43], [36], [45], [46]). The aim of this paper is to characterize T V (u, Ω). In [27] Fonseca and Marcellini accomplished this for u (x) = x/ |x|. Fonseca and Malý [25], and Bouchitté, Fonseca and Malý [8] set up the problem into a broader context. Precisely, if f : Ω × ℝ^{n×n} → ℝ is a Carathéodory function, then the effective (or relaxed) energy is defined as

F_{p,q}(u, Ω) := inf {lim inf_{h→∞} ∫_{Ω} f(x, Du_{h}) dx : u_{h} ∈ W_{loc}^{1,q}, u_{h} ⇀ u in W^{1,p}} .  (41)

In the case, where f(x, ξ) := g (det ξ) and g :→ [0, +∞) is a convex function, then (see [14], [21], [25])

F_{p,n}(u, Ω) ≥ ∫_{Ω} g (det Du (x)) dx if p ≥ n − 1,

and if p > n − 1 then (see [8])

F_{p,n}(u, Ω) = ∫_{Ω} g (det Du (x)) dx + μ_{s}(Ω) ,

for some Radon measure μ_{s}, singular with respect to the Lebesgue measure L^{n}. For a general integrand f, and under the growth condition 0 ≤ f(x, ξ) ≤ C(1 + |ξ|^{q}), with p > \frac{n−1}{n}q, we have

F_{p,q}(u, Ω) = h_{u} L^{n}⌊Ω + λ_{s},  (42)

where (see [1]) h_{u} ≤ Qf(x, Du), and λ_{s} is a singular measure. If f = f(ξ) then it can be shown that (see [8], [25])

h_{u} = Qf(Du) ,  (43)

where Qf stands for the quasiconvexification of f, precisely (see [18], [48])

Qf(ξ) := inf {∫_{(0,1)^{n}} f(ξ + Dφ(x)) dx : φ ∈ C_{0}^{1} (Ω; ℝ^{n})} .

This may no longer be true when f depends also on x and p < q (although it is still valid if f(x, ·) is convex, see [1]). Indeed, Gangbo [29] constructed an example where f(x, ξ) = χ_{K}(x) |det ξ|, and h_{u} = f if and only if L^{N}(∂K) = 0. Hence, in general, (43) fails and f^{∗∗}(x, ∇u) ≤ h_{u} is the only known lower bound (see also [1], [8], [25], [27], [43], [44]).

Further understanding of the total variation T V (u, Ω) asks for mastery of weak convergence of minors for p < n. Works by Ball [4], Dacorogna and Murat [20], Giaquinta, Modica and Souček [36], and Reshetnyak [56], established that

u_{h} ⇀ u in W^{1,n} (Ω; ℝ^{n}) ⟹ det Du ⇀ det Du

in the sense of measures, where we recall that a sequence {μ_{h}} of Radon measures is said to converge in the sense of measures to a Radon measure μ in Ω if for every φ ∈ C_{c} (Ω; ℝ) we have

∫_{Ω} φ dμ_{h} → ∫_{Ω} φ dμ .

Müller [49] has shown that, if in addition det Du_{h} ≥ 0, then det Du_{h} ⇀ det Du weakly in L^{1} (Ω). Moreover, if u_{h} ⇀ u in W^{1,p} (Ω; ℝ^{n}) and {adj Du_{h}} is bounded in L^{q} (Ω; ℝ^{d×n}) with p ≥ n − 1, q ≥ n/(n − 1), one of these two inequalities being strict, then

det Du_{h} ⇀ det Du in the sense of measures.

Also, if u_{h} ∈ W^{1,n} (Ω; ℝ^{n}), u_{h} ⇀ u in W^{1,p} (Ω; ℝ^{n}) and p > n − 1, then

adj Du_{h} ⇀ adj Du  in L^{p/(n−1)} ∀ p > n − 1.  (44)

A complete characterization of weak convergence of the determinant has been obtained by Fonseca, Leoni and Malý in [26], where it was shown that, if the sequence {u_{h}} ⊂ W^{1,n} (Ω; ℝ^{n}) converges to a function u in L^{1} (Ω; ℝ^{n}), if {u_{h}} is bounded in W^{1,n−1} (Ω; ℝ^{n}), and if det Du_{h} ⇀ μ for some Radon measure μ, then

\frac{dμ}{dL^{n}} = det Du, a.e. x ∈ Ω.  (45)

For related works we refer to [2], [4], [14], [18], [19], [21], [28], [29], [32], [36], [42], [48], [49], [54].

What can we then say about the singular measure μ_{s} in (40), its significance and interpretation, and what are the relations, if any, between the total variation of Det Du, i.e. | Det Du|(Ω), and T V (u, Ω)? An answer is given by Theorem 6, which contemplates a general framework where only integrability assumptions are considered, and no structural properties of the function u are prescribed. Next we present the proof of this result.

Proof of Theorem 6. Since T V (u, Ω) < +∞, by (42) and (43) T V (u, ·) is a finite Radon measure, and it admits the Radon-Nikodym decomposition (14). In particular, it follows that det Du ∈ L^{1} (Ω).

Let δ > 0 be fixed and consider a sequence {u_{h}}_{h∈ℕ} ⊂ C^{1} (Ω; ℝ^{n}) such that u_{h} ⇀ u in W^{1,p} (Ω; ℝ^{n}), with p > n − 1, and

T V (u, Ω) + δ ≥ lim_{h→+∞} ∫_{Ω} |det Du_{h}| dx .  (46)

We first observe that, without loss of generality, we may assume that the sequence of the first components {u_{h}^{1}}_{h∈ℕ} is bounded in L^{∞} (Ω). Indeed, under the notation M := ∥u^{1}∥_{∞}, it suffices to consider the truncation

w_{h}^{1} (x) := \{ \begin{matrix}−M  if u_{h}^{j} (x) ≤ −M \\ u_{h}^{j} (x)  if − M ≤ u_{h}^{j} (x) ≤ M \\ M  if u_{h}^{j} (x) ≥ M \end{matrix} ,

and to set w_{h} := (w_{h}^{1}, u_{h}^{2}, …, u_{h}^{n}), for every h ∈ ℕ. It is easy to verify that, as h → +∞, w_{h} converges to u in the weak topology of W^{1,p} (Ω; ℝ^{n}) and, since |det Dw_{h}| ≤ |det Du_{h}|, for almost every x ∈ Ω, inequality (46) still holds with {u_{h}}_{h∈ℕ} replaced by {w_{h}}_{h∈ℕ}.

Since T V (u, Ω) < +∞, by (46) the sequence {det Du_{h}}_{h∈ℕ} is bounded in L^{1} (Ω), therefore, up to a subsequence (not relabeled) det Du_{h} ⇀^{∗} μ̃ as h → +∞, where μ̃ is a finite Radon measure. By (45) we have

\frac{dμ̃}{dL^{n}} = det Du, a.e. x ∈ Ω .  (47)

Next we prove that the distribution Det Du coincides with μ̃ on C_{0}^{1} (Ω) and hence, by regularization and density, on C_{0}^{0} (Ω). To prove this, for fixed φ ∈ C_{0}^{1} (Ω) we have

⟨Det Du, φ⟩ = − ∫_{Ω} ∑_{i=1}^{n} u^{1}(adj Du)_{1}^{i} \frac{∂φ}{∂x_{i}} dx

= − lim_{h→∞} ∫_{Ω} ∑_{i=1}^{n} u_{h}^{1} (adj Du_{h})_{1}^{i} \frac{∂φ}{∂x_{i}} dx = lim_{h→∞} ∫_{Ω} det Du_{h} φ dx = ⟨μ̃, φ⟩ .

Here we have used the facts that, since u_{h} ⇀ u in W^{1,p} (Ω; ℝ^{n}) for p > n − 1, then adj Du_{h} weakly converge to adj Du in L^{p/(n−1)}, and since the sequence {u_{h}^{1}}_{h∈ℕ} is bounded in L^{∞} (Ω) and converges as h → +∞ to u^{1} in L^{p} (Ω), it also converges to u^{1} in L^{q} (Ω), for every q < +∞, in particular for q = \frac{p}{p−(n−1)}, the conjugate exponent of \frac{p}{n−1}. Therefore, in view of (47), we deduce the Radon-Nikodym decomposition for Det Du as asserted in (15).

Let A be an open subset of Ω and let φ ∈ C_{0}^{1} (A; ℝ) be such that ∥φ∥_{∞} ≤ 1. By (46), a similar argument yields

|⟨Det Du, φ⟩| = |∫_{A} ∑_{i=1}^{n} u^{1}(adj Du)_{1}^{i} \frac{∂φ}{∂x_{i}} dx| = lim_{h→∞} |∫_{A} ∑_{i=1}^{n} u_{h}^{1} (adj Du_{h})_{1}^{i} \frac{∂φ}{∂x_{i}} dx|

= lim_{h→∞} |∫_{A} det Du_{h} φ dx| ≤ lim sup_{k→∞} ∥φ∥_{∞} ∫_{A} |det Du_{h}| dx ≤ T V (u, A) + δ .

It suffices to let δ → 0^{+}, and to take the supremum over all such functions φ, to conclude (16), i.e., |Det Du| (A) ≤ T V (u, A).

Suppose now, in addition, that u ∈ W_{loc}^{1,n} (Ω\ {0}; ℝ^{n}). Let A be an open subset of Ω such that 0 ∉ A. We recall that for every sequence u_{h} which converges to u in the weak topology of W^{1,p} (A; ℝ^{n}) for some p > n − 1, with u, u_{h} ∈ W_{loc}^{1,n} (A; ℝ^{n}) for every h ∈ ℕ, we have (see [19]; see also Theorem 35)

lim inf_{h→+∞} ∫_{A} |det Du_{h}| dx ≥ ∫_{A} |det Du| dx .

Hence

T V (u, A) = ∫_{A} |det Du| dx ,

whenever A is an open subset of Ω and 0 ∉ A. Therefore we conclude that supp λ_{s} ⊂ {0}, and thus λ_{s} = λδ_{0} for some constant λ ≥ 0, where δ_{0} is the Dirac measure at the origin.

On the other hand, in view of the inequality |Det Du| (A) ≤ T V (u, A) in (16), it follows that supp μ_{s} ⊂ supp λ_{s} ⊂ {0}, therefore μ_{s} = μδ_{0}, for some constant μ ∈ ℝ, with |μ| ≤ λ, where we have used (16).once more.

Remark 19 The result stated in Theorem 6 holds also under the assumption that u ∈ W^{1,p} (Ω; ℝ^{n}) for some p > n^{2}/(n + 1). Indeed, in this case, instead of truncating the sequence {u_{h}^{1}}_{h∈ℕ} we use the fact that, by Kondrachov’s Compact Embedding Theorem, u_{h} → u strongly in L^{n^{2}}, with n^{2} being the conjugate exponent of n^{2}/(n^{2} − 1). Again, {Du_{h}}_{h∈ℕ} weakly converges in L^{p} (Ω; ℝ^{n×n}) and the sequence {adj Du_{h}}_{h∈ℕ} weakly converges in L^{n^{2}/(n^{2}−1)}.

5 The general 2−d case

Let n = 2. For every ξ = (ξ^{1}, ξ^{2}) ∈ ℝ^{2}, ξ ≠ 0, we denote by Arg ξ the unique angle in [−π, π) such that

cos Arg ξ = \frac{ξ^{1}}{|ξ|} , sin Arg ξ = \frac{ξ^{2}}{|ξ|} .

As before, we denote by B_{r} the circle in ℝ^{2} with center in 0 and radius r > 0. Then B_{1} is the circle of radius r = 1 and ∂B_{1} = S^{1} is its boundary. If α, β ∈ [0, 2π], α < β, then S (α, β) stands for the polar sector given by

S (α, β) := {ξ = ϱ (cos ϑ, sin ϑ) ∈ ℝ^{2} : ϱ ≤ 1, ϑ ∈ [α, β]} .

In the sequel v : [0, 2π] → ℝ^{2} is a Lipschitz-continuous closed curve, i.e., v (0) = v (2π), that we represent as v = (v^{1}, v^{2}) = (v^{1} (ϑ) , v^{2} (ϑ)), with ϑ ∈ [0, 2π]. We shall denote by v_{ϑ} := (v_{ϑ}^{1}, v_{ϑ}^{2}) the gradient of v, which exists for almost every ϑ ∈ [0, 2π]. If v (ϑ) ≠ 0 for every ϑ ∈ [0, 2π], then we denote by A_{v} (ϑ) the quantity

A_{v} (ϑ) := Arg v (0) + ∫_{0}^{ϑ} \frac{v^{1} (t) v_{ϑ}^{2} (t) − v^{2} (t) v_{ϑ}^{1} (t)}{|v (t)|^{2}} dt .

There exists a simple relation between A_{v} and Arg v, which is inferred from the next lemma.

Lemma 20 If v : [0, 2π] → ℝ^{2} is a Lipschitz-continuous curve such that v (ϑ) ≠ 0 for every ϑ ∈ [0, 2π], then, for every α, β ∈ [0, 2π] with α < β, there exists k ∈ ℤ such that

A_{v} (β) − A_{v} (α) = Arg v (β) − Arg v (α) + 2kπ .  (48)

Proof. Assume first that v ∈ C^{1} ([0, 2π] ; ℝ^{2}) and that there exist at most a finite number of angles ϑ_{i} ∈ [0, 2π) such that either v^{1} (ϑ_{i}) = 0 or v^{2} (ϑ_{i}) = 0. Then, for every ϑ ≠ ϑ_{i} (since v^{1} (ϑ_{i}) ≠ 0) we have

\frac{d}{dϑ} Arg v (ϑ) = \frac{d}{dϑ} arctan \frac{v^{2} (ϑ)}{v^{1} (ϑ)} = \frac{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)}{|v (ϑ)|^{2}} .

The result then follows by integrating this equality and recalling that, each time that v (ϑ) crosses the half line {(x, y) ∈ ℝ^{2} : x < 0, y = 0}, and this may happen at most a finite number of times (necessarily for ϑ equal to some ϑ_{i}, where v^{2} (ϑ_{i}) = 0), the function Arg v (ϑ) has a jump of ±2π.

In the general case, we approximate v by a sequence {v_{j}}_{j∈ℕ} of curves of class C^{1} ([0, 2π] ; ℝ^{2}) such that {v_{j}}_{j∈ℕ} uniformly converges to v and {dv_{j}/dϑ}_{j∈ℕ} converges to dv/dϑ in L^{p} ([0, 2π]) for every p ∈ [1, +∞). We may construct the curves v_{j} so that v_{j} (ϑ) ≠ 0 for all ϑ ∈ [0, 2π] and either v^{1} (ϑ_{i}) = 0 or v^{2} (ϑ_{i}) = 0 only for finitely many i. Moreover, if Arg v (ϑ) ≠ −π, then Arg v_{j} (ϑ) → Arg v (ϑ), while, if Arg v (ϑ) = −π, then, up to a subsequence, Arg v_{j} (ϑ) → Arg v (ϑ) = −π or Arg v_{j} (ϑ) → π. Finally, the quantity

A_{v_{j}} (β) − A_{v_{j}} (α) = ∫_{α}^{β} \frac{v_{j}^{1}v_{j,ϑ}^{2} − v_{j}^{2}v_{j,ϑ}^{1}}{|v_{j}|^{2}} dt

converges, as j → +∞, to A_{v} (β) − A_{v} (α). From the relation

A_{v_{j}} (β) − A_{v_{j}} (α) = Arg v_{j} (β) − Arg v_{j} (α) + 2k_{j}π ,

valid for every j ∈ ℕ and for some k_{j} ∈ ℤ, we see that the sequence k_{j} is bounded, since Arg v_{j} (β) , Arg v_{j} (α) ∈ [−π, π). Then, up to a subsequence, we obtain the conclusion (48) as j → +∞.

As in Section 2, we denote by Γ a curve in ℝ^{2} parametrized in the following way

Γ := {ξ + r (ϑ) (cos ϑ, sin ϑ) : ϑ ∈ [0, 2π]} ,  (49)

where r (ϑ) is a piecewise C^{1} function such that r (0) = r (2π), and r (ϑ) ≥ r_{0} for every ϑ ∈ [0, 2π] and for some r_{0} > 0. Condition (49) means that Γ is the Lipschitz-continuous boundary of a domain

D := {ξ + ϱ (cos ϑ, sin ϑ) : ϑ ∈ [0, 2π] , 0 ≤ ϱ ≤ r (ϑ)} ,

starshaped with respect to a point ξ in the interior of D. In the sequel it is understood that the function r (ϑ) is extended to ℝ by periodicity.

Lemma 21 Let Γ be as in (49) and let v : [0, 2π] → Γ be a Lipschitz-continuous map. If Arg (v (0) − ξ) = 0, then the curve v may be represented in the form

v (ϑ) = ξ + r (A_{v−ξ} (ϑ)) (cos A_{v−ξ} (ϑ) , sin A_{v−ξ} (ϑ))  (50)

for all ϑ ∈ [0, 2π].

Proof. Since A_{v−ξ} (0) = Arg (v (0) − ξ) = 0, by Lemma 20 for every ϑ ∈ [0, 2π] there exists k ∈ ℤ such that A_{v−ξ} (ϑ) = Arg (v (ϑ) − ξ) + 2kπ. Also, as v (ϑ) ∈ Γ for all ϑ, we have

r (A_{v−ξ} (ϑ)) = r (Arg (v (ϑ) − ξ)) = |v (ϑ) − ξ| ,

and we obtain

r (A_{v−ξ} (ϑ)) (cos A_{v−ξ} (ϑ) , sin A_{v−ξ} (ϑ))
= |v (ϑ) − ξ| (cos Arg (v (ϑ) − ξ) , sin Arg (v (ϑ) − ξ)) = v (ϑ) − ξ .

Remark 22 Under the assumptions of Lemma 21, from the representation formula (50) for v (ϑ) it follows that, if A_{v−ξ} (α) = A_{v−ξ} (β), then v (α) = v (β). Conversely, if v (α) = v (β), then there exists k ∈ ℤ such that A_{v−ξ} (α) = A_{v−ξ} (β) + 2kπ. However, notice that if Γ is the boundary of a simply connected domain which is not starshaped with respect to ξ, then the conclusion of Lemma 21 may not be true. In particular, the condition A_{v−ξ} (α) = A_{v−ξ} (β) may not imply that v (α) = v (β).

The next Lemma 23 plays a central role in this section.

Lemma 23 (The “umbrella” lemma) Let Γ = {ξ + r (ϑ) (cos ϑ, sin ϑ)} and let v : [0, 2π] → Γ be a Lipschitz-continuous map. If α, β ∈ [0, 2π], α < β, are such that A_{v−ξ} (α) = A_{v−ξ} (β), then for every ε > 0 there exists a Lipschitz-continuous map w : S (α, β) → ℝ^{2} satisfying the boundary conditions

\{ \begin{matrix}w (1, ϑ) = v (ϑ) ,  ∀ ϑ ∈ [α, β] \\ w (ϱ, α) = w (ϱ, β) = ξ + ϱ (v (α) − ξ) , ∀ ϱ ∈ [0, 1] \end{matrix}  (51)

and such that

∫_{S(α,β)} |det Dw (x)| dx < ε .  (52)

Proof. We can assume, without loss of generality, that Arg (v (0) − ξ) = 0. Fix h ∈ ℕ and set

w_{h} (ϱ, ϑ) := ξ + ϱr (φ_{h} (ϱ, ϑ)) (cos φ_{h} (ϱ, ϑ) , sin φ_{h} (ϱ, ϑ)) ,  (53)

where, for every ϱ ∈ [0, 1] and for every ϑ ∈ [α, β],

φ_{h} (ϱ, ϑ) = ϱ^{h}A_{v−ξ} (ϑ) + (1 − ϱ^{h}) A_{v−ξ} (α) .

Since φ_{h} (1, ϑ) = A_{v−ξ} (ϑ), φ_{h} (ϱ, α) = φ_{h} (ϱ, β) = A_{v−ξ} (α), by the representation formula (50) of Lemma 21 we obtain the validity of the boundary conditions (51).

Now we evaluate the left hand side in (52). We observe that, for a generic function u (x) = (u^{1} (ϱ, ϑ) , u^{2} (ϱ, ϑ)), with the notation \frac{∂u^{i}}{∂ϱ} = u_{ϱ}^{i}, \frac{∂u^{i}}{∂ϑ} = u_{ϑ}^{i} (i = 1, 2), in general we have

det Du (x) = \frac{1}{ϱ} | \begin{matrix}u_{ϱ}^{1} (ϱ, ϑ) & u_{ϑ}^{1} (ϱ, ϑ) \\ u_{ϱ}^{2} (ϱ, ϑ) & u_{ϑ}^{2} (ϱ, ϑ) \end{matrix} | .  (54)

For the function w_{h} we obtain

∫_{S(α,β)} |det Dw_{h} (x)| dx = ∫_{0}^{1} dϱ ∫_{α}^{β} |\frac{∂ (w_{h}^{1}, w_{h}^{2})}{∂ (ϱ, ϑ)}| dϑ .

Let us compute the partial derivatives of w_{h}

\frac{∂w_{h}^{1}}{∂ϱ} = r (φ_{h}) cos φ_{h} + ϱ\frac{∂φ_{h}}{∂ϱ} [r′ (φ_{h}) cos φ_{h} − r (φ_{h}) sin φ_{h}]
\frac{∂w_{h}^{1}}{∂ϑ} = ϱ\frac{∂φ_{h}}{∂ϑ} [r′ (φ_{h}) cos φ_{h} − r (φ_{h}) sin φ_{h}]
\frac{∂w_{h}^{2}}{∂ϱ} = r (φ_{h}) sin φ_{h} + ϱ\frac{∂φ_{h}}{∂ϱ} [r′ (φ_{h}) sin φ_{h} + r (φ_{h}) cos φ_{h}]
\frac{∂w_{h}^{2}}{∂ϑ} = ϱ\frac{∂φ_{h}}{∂ϑ} [r′ (φ_{h}) sin φ_{h} + r (φ_{h}) cos φ_{h}]

and the Jacobian determinant of w_{h}

\frac{∂ (w_{h}^{1}, w_{h}^{2})}{∂ (ϱ, ϑ)} = ϱr^{2} (φ_{h}) \frac{∂φ_{h}}{∂ϑ} = ϱ^{h+1}r^{2} (φ_{h}) A′_{v−ξ} (ϑ) .

Thus we obtain

∫_{S(α,β)} |det Dw_{h} (x)| dx = ∫_{0}^{1} ϱ^{h+1} dϱ ∫_{α}^{β} r^{2} (φ_{h}) |A′_{v−ξ} (ϑ)| dϑ = \frac{c}{h + 2} ,

where we denote by c the constant

c := sup_{h∈ℕ} {∫_{α}^{β} r^{2} (φ_{h}) |A′_{v−ξ} (ϑ)| dϑ} .

The conclusion follows by choosing h ∈ ℕ sufficiently large.

Remark 24 As we pointed out already, the previous Lemma 23 plays a central role in our analysis. We informally call Lemma 23 as “the umbrella lemma”, since the geometric representation of the graph of the map w : ℝ^{2} → ℝ^{2} considered in Lemma 23 is some sort of “umbrella” (under some mathematical tolerance and human imagination!). In fact, let us consider for simplicity the case where the image Γ of the map v is the unit circle [0, 2π] ⊂ ℝ^{2} centered around ξ = 0. Then the graph of w is a subset of S^{1}: it “starts” from the center ξ = 0 (the starting point of the “umbrella-stick”, in correspondence to ϱ = 0) and it “ends” for ϱ = 1, at the surface {w (1, ϑ) = v (ϑ) : ϑ ∈ [α, β]} ⊂ S^{1}, which can be interpreted as the upper surface of the open umbrella, to protect one from the rain. Moreover, by (52), like an umbrella, the total volume of the image of w is small (large upper surface, small volume! In our 2−d case, we have a 2−dimensional “picture” of an umbrella, with large upper length and small area).

We refer to Figures 1, 2 and 3, where we represented the image of the map w_{h} (ϱ, ϑ) in (53) under three particular choices of the parameters. Precisely, for fixed h ∈ ℕ we considered w_{h} : S (α, β) → B_{1} (i.e., r (φ_{h} (ϱ, ϑ)) in (53) identically equal to 1 and ξ = 0) given by

\{ \begin{matrix}w_{h} (ϱ, ϑ) = ϱ (cos φ_{h} (ϱ, ϑ) , sin φ_{h} (ϱ, ϑ)) \\ φ_{h} (ϱ, ϑ) = ϱ^{h}A_{v} (ϑ) + (1 − ϱ^{h}) A_{v} (α) \end{matrix} ,  (55)

where A_{v} : [α, β] → ℝ is a function such that A_{v} (α) = A_{v} (β). The common value of A_{v} at ϑ = α and ϑ = β is the asymptotic value of the angle φ_{h} (ϱ, ϑ) as ϱ → 0^{+} and it represents the angle which the umbrella-stick forms with the x−axis. At ϱ = 1 the angle φ_{h} (1, ϑ) holds A_{v} (ϑ); therefore the maximum M and the minimum m of A_{v} (ϑ) represent the bounds for the angle φ_{h} (1, ϑ) of the image w (1, ϑ) at the surface S^{1} of the ball B_{1}. These pictures has been made by Emanuele Paolini, starting from the analytic expression of w in (55). We thank him for the beautiful job.

Lemma 25 Let f : [a, b] → ℝ be a continuous function, piecewise strictly monotone in [a, b] (with a finite number of monotonicity intervals) and such that f (a) < f (b). Then there exists a partition a = α_{0} < α_{1} < … < α_{N} = b of [a, b] such that, for every i = 1, 2, … N, either f is strictly increasing in [α_{i−1}, α_{i}], or f (α_{i−1}) = f (α_{i}).

Figure 1: A 2 − d image of the map w defined in (55), with h = 4, for a particular (piecewise linear) function A_{u} (ϑ). The angle which the umbrella-stick forms with the x−axis is given by A_{u} (α) = A_{u} (β) = π/2. The maximun M and the minimum m of A_{u} (ϑ) = φ (1, ϑ), which give the bounds for the angles of the image w (1, ϑ) at the surface S^{1} of the ball B_{1}, in this case are equal to m = π/6, M = 5π/6, respectively. Note that the map is radially linear when ϑ = α and ϑ = β, where the angle of the image is equal to π/2.

Proof. Let α_{0} = a. If f is (strictly) decreasing in a right neighborhood of α_{0}, then we define

α_{1} := max {ϑ ∈ [α_{0}, b] : f (ϑ) = f (α_{0})} .

Since f is continuous, α_{1} is well defined. Moreover, since f (α_{1}) = f (α_{0}) < f (b), then α_{1} < b and f is (strictly) increasing in a right neighborhood of α_{1}. Next we define

α_{2} := max {ϑ ∈ [α_{1}, b] : f (ϑ) ≤ f (b) and f is strictly increasing in [α_{1}, ϑ]} .

If α_{2} = b then the lemma follows. Otherwise, if α_{2} < b and f (α_{2}) = f (b), then we set α_{3} = b and again the lemma follows. Finally, if α_{2} < b and f (α_{2}) < f (b), then f is strictly decreasing in a right neighborhood of α_{2}, and we may argue as we did before starting from α_{0}. If, instead, f is (strictly) increasing in a right neighborhood of α_{0}, then we may use the same argument as we did before starting from α_{1}.

Since the number of intervals of monotonicity is finite, after a finite number of iterations we reach the conclusion.

Lemma 26 Let v : [0, 2π] → Γ be a Lipschitz-continuous map. Let α, β ∈ [0, 2π], α < β, be such that A_{v−ξ} (α) = A_{v−ξ} (β). If A_{v−ξ} (ϑ) is piecewise strictly monotone in [α, β] (with a finite number of monotonicity intervals) then

∫_{α}^{β} {(v^{1} (ϑ) − ξ^{1}) v_{ϑ}^{2} (ϑ) − (v^{2} (ϑ) − ξ^{2}) v_{ϑ}^{1} (ϑ)} dϑ = 0 .

Proof. Without loss of generality we assume that ξ = (0, 0). Since A_{v} (ϑ) is piecewise strictly monotone in [α, β] and A_{v} (α) = A_{v} (β), there exists a partition of the interval [α, β], α = ϑ_{0} < ϑ_{1} < … < ϑ_{N} = β, N ≥ 2, such that, for every i = 1, 2, … N, the real function A_{v} (ϑ) is strictly increasing Figure 2: Another 2 − d image of the map w defined in (55), with h = 4, for a different choise of the piecewise linear function A_{u} (ϑ). In this case we obtain an asymmetric umbrella. Again A_{u} (α) = A_{u} (β) = π/2, while in this case m = −π/6, M = 5π/6. The map is not one-to-one: only the image points with angles −π/6 and 5π/6 may be assumed once; all the other points are hit at least twice; the points with angle π/2 are hit at least three times.

in [ϑ_{i−1}, ϑ_{i}] and is strictly decreasing in [ϑ_{i}, ϑ_{i+1}] (or viceversa). We will prove the lemma by an induction argument based on the number N of these maximal intervals of monotonicity.

Let us first assume that N = 2. Hence there exists ϑ_{1} ∈ (α, β) such that A_{v} (ϑ) is strictly increasing in [α, ϑ_{1}] and is strictly decreasing in [ϑ_{1}, β], or conversely. To fix the ideas, let us assume that A_{v} (ϑ) is strictly increasing in [α, ϑ_{1}]. For every (ϱ, ϑ) ∈ S (α, β) let us define ṽ(ϱ, ϑ) := ϱ v (ϑ) . If A_{v} (ϑ_{1}) − A_{v} (α) ≤ 2π, then ṽ restricted to the interior of S (α, ϑ_{1}) and S (ϑ_{1}, β) is one-to-one. Moreover the images ṽ(S (α, ϑ_{1})) and ṽ(S (ϑ_{1}, β)) are equal. Therefore, by the area formula,

∫_{S(α,ϑ_{1})} |det Dṽ(x)| dx = area (ṽ(S (α, ϑ_{1}))) = area (ṽ(S (ϑ_{1}, β))) = ∫_{S(ϑ_{1},β)} |det Dṽ(x)| dx .

Since det Dṽ ≥ 0 in S (α, ϑ_{1}) and det Dṽ ≤ 0 in S (ϑ_{1}, β), we obtain

∫_{S(α,ϑ_{1})} det Dṽ(x) dx = area (ṽ(S (α, ϑ_{1}))) = area (ṽ(S (ϑ_{1}, β))) = − ∫_{S(ϑ_{1},β)} det Dṽ(x) dx .

By using again (54), we have

det Dṽ(ϱ, ϑ) = \frac{1}{ϱ} | \begin{matrix}v^{1} (ϑ) & ϱv_{ϑ}^{1} (ϑ) \\ v^{2} (ϑ) & ϱv_{ϑ}^{2} (ϑ) \end{matrix} | = v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ) = A_{v} (ϑ) |v (ϑ)|^{2} .  (56)

Therefore, as claimed,

0 = ∫_{S(α,β)} det Dṽ(x) dx = ∫_{0}^{1} ϱ dϱ ∫_{α}^{β} {v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)} dϑ

= \frac{1}{2} ∫_{α}^{β} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ .

Figure 3: Map w : S (α, β) → B_{1} in (55). Here we fixed h = 3, while the angle which the umbrella-stick forms with the x−axis is equal to A_{u} (α) = A_{u} (β) = 0. The bounds for the angle of the image w (1, ϑ), at the surface S^{1} of the unit ball B_{1}, in this case are equal to m = 0, M = 2π + π/2. The map is radially linear when the angle A_{u} (ϑ) of the image is 0 (and this happens only if ϑ = α = β, when φ (1, ϑ) = A_{u} (ϑ) = 0). The map w is not one-to-one: due to the overlaping phenomenon, some points with ϱ close to 1 and 0 ≤ φ < π/2 are assumed at least four times.

If 2kπ < A_{v} (ϑ_{1}) − A_{v} (α) ≤ 2π (k + 1) for some k ≥ 1, then we denote by ϑ′ ∈ (α, ϑ_{1}), ϑ′′ ∈ (ϑ_{1}, β) the points such that A_{v} (ϑ′) = A_{v} (ϑ′′) = 2kπ. Again, using the area formula, we have

∫_{S(α,ϑ_{1})} |det Dṽ(x)| dx = ∫_{S(α,ϑ′)} |det Dṽ(x)| dx + ∫_{S(ϑ′,ϑ_{1})} |det Dṽ(x)| dx = k area D + area E ,

where D is the domain in (7) enclosed by Γ and E is the domain represented in polar coordinates by

E = {ϱ (cos A_{v} (ϑ) , sin A_{v} (ϑ)) : ϑ ∈ [ϑ′, ϑ_{1}] , 0 ≤ ϱ ≤ r (ϑ)}
= {ϱ (cos A_{v} (ϑ) , sin A_{v} (ϑ)) : ϑ ∈ [ϑ_{1}, ϑ′′] , 0 ≤ ϱ ≤ r (ϑ)} .

Therefore, we also have

∫_{S(ϑ_{1},β)} |det Dṽ(x)| dx = ∫_{S(ϑ_{1},ϑ′′)} |det Dṽ(x)| dx + ∫_{S(ϑ′′,β)} |det Dṽ(x)| dx = area E + k area D .

Arguing as before we get the thesis (with N = 2)

\frac{1}{2} ∫_{α}^{β} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ = ∫_{S(α,β)} det Dṽ(x) dx

= ∫_{S(α,ϑ_{1})} |det Dṽ(x)| dx − ∫_{S(ϑ_{1},β)} |det Dṽ(x)| dx = 0 .

By induction, we assume that the result is true if there are N −1 maximal intervals of monotonicity for the function A_{v} (ϑ). Then we consider the case where there are N of such intervals, with endpoints α = ϑ_{0} < ϑ_{1} < … < ϑ_{N} = β. Without loss of generality, we can assume that A_{v} (ϑ) is strictly increasing in [α, ϑ_{1}] and is strictly decreasing in [ϑ_{1}, ϑ_{2}]. If A_{v} (α) > A_{v} (ϑ_{2}), then there exists γ ∈ (ϑ_{1}, ϑ_{2}) such that A_{v} (γ) = A_{v} (α); since the thesis holds for the case of two intervals [α, ϑ_{1}], [ϑ_{1}, γ], we obtain

∫_{α}^{γ} {v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}} dϑ = 0 ;  (57)

the thesis also holds for the N − 1 intervals [γ, ϑ_{2}] , [ϑ_{2}, ϑ_{3}] , …, [ϑ_{N−1}, β], and so we have

∫_{γ}^{β} {v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}} dϑ = 0 ,

which, together with (57), yields the conclusion if A_{v} (α) > A_{v} (ϑ_{2}).

If A_{v} (α) = A_{v} (ϑ_{2}), the same argument works with γ = ϑ_{2}. If A_{v} (α) < A_{v} (ϑ_{2}) then there exists δ ∈ (α, ϑ_{1}) such that A_{v} (δ) = A_{v} (ϑ_{2}) and, as before, by considering the two intervals [δ, ϑ_{1}], [ϑ_{1}, ϑ_{2}], we have

∫_{δ}^{ϑ_{2}} {v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}} dϑ = 0 .  (58)

Then we “modify” the function v (ϑ) by “cutting out” the interval (δ, ϑ_{2}) from [α, β]; more precisely, we define in the interval [α + [ϑ_{2} − δ] , β]

w (ϑ) := \{ \begin{matrix}v (ϑ − [ϑ_{2} − δ]) , if α + [ϑ_{2} − δ] ≤ ϑ ≤ ϑ_{2} \\ v (ϑ) , if ϑ_{2} ≤ ϑ ≤ β \end{matrix} .

Then A_{w} (ϑ) is piecewise strictly monotone in [α + [ϑ_{2} − δ] , β], with N −1 monotonicity intervals. By the induction assumption we have

0 = ∫_{α+[ϑ_{2}−δ]}^{β} {w^{1}w_{ϑ}^{2} − w^{2}w_{ϑ}^{1}} dϑ = ∫_{α}^{δ} {v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}} dϑ + ∫_{ϑ_{2}}^{β} {v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}} dϑ ,

which, together with (58), yields the conclusion.

Lemma 27 Let v : [0, 2π] → Γ be a Lipschitz-continuous map. Let A_{v−ξ} (ϑ) be piecewise strictly monotone in [a, b] (with a finite number of monotonicity intervals). For every ε > 0 there exists a Lipschitz-continuous map w : B_{1} → ℝ^{2} such that w (1, ϑ) = v (ϑ) for every ϑ ∈ [0, 2π], and

∫_{B_{1}} |det Dw (x)| dx < ε + \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .

Proof. If A_{v−ξ} (0) = A_{v−ξ} (2π) then the result follows from Lemma 23. Otherwise, we may assume, without loss of generality, that A_{v−ξ} (0) < A_{v−ξ} (2π). By Lemma 25 we can consider a partition of [0, 2π] by means of points 0 = α_{0} < α_{1} < … < α_{N} = 2π such that, for every i = 1, 2, … N, either A_{v−ξ} is strictly increasing in [α_{i−1}, α_{i}], or A_{v−ξ} (α_{i−1}) = A_{v−ξ} (α_{i}). Denote by I the set of indices

I := {i ∈ {1, 2, … N} : A_{v−ξ} (α_{i−1}) = A_{v−ξ} (α_{i})} .

Given ε > 0, if i ∈ I we denote by w_{i} : S (α_{i−1}, α_{i}) → ℝ^{2} the Lipschitz-continuous map provided by Lemma 23, satisfying the boundary conditions

\{ \begin{matrix}w_{i} (1, ϑ) = v (ϑ) ,  ∀ ϑ ∈ [α_{i−1}, α_{i}] \\ w_{i} (ϱ, α_{i−1}) = w_{i} (ϱ, α_{i}) = ξ + ϱ (v (α_{i−1}) − ξ) , ∀ ϱ ∈ [0, 1] \end{matrix}

and the bound

∫_{S(α_{i−1},α_{i})} |det Dw_{i} (x)| dx < ε .  (59)

We then define the Lipschitz-continuous map w : B_{1} → ℝ^{2} by setting

w (ϱ, ϑ) := \{ \begin{matrix}ξ + ϱ (v (ϑ) − ξ) , ∀ ϑ ∈ [α_{i−1}, α_{i}] , if i ∉ I \\ w_{i} (ϱ, ϑ) , ∀ ϑ ∈ [α_{i−1}, α_{i}] , if i ∈ I \end{matrix} ,

and for every ϱ ∈ [0, 1]. In particular, w satisfies the boundary condition w (1, ϑ) = v (ϑ). Moreover, if ϑ ∈ [α_{i−1}, α_{i}] for some i ∉ I, in view of (56), we have

|det Dw (x)| = |det D [ξ + ϱ (v (ϑ) − ξ)]|  (60)
= (v^{1} (ϑ) − ξ^{1}) v_{ϑ}^{2} (ϑ) − (v^{2} (ϑ) − ξ^{2}) v_{ϑ}^{1} (ϑ) ,

where we have used the fact that A_{v−ξ} (ϑ) is strictly increasing for ϑ ∈ [α_{i−1}, α_{i}]. By (59) and (60) we obtain

∫_{B_{1}} |det Dw (x)| dx = ∑_{i∈I} ∫_{S(α_{i−1},α_{i})} |det Dw_{i} (x)| dx + ∑_{i∉I} ∫_{S(α_{i−1},α_{i})} |det Dw (x)| dx

≤ ε · # (I) + ∑_{i∉I} ∫_{0}^{1} ϱ dϱ ∫_{α_{i−1}}^{α_{i}} {(v^{1} (ϑ) − ξ^{1}) v_{ϑ}^{2} (ϑ) − (v^{2} (ϑ) − ξ^{2}) v_{ϑ}^{1} (ϑ)} dϑ

= ε · # (I) + ∑_{i∉I} \frac{1}{2} ∫_{α_{i−1}}^{α_{i}} \{(v^{1} (ϑ) − ξ^{1}) v_{ϑ}^{2} (ϑ) − (v^{2} (ϑ) − ξ^{2}) v_{ϑ}^{1} (ϑ)\} dϑ .

By Lemma 26, for every i ∈ I we have

∫_{α_{i−1}}^{α_{i}} {(v^{1} (ϑ) − ξ^{1}) v_{ϑ}^{2} (ϑ) − (v^{2} (ϑ) − ξ^{2}) v_{ϑ}^{1} (ϑ)} dϑ = 0 ,

hence

∫_{B_{1}} |det Dw (x)| dx ≤ ε · N + \frac{1}{2} ∫_{0}^{2π} \{(v^{1} (ϑ) − ξ^{1}) v_{ϑ}^{2} (ϑ) − (v^{2} (ϑ) − ξ^{2}) v_{ϑ}^{1} (ϑ)\} dϑ

= ε · N + \frac{1}{2} ∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ .

Next we consider maps u = u (ϱ, ϑ) depending explicitly on ϱ as well. We assume first that u is a smooth map in the unit ball B_{1} ⊂ ℝ^{2}.

Lemma 28 (The integral of the Jacobian for smooth maps) Let u ∈ W^{1,∞} (B_{1}; ℝ^{2}). For every r ∈ (0, 1] we have

∫_{B_{r}} det Du (x) dx = \frac{1}{2} ∫_{0}^{2π} \{u^{1} (r, ϑ) \frac{∂u^{2}}{∂ϑ} (r, ϑ) − u^{2} (r, ϑ) \frac{∂u^{1}}{∂ϑ} (r, ϑ)\} dϑ .  (61)

Proof. If first u ∈ C^{2} (B_{1}; ℝ^{2}), by the divergence theorem, we have for every r ∈ (0, 1)

∫_{B_{r}} det Du (x) dx = ∫_{B_{r}} div (u^{1} \frac{∂u^{2}}{∂x_{2}} , −u^{1} \frac{∂u^{2}}{∂x_{1}} ) dx = ∫_{∂B_{r}} \{u^{1} \frac{∂u^{2}}{∂x_{2}} ν^{1} − u^{1} \frac{∂u^{2}}{∂x_{1}} ν^{2}\} dH^{1} , (62)

where ν = (ν^{1}, ν^{2}) is the exterior normal to ∂B_{r} and dH^{1} = ds = rdϑ is the element of archlenght. A standard approximation argument yields formula (62) for every u ∈ W^{1,∞} (B_{1}; ℝ^{2}).and for every r ∈ (0, 1) (since u ∈ W^{1,∞} (∂B_{r}; ℝ^{2}) for for every r ∈ (0, 1) too).

With an obvious abuse of notation we write u in polar coordinates (ϱ, ϑ), i.e., u (x_{1}, x_{2}) = u (ϱ, ϑ). We have

\frac{∂u}{∂x_{1}} = \frac{∂u}{∂ρ} \frac{∂ϱ}{∂x_{1}} + \frac{∂u}{∂ϑ} \frac{∂ϑ}{∂x_{1}} = \frac{∂u}{∂ρ} cos ϑ − \frac{∂u}{∂ϑ} \frac{sin ϑ}{ϱ} ,

\frac{∂u}{∂x_{2}} = \frac{∂u}{∂ρ} \frac{∂ϱ}{∂x_{2}} + \frac{∂u}{∂ϑ} \frac{∂ϑ}{∂x_{2}} = \frac{∂u}{∂ρ} sin ϑ + \frac{∂u}{∂ϑ} \frac{cos ϑ}{ϱ} .

Since on ∂B_{r} the exterior normal reduces to ν = (ν^{1}, ν^{2}) = (cos ϑ, sin ϑ), from (62) we obtain

\frac{∂u^{2}}{∂x_{2}} ν^{1} − \frac{∂u^{2}}{∂x_{1}} ν^{2} = (\frac{∂u^{2}}{∂ρ} sin ϑ + \frac{∂u^{2}}{∂ϑ} \frac{cos ϑ}{ϱ}) cos ϑ − (\frac{∂u^{2}}{∂ρ} cos ϑ − \frac{∂u^{2}}{∂ϑ} \frac{sin ϑ}{ϱ}) sin ϑ

= \frac{∂u^{2}}{∂ρ} sin ϑ cos ϑ + \frac{1}{ϱ} \frac{∂u^{2}}{∂ϑ} cos^{2} ϑ − \frac{∂u^{2}}{∂ρ} sin ϑ cos ϑ + \frac{1}{ϱ} \frac{∂u^{2}}{∂ϑ} sin^{2} ϑ = \frac{1}{ϱ} \frac{∂u^{2}}{∂ϑ} .

Thus on ∂B_{r}, since dH^{1} = rdϑ, we get

∫_{B_{r}} det Du (x) dx = ∫_{∂B_{r}} \{u^{1} \frac{∂u^{2}}{∂x_{2}} ν^{1} − u^{1} \frac{∂u^{2}}{∂x_{1}} ν^{2}\} dH^{1} = ∫_{0}^{2π} u^{1} \frac{∂u^{2}}{∂ϑ} dϑ .

For symmetric reasons, starting now from det Du (x) = − (u^{2} · u_{x_{2}}^{1})_{x_{1}} + (u^{2} · u_{x_{1}}^{1})_{x_{2}}, we also obtain

∫_{B_{r}} det Du (x) dx = ∫_{0}^{2π} −u^{2} \frac{∂u^{1}}{∂ϑ} dϑ ,

and thus, for every value of a real parameter λ,

∫_{B_{r}} det Du (x) dx = λ ∫_{0}^{2π} u^{1} \frac{∂u^{2}}{∂ϑ} dϑ + (1 − λ) ∫_{0}^{2π} −u^{2} \frac{∂u^{1}}{∂ϑ} dϑ

and, in particular, for λ = 1/2, the conclusion in (61).

Now we start to consider functions u of class W^{1,p} (Ω; ℝ^{2})∩W_{loc}^{1,∞} (Ω\ {0}; ℝ^{2}) for some p ∈ (1, 2).

Lemma 29 Let u be a map satisfying the assumptions of Theorem 1. There exists a sequence ϱ_{j} → 0 such that

lim_{j→+∞} ∫_{0}^{2π} \{u^{1} (ϱ_{j}, ϑ) \frac{∂u^{2}}{∂ϑ} (ϱ_{j}, ϑ) − u^{2} (ϱ_{j}, ϑ) \frac{∂u^{1}}{∂ϑ} (ϱ_{j}, ϑ)\} dϑ

= ∫_{0}^{2π} \{v^{1} (ϑ) \frac{dv^{2}}{dϑ} (ϑ) − v^{2} (ϑ) \frac{dv^{1}}{dϑ} (ϑ)\} dϑ .

Proof. First we use assumption (9). Recall that u ∈ W^{1,p} (Ω; ℝ^{2}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{2}) for some p ∈ (1, 2).. By Lemma 38 (valid in the general n−dimensional case) there exists a sequence ϱ_{j} → 0 such that

(ϱ_{j})^{p−1} ∫_{∂B_{ϱ_{j}}} |D_{τ} u|^{p} dH^{1} ≤ cM_{0} .

Since

(ϱ_{j})^{p−1} ∫_{∂B_{ϱ_{j}}} |D_{τ} u|^{p} dH^{1} = ∫_{0}^{2π} |\frac{∂u}{∂ϑ} (ϱ_{j}, ϑ)|^{p} dϑ ,

we deduce that \{\frac{∂u}{∂ϑ} (ϱ_{j}, ·)\}_{j∈ℕ} is a bounded sequence in L^{p} (0, 2π). By assumption (8) \{u (ϱ_{j}, ·)\}_{j∈ℕ} converges to v (·) in L^{∞} ((0, 2π) ; ℝ^{2}). Since \{\frac{∂u}{∂ϑ} (ϱ_{j}, ·)\}_{j∈ℕ} remains bounded in L^{p} ((0, 2π) ; ℝ^{2}), then it converges to \frac{∂v}{∂ϑ} in the weak topology of L^{p} ((0, 2π) ; ℝ^{2}) as j → +∞. We obtain the conclusion

lim_{j→+∞} ∫_{0}^{2π} \{u^{1} (ϱ_{j}, ϑ) \frac{∂u^{2}}{∂ϑ} (ϱ_{j}, ϑ) − u^{2} (ϱ_{j}, ϑ) \frac{∂u^{1}}{∂ϑ} (ϱ_{j}, ϑ)\} dϑ

= ∫_{0}^{2π} \{v^{1} (ϑ) \frac{dv^{2}}{dϑ} (ϑ) − v^{2} (ϑ) \frac{dv^{1}}{dϑ} (ϑ)\} dϑ .

We are ready to give the proof of Theorem 1. We divide the proof into four steps, and we will refer to the preliminary lemmas above and to Lemma 37, valid in the general n−dimensional case.

Proof of Theorem 1. Step 1 (lower bound). Let u be a function of class W^{1,p} (Ω; ℝ^{2}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{2}) for some p ∈ (1, 2). Let us observe that, by assumption (8), there exists r > 0 such that B_{r} ⊂ Ω and u ∈ L^{∞} (B_{r}; ℝ^{2}). Let ϱ_{j} → 0 be the sequence of the previous Lemma 29 and let j ∈ ℕ be sufficiently large so that B_{ϱ_{j}} ⊂ B_{r}. For such values of j ∈ ℕ, we use the estimate (97) of Lemma 37 to obtain

T V (u, Ω) ≥ ∫_{Ω\B_{ϱ_{j}}} |det Du (x)| dx + |∫_{B_{ϱ_{j}}} det Dũ (x) dx| ,

where ũ : B_{ϱ_{j}} → ℝ^{2} is any Lipschitz-continuous map such that ũ (x) = u (x) on ∂B_{ϱ_{j}}. By formula (61) of Lemma 28 (valid on each ball B_{ϱ_{j}}), since ũ = u, ∂ũ/∂ϑ = ∂u/∂ϑ on ∂B_{ϱ_{j}}, we have

T V (u, Ω) ≥ ∫_{Ω\B_{ϱ_{j}}} |det Du (x)| dx

+ | \frac{1}{2} ∫_{0}^{2π} \{u^{1} (ϱ_{j}, ϑ) \frac{∂u^{2}}{∂ϑ} (ϱ_{j}, ϑ) − u^{2} (ϱ_{j}, ϑ) \frac{∂u^{1}}{∂ϑ} (ϱ_{j}, ϑ)\} dϑ| .

Letting j → +∞, by Lemma 29 we obtain the lower bound

T V (u, Ω) ≥ ∫_{Ω} |det Du (x)| dx + \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) \frac{dv^{2}}{dϑ} (ϑ) − v^{2} (ϑ) \frac{dv^{1}}{dϑ} (ϑ)\} dϑ| .  (63)

Step 2 (upper bound - first part). To assert the opposite inequality in (63), let us first assume that u is radially symmetric, i.e., u = v = v (ϑ) = u (ϑ), and that it satisfies the assumptions of Lemma 27. By the conclusion of Lemma 27, given ε > 0 there exists a Lipschitz-continuous map w : B_{1} → ℝ^{2} such that w (1, ϑ) = v (ϑ) for every ϑ ∈ [0, 2π] and

∫_{B_{1}} |det Dw (x)| dx < ε + \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .

For every h ∈ ℕ we set

u_{h} (ϱ, ϑ) := \{ \begin{matrix}w (ϱh, ϑ) ,  if 0 ≤ ϱ ≤ 1/h \\ v (ϑ) ,  if ϱ ≥ 1/h \end{matrix} .

Then {u_{h}}_{h∈ℕ} converges to u in L^{p} (Ω) and

∫_{Ω} |Du_{h} (x) − Du (x)|^{p} dx = ∫_{B_{1/h}} |hDw (ϱh, ϑ)|^{p} dx = h^{p−2} ∫_{B_{1}} |Dw (x)|^{p} dx

and so, since 1 ≤ p < 2, {Du_{h}}_{h∈ℕ} converges to Du strongly in L^{p} (B_{1}; ℝ^{2×2}) and finally

∫_{Ω} |det Du_{h} (x)| dx = ∫_{B_{1/h}} |h^{2} det Dw (ϱh, ϑ)| dx = ∫_{B_{1}} |det Dw (x)| dx

< ε + \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .

Therefore, making use of the definition (26) of the total variation of the Jacobian T V^{s} (u, Ω) in the strong topology, we can conclude that

T V^{s} (u, Ω) ≤ \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .

This inequality, together with

T V (u, Ω) ≤ T V^{s} (u, Ω) , ∀ u ∈ L^{∞} (Ω; ℝ^{n}) ∩ W^{1,p} (Ω; ℝ^{n}) ,

and with Step 1, yields the conclusion

T V (u, Ω) = T V^{s} (u, Ω) = \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ|  (64)

when u = v = v (ϑ) = u (ϑ) satisfies the assumptions of Lemma 27.

Step 3 (upper bound - second part). As in the previous Step 2, we still assume that u = v = v (ϑ) = u (ϑ), but we no longer require that the conditions of Lemma 27 are satisfied. Without loss of generality, we can assume that Arg (v (0) − ξ) = 0 and thus, by Lemma 21, the map v (ϑ) may be represented as

v (ϑ) = ξ + r (A_{v−ξ} (ϑ)) (cos A_{v−ξ} (ϑ) , sin A_{v−ξ} (ϑ)) .

Construct a sequence {A_{k} (ϑ)}_{k∈ℕ} of piecewise affine functions, Lipschitz-continuous with bounded Lipschitz constants, satisfying the conditions

A_{k} (0) = 0, ∀ k ∈ ℕ,
A′_{k} (ϑ) ≠ 0, a.e. ϑ ∈ [0, 2π] , ∀ k ∈ ℕ,
A_{k} → A_{v−ξ} in C^{0} ([0, 2π]) ,
A′_{k} (ϑ) → A′_{v−ξ} (ϑ) , a.e. ϑ ∈ [0, 2π] ,
|A′_{k} (ϑ)| ≤ L_{0}, a.e. ϑ ∈ [0, 2π] , ∀ k ∈ ℕ,

and define

v_{k} (ϑ) := ξ + r (A_{k} (ϑ)) (cos A_{k} (ϑ) , sin A_{k} (ϑ)) .

Then the map v_{k} (ϑ) satisfies all the assumptions of the previous Step 2 and

\{ \begin{matrix}v_{k} → v  in C^{0} ([0, 2π]) , \\ ∥\frac{dv_{k}}{dϑ} ∥_{L^{∞}(0,2π)} ≤ L_{0},  ∀ k ∈ ℕ, \end{matrix} .

We prove that \{\frac{dv_{k}}{dϑ} \}_{k∈ℕ} converges to \frac{dv}{dϑ} for almost every ϑ ∈ [0, 2π], as k → +∞. To this aim, let us recall that

v, v_{k} : [0, 2π] → Γ = {ξ + r (ϑ) (cos ϑ, sin ϑ) : ϑ ∈ [0, 2π]} ,

where r (ϑ) is a piecewise C^{1}-function, i.e., there exist a finite number of points 0 ≤ a_{0} < a_{1} < … < a_{N} ≤ 2π, such that r (ϑ) is a function of class C^{1} ([a_{j−1}, a_{j}]) for every j = 1, 2, …, N. Define

E := {ϑ ∈ [0, 2π] : ∃ j = 1, 2, …, N : A_{v−ξ} (ϑ) = a_{j}} .

Then

A′_{v−ξ} (ϑ) = 0 and v′ (ϑ) = 0, a.e. ϑ ∈ E ,  (65)

and for almost every ϑ ∈ [0, 2π] we have

\frac{dv_{k}}{dϑ} = A′_{k} (ϑ) \{r′ (A_{k} (ϑ)) (cos A_{k} (ϑ) , sin A_{k} (ϑ)) + r (A_{k} (ϑ)) (− sin A_{k} (ϑ) , cos A_{k} (ϑ))\}  (66)

As k → +∞, {r′ (A_{k} (ϑ))}_{k∈ℕ} converges to r′ (A_{v−ξ}) for every ϑ ∉ E. Thus, by (66), we have that \{\frac{dv_{k}}{dϑ} \}_{k∈ℕ} converges to \frac{dv}{dϑ} for almost every ϑ ∈ [0, 2π] \E. On the other hand, by (65), for almost every ϑ ∈ E, {A′_{k} (ϑ)}_{k∈ℕ} converges to A′_{v−ξ} (ϑ) = 0 and, since {r′ (A_{k} (ϑ))}_{k∈ℕ} is uniformly bounded,

by (66) we can conclude that \{\frac{dv_{k}}{dϑ} \}_{k∈ℕ} converges to 0 = \frac{dv}{dϑ} for almost every ϑ ∈ E, as k → +∞.

Therefore {v_{k} (ϑ)}_{k∈ℕ} converges to v (ϑ) in the strong topology of W^{1,q} (B_{1}; ℝ^{2}) for every q ≥ 1, as k → +∞, and this implies that the map u_{k} (ϱ, ϑ) := v_{k} (ϑ) (independent of ϱ), which belongs to W^{1,p} (B_{1}; ℝ^{2}) for every p ∈ [1, 2) converges to u (ϱ, ϑ) = v (ϑ), as k → +∞, in the strong topology of W^{1,p} (B_{1}; ℝ^{2}) for every p ∈ [1, 2). From Step 2, and in particular from (64), we deduce that

T V (u, Ω) ≤ T V^{s} (u, Ω) ≤ lim inf_{k→+∞} T V^{s} (u_{k}, Ω) = lim inf_{k→+∞} T V^{s} (u_{k}, Ω)

= lim_{k→+∞} \frac{1}{2} |∫_{0}^{2π} \{v_{k}^{1} \frac{∂v_{k}^{2}}{∂ϑ} − v_{k}^{2} \frac{∂v_{k}^{1}}{∂ϑ} \} dϑ| = \frac{1}{2} |∫_{0}^{2π} \{v^{1} \frac{dv^{2}}{dϑ} − v^{2} \frac{dv^{1}}{dϑ} \} dϑ| .

By (63) of Step 1 we finally obtain

T V (u, Ω) = T V^{s} (u, Ω) = \frac{1}{2} |∫_{0}^{2π} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| .

Step 4 (upper bound - third part). Here we study the general case, with u = u (ϱ, ϑ) explicitly depending on ϱ too. The proof of this step, in n = 2 dimensions, is similar to the proof given in the n−dimensional case in Step 3 of Section 7. However we are able to make some simplifications and, in particular, we do not need the use of Lemma 39. For this reason, and also to make the idea of the proof clearer, we provide some details of the proof in this setting.

Using the argument of Lemma 36, for every u ∈ L^{∞} (Ω; ℝ^{2}) ∩ W^{1,p} (Ω; ℝ^{2}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{2}) with p > 1, it can be shown that admissible sequences for T V^{s} (u, Ω), defined in (26), may be required to assume prescribed boundary values, precisely

T V^{s} (u, Ω) = inf {lim inf_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx :  (67)

u_{h} → u strongly in W^{1,p} (Ω; ℝ^{2}) , u_{h} ∈ u + W_{0}^{1,∞} (Ω; ℝ^{2})} .

Thus, for every ε > 0, there exists a map w ∈ v + W_{0}^{1,∞} (B_{1}; ℝ^{2}) such that

∫_{B_{1}} |det Dw (x)| dx < ε + T V^{s} (v, B_{1}) .  (68)

In view of Lemma 38 (valid in the general n−dimensional case) there exists a sequence (ϱ_{h})_{h∈ℕ} converging to zero, such that

(ϱ_{j})^{p−1} ∫_{∂B_{ϱ_{j}}} |D_{τ} u|^{p} dH^{1} = ∫_{0}^{2π} |u_{ϑ} (ϱ_{h}, ϑ)|^{p} dϑ ≤ M_{0} , ∀ h ∈ ℕ.  (69)

Introduce the sequence of functions

u_{h} (ϱ, ϑ) := \{ \begin{matrix}w (\frac{ϱ}{ϱ_{h}(1−σ_{h})}, ϑ) ,  if 0 ≤ ϱ ≤ ϱ_{h} (1 − σ_{h}) \\ η_{h} (ϱ) v (ϑ) + [1 − η_{h} (ϱ)] u (ϱ_{h}, ϑ) , if ϱ_{h} (1 − σ_{h}) < ϱ < ϱ_{h} \\ u (ϱ, ϑ) ,  if ϱ_{h} ≤ ϱ \end{matrix} ,

where σ_{h} := (ϱ_{h})^{\frac{2−p}{p−1}} and η_{h} is a cut-off function, i.e., η_{h} (ϱ) = 1 if 0 ≤ ϱ ≤ ϱ_{h} (1 − σ_{h}), η_{h} (ϱ) = 0 if ϱ_{h} ≤ ϱ ≤ 1, η_{h} (ϱ) is linear in the interval [ϱ_{h} (1 − σ_{h}) , ϱ_{h}]. Notice that u_{h} → u in L^{p} (B_{1}; ℝ^{2}), as h → +∞, and that the sequence of gradients (Du_{h})_{h∈ℕ} converges in L^{p} to Du. Indeed

∫_{Ω} |Du_{h} − Du|^{p} dx ≤ c_{1} ∫_{B_{ϱ_{h}(1−σ_{h})}} |Dw (\frac{x}{ϱ_{h} (1 − σ_{h})})|^{p} dx + c_{1} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} \begin{matrix}|v_{ϑ}| & p \\ |x| & p \end{matrix} dx

+c_{1} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} \begin{matrix}|u_{ϑ} (ϱ_{h}, ϑ)| & p \\ |x| & p \end{matrix} dx+ \frac{c_{1}}{ϱ_{h}^{p}σ_{h}^{p}} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |u (ϱ_{h} \frac{x}{|x|}) − v (\frac{x}{|x|})|^{p} dx+c_{1} ∫_{B_{ϱ_{h}}} |Du|^{p} dx

≤ c_{2}ϱ_{h}^{2−p} + c_{2}ϱ_{h}^{2−p}σ_{h} ∫_{0}^{2π} |u_{ϑ} (ϱ_{h}, ϑ)|^{p} dϑ + c_{2} \frac{ϱ_{h}^{2−p}}{σ_{h}^{p−1}} ∥u (ϱ_{h}, ϑ) − v (ϑ)∥_{L^{∞}(0,2π)}^{p} + c_{1} ∫_{B_{ϱ_{h}}} |Du|^{p} dx ,

and this quantity goes to zero since σ_{h} = (ϱ_{h})^{\frac{2−p}{p−1}} . Therefore, by (68) we get

T V^{s} (u, Ω) ≤ lim inf_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx

= ∫_{B_{1}} |det Dw (x)| dx + lim inf_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx + ∫_{Ω} |det Du (x)| dx

≤ ε + T V^{s} (v, B_{1}) + lim inf_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx + ∫_{Ω} |det Du (x)| dx .  (70)

We evaluate the last integral in the right hand side. For ϱ_{h} (1 − σ_{h}) < ϱ < ϱ_{h} we have

det Du_{h} = \frac{1}{ϱ} | \begin{matrix}\frac{∂u_{h}^{1}(ϱ,ϑ)}{∂ϱ} & \frac{∂u_{h}^{1}(ϱ,ϑ)}{∂ϑ} \\ \frac{∂u_{h}^{2}(ϱ,ϑ)}{∂ϱ} & \frac{∂u_{h}^{2}(ϱ,ϑ)}{∂ϑ} \end{matrix} |

= \frac{1}{ϱ} | \begin{matrix}η′_{h} (ϱ) [v^{1} (ϑ) − u^{1} (ϱ_{h}, ϑ)] & η_{h} (ϱ) \frac{∂v^{1}(ϑ)}{∂ϑ} + [1 − η_{h} (ϱ)] \frac{∂u_{ϑ}^{1}(ϱ_{h},ϑ)}{∂ϑ} \\ η′_{h} (ϱ) [v^{2} (ϑ) − u^{2} (ϱ_{h}, ϑ)] & η_{h} (ϱ) \frac{∂v^{2}(ϑ)}{∂ϑ} + [1 − η_{h} (ϱ)] \frac{∂u^{2}(ϱ_{h},ϑ)}{∂ϑ} \end{matrix} | ,

and thus, since |η′_{h} (ϱ)| ≤ \frac{c}{σ_{h}ϱ_{h}} for some constant c_{1}, we have

∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h}| dx

≤ \frac{c_{1}}{σ_{h}ϱ_{h}} ∫_{ϱ_{h}(1−σ_{h})}^{ϱ_{h}} dϱ ∫_{0}^{2π} |v (ϑ) − u (ϱ_{h}, ϑ)| · \{|\frac{∂v (ϑ)}{∂ϑ}| + |\frac{∂u (ϱ_{h}, ϑ)}{∂ϑ}|\} dϑ

≤ c_{1} sup \{|v (ϑ) − u (ϱ_{h}, ϑ)| : ϑ ∈ [0, 2π]\} · ∫_{0}^{2π} \{|\frac{∂v (ϑ)}{∂ϑ}| + |\frac{∂u (ϱ_{h}, ϑ)}{∂ϑ}|\} dϑ .

By (69) there exists a new constant c_{3} such that

∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h}| dx ≤ c_{3} sup {|v (ϑ) − u (ϱ, ϑ)| : ϑ ∈ [0, 2π]}

and thus, by assumption (8) and by (70), letting ε → 0 we obtain

T V (u, Ω) ≤ T V^{s} (u, Ω) ≤ ∫_{Ω} |det Du| dx + T V (v, B_{1}) .

This upper bound, together with the lower bound of Step 1, yields the conclusion.

6 The “eight” curve

Let us denote by γ the image of the “eight” curve, i.e., the union of the two circles γ^{+} and γ^{−} of radius 1, respectively of center at (1, 0) and at (−1, 0). Below we will use some elementary representation formulas for γ^{+} and γ^{−}. Precisely, for γ^{+} we will use the representation formulas

γ^{+} := {(ξ_{1}, ξ_{2}) ∈ ℝ^{2} : ξ_{1}^{2} + ξ_{2}^{2} − 2ξ_{1} = 0} ,  (71)
ξ ∈ γ^{+}\ (0, 0) ⟺ \{ \begin{matrix}ξ_{1} = 2 cos^{2} Arg ξ \\ ξ_{2} = 2 cos Arg ξ · sin Arg ξ \end{matrix} .  (72)

With the aim to prove Theorem 4, we start with some preliminary results concerning a map w with values in the circle γ^{+}.

Lemma 30 Let w : [0, 2π] → γ^{+} be a Lipschitz-continuous curve such that w (0) = (2, 0). The real function R (ϑ) defined by R (ϑ) := 0 if w (ϑ) = (0, 0) and by

R (ϑ) := \frac{w^{1} (ϑ) w_{ϑ}^{2} (ϑ) − w^{2} (ϑ) w_{ϑ}^{1} (ϑ)}{|w (ϑ)|^{2}} , if w (ϑ) ≠ (0, 0) ,  (73)

is bounded in [0, 2π] by a constant depending only on the Lipschitz constant of w. Moreover, if

A_{w} (ϑ) = ∫_{0}^{ϑ} \frac{w^{1} (t) w_{ϑ}^{2} (t) − w^{2} (t) w_{ϑ}^{1} (t)}{|w (t)|^{2}} dt  (74)

then, for every α, β ∈ [0, 2π] such that w (α) ≠ (0, 0) and w (β) ≠ (0, 0), there exists k ∈ ℤ such that

A_{w} (β) − A_{w} (α) = Arg w (β) − Arg w (α) + kπ .  (75)

Proof. Step 1 (boundedness of R (ϑ)). Let L be the Lipschitz constant of w. If |w (ϑ)| ≥ \frac{1}{2}, then there exists a constant c such that

|R (ϑ)| ≤ cL .  (76)

On the other hand, if |w (ϑ)| < \frac{1}{2} then, since [w^{1} (ϑ)]^{2} + [w^{2} (ϑ)]^{2} − 2w^{1} (ϑ) = 0, we deduce that

|w (ϑ)|^{2} = 2w^{1} (ϑ) and w^{1} (ϑ) = 1 − \sqrt{1 − [w^{2} (ϑ)]^{2}} . Taking the derivative of both sides we obtain

w_{ϑ}^{1} (ϑ) = \frac{w^{2} (ϑ) w_{ϑ}^{2} (ϑ)}{\sqrt{1 − [w^{2} (ϑ)]^{2}}} .

Therefore, if w (ϑ) ≠ (0, 0), for almost every ϑ we also have

R (ϑ) = \frac{w^{1} (ϑ) w_{ϑ}^{2} (ϑ) − w^{2} (ϑ) w_{ϑ}^{1} (ϑ)}{|w (ϑ)|^{2}} = \frac{w_{ϑ}^{2} (ϑ)}{2} − \frac{w^{2} (ϑ) w_{ϑ}^{1} (ϑ)}{2w^{1} (ϑ)}

= \frac{w_{ϑ}^{2} (ϑ)}{2} (1 − \frac{[w^{2} (ϑ)]^{2}}{(1 − \sqrt{1 − [w^{2} (ϑ)]^{2}} ) · \sqrt{1 − [w^{2} (ϑ)]^{2}}} ) .

The derivative of the real function g (t) = 1 − \sqrt{1 − t} satisfies the condition g′ (t) ≥ 1/2 for every t ∈ [0, 1); thus we have

1 − \sqrt{1 − [w^{2} (ϑ)]^{2}} ≥ \frac{1}{2} [w^{2} (ϑ)]^{2} .

We deduce that

|R (ϑ)| ≤ \frac{1}{2} |w_{ϑ}^{2} (ϑ)| (1 + \frac{2}{\sqrt{1 − [w^{2} (ϑ)]^{2}}} ) ,

and again (76) holds for an appropriate constant c since |w^{2} (ϑ)| < \frac{1}{2}. This proves the first assertion of the lemma.

Step 2 (proof of (75) under special assumptions). To prove assertion (75) we first make the further assumption that there exist N disjoint open intervals (α_{i}, β_{i}) such that

0 = α_{1} < β_{1} ≤ α_{2} < β_{2} ≤ … ≤ α_{N} < β_{N} = 2π ,

and w (ϑ) = (0, 0) if and only if ϑ ∈ [0, 2π] \ ∪_{i=1}^{N} (α_{i}, β_{i}). Fix α, β ∈ (0, 2π) such that w (α) ≠ (0, 0) and w (β) ≠ (0, 0). If α, β ∈ (α_{i}, β_{i}) for some i ∈ {1, 2, …, N}, then, using an argument similar to that of the first part of Lemma 20, we have

A_{w} (β) − A_{w} (α) = Arg w (β) − Arg w (α) .  (77)

Otherwise, if there exists i ∈ {1, 2, …, N} such that

α_{i} < α < β_{i} ≤ α_{i+1} < β < β_{i+1} ,  (78)

then we apply (77) to the interval (α, β_{i} − ε) to obtain

A_{w} (β_{i} − ε) − A_{w} (α) = Arg w (β_{i} − ε) − Arg w (α) .

In the limit as ε → 0^{+}, since when w (ϑ) ∈ γ^{+}\ \{(0, 0)\} then Arg w (ϑ) ∈ (−\frac{π}{2} , \frac{π}{2} ), we obtain

A_{w} (β_{i}) − A_{w} (α) = ±\frac{π}{2} − Arg w (α) ,  (79)

where the sign + holds if w^{2} (ϑ) > 0 as ϑ → β_{i}^{−}, and the sign − holds otherwise. Similarly, we have

A_{w} (β) − A_{w} (α_{i+1}) = Arg w (β) − (±\frac{π}{2} )  (80)

and, adding side by side (79) and (80), yields

A_{w} (β) − A_{w} (α) = Arg w (β) − Arg w (α) + kπ,

where k ∈ {−1, 0, 1}. The general case, when (78) is not necessarily satisfied, follows from the previous case by iteration.

Step 3 (proof of (75)). Let w : [0, 2π] → γ^{+} be a Lipschitz-continuous map. Let {I_{j}}_{j∈ℕ} be a sequence of disjoint open intervals (possibly empty) such that w (ϑ) ≠ (0, 0) if and only if ϑ ∈ ∪_{j∈ℕ}I_{j}. For every h ∈ ℕ we define

w_{h} (ϑ) := \{ \begin{matrix}w (ϑ) if ϑ ∈ ∪_{j=1}^{h}I_{j} \\ (0, 0) if ϑ ∉ ∪_{j=1}^{h}I_{j} \end{matrix} .

Then the sequence of Lipschitz constants L_{h} of w_{h} is bounded. Moreover w_{h} converges uniformly to w in [0, 2π], as h → +∞, and the corresponding sequence {R_{h} (ϑ)}_{h∈ℕ} converge to R (ϑ) almost everywhere in [0, 2π]. Therefore, integrating (73), we deduce that {A_{w_{h}} (ϑ)}_{h∈ℕ} converges to A_{w} (ϑ) uniformly in [0, 2π]. Let α, β ∈ [0, 2π] be such that w (α) ≠ (0, 0) and w (β) ≠ (0, 0). For h large enough we also have w_{h} (α) = w (α) ≠ (0, 0) and w_{h} (β) = w (β) ≠ (0, 0) and, by the previous step,

A_{w_{h}} (β) − A_{w_{h}} (α) = Arg w (β) − Arg w (α) + k_{h}π .

Since the sequence k_{h} is bounded, we can pass to the limit in a subsequence and we arrive at the conclusion (75).

Lemma 31 Under the same assumptions of the previous Lemma 30, for every ϑ ∈ [0, 2π] we have

w (ϑ) = 2 cos A_{w} (ϑ) (cos A_{w} (ϑ) , sin A_{w} (ϑ)) .  (81)

Proof. Recall that w (0) = (2, 0) and so Arg w (ϑ) = 0. By Lemma 30, if w (ϑ) ≠ (0, 0), then there exists k_{ϑ} ∈ ℤ such that Arg w (ϑ) = A_{w} (ϑ) + k_{ϑ}π. By (72) we deduce the conclusion

\{ \begin{matrix}w^{1} (ϑ) = 2 cos^{2} Arg w (ϑ) = 2 cos^{2} A_{w} (ϑ) \\ w^{2} (ϑ) = sin 2 Arg w (ϑ) = sin 2 A_{w} (ϑ) = 2 sin A_{w} (ϑ) · cos A_{w} (ϑ) \end{matrix} .

If w (ϑ_{0}) = (0, 0) and there exists a sequence ϑ_{i} → ϑ_{0} such that w (ϑ_{i}) ≠ (0, 0) for every i ∈ ℕ, then (81) holds for ϑ = ϑ_{i}. Since A_{w} (ϑ) is a continuous function, (81) holds for ϑ = ϑ_{0} as well.

If w (ϑ_{0}) = (0, 0) and a sequence ϑ_{i} → ϑ_{0} such that w (ϑ_{i}) ≠ (0, 0) for every i ∈ ℕ does not exist, then there exists an interval (ϑ_{0} − δ, ϑ_{0} + δ), with δ > 0, such that w (ϑ) is identically equal to (0, 0) in (ϑ_{0} − δ, ϑ_{0} + δ). In this case let us denote by (α, β) the largest interval containing ϑ_{0} with this property; since R (ϑ) = 0 in (α, β) we have A_{w} (α) = A_{w} (ϑ_{0}). On the other hand (81) holds for ϑ = α since (α, β) is an extremal interval; hence

w (ϑ_{0}) = (0, 0) = w (α) = 2 cos A_{w} (α) (cos A_{w} (α) , sin A_{w} (α))
= 2 cos A_{w} (ϑ_{0}) (cos A_{w} (ϑ_{0}) , sin A_{w} (ϑ_{0})) .

The next lemma is similar to the “umbrella” Lemma 23, with the main difference that here the starting point of the “umbrella-stick” is placed at a boundary point of the circle γ^{+}.

Lemma 32 (The “umbrella” lemma for the “eight” curve) Let w : [0, 2π] → γ^{+} be a Lipschitzcontinuous curve. Assume that there exist α, β ∈ [0, 2π], α < β, such that A_{w} (α) = A_{w} (β). Then, for every ε > 0, there exists a Lipschitz-continuous map w̃ : S (α, β) → ℝ^{2} satisfying the boundary conditions

\{ \begin{matrix}w̃ (1, ϑ) = w (ϑ) , ∀ ϑ ∈ [α, β] \\ w̃ (ϱ, α) = ϱw (α) , ∀ ϱ ∈ [0, 1] \\ w̃ (ϱ, β) = ϱw (β) , ∀ ϱ ∈ [0, 1] \end{matrix}

(note that w (α) = w (β)) and such that

∫_{S(α,β)} |det Dw̃ (x)| dx < ε .

Proof. For fixed h ∈ ℕ we set

w̃_{h} (ϱ, ϑ) := 2ϱ cos φ_{h} (ϱ, ϑ) (cos φ_{h} (ϱ, ϑ) , sin φ_{h} (ϱ, ϑ)) ,

where

φ_{h} (ϱ, ϑ) := ϱ^{h}A_{w} (ϑ) + (1 − ϱ^{h}) A_{w} (α) .

Let us test the boundary conditions of w̃ (ϱ, ϑ). By Lemma 31, for every ϑ ∈ [α, β] we have

w̃_{h} (1, ϑ) = 2 cos A_{w} (ϑ) (cos A_{w} (ϑ) , sin A_{w} (ϑ)) = (w_{h}^{1} (ϑ) , w_{h}^{2} (ϑ)) = w (ϑ) ,

and, for every ϱ ∈ [0, 1],

w̃_{h} (ϱ, α) = 2ϱ cos A_{w} (α) (cos A_{w} (α) , sin A_{w} (α)) = ϱ (w_{h}^{1} ((α)) , w_{h}^{2} ((α))) = ϱw (α) .

Similarly w̃_{h} (ϱ, β) = ϱw (β) for every ϱ ∈ [0, 1]. Using an argument similar to the one used in Lemma 23, we can see that (we do not denote in the matrix the dependence on h)

det Dw̃_{h} (x) = \frac{1}{ϱ} | \begin{matrix}w_{ϱ}^{1} (ϱ, ϑ) & w_{ϑ}^{1} (ϱ, ϑ) \\ w_{ϱ}^{2} (ϱ, ϑ) & w_{ϑ}^{2} (ϱ, ϑ) \end{matrix} | = 4ϱ^{h} cos^{2} φ_{h} (ϱ, ϑ) A′_{w} (ϑ) .

By Lemma 30 the function

A′_{w} (ϑ) = \frac{w^{1} (ϑ) w_{ϑ}^{2} (ϑ) − w^{2} (ϑ) w_{ϑ}^{1} (ϑ)}{|w (ϑ)|^{2}}

is bounded; thus there exists a constant c such that

∫_{S(α,β)} |det Dw̃_{h} (x)| dx ≤ c ∫_{0}^{1} ϱ^{h+1} dϱ = \frac{c}{h + 2} ,

and this concludes the proof of our lemma.

Lemma 33 Let w : [0, 2π] → γ^{+} be a Lipschitz-continuous map. If α, β ∈ [0, 2π], α < β, are such that A_{w} (α) = A_{w} (β), and if the function A_{w} (ϑ) is piecewise strictly monotone in [α, β] (with a finite number of monotonicity intervals), then

∫_{α}^{β} {w^{1} (ϑ) w_{ϑ}^{2} (ϑ) − w^{2} (ϑ) w_{ϑ}^{1} (ϑ)} dϑ = 0 .

Proof. This result can be proved just as in Lemma 26.

Lemma 34 Let u : [0, 2π] → γ = γ^{+} ∪ γ^{−} be a Lipschitz-continuous map. Assume that there exist N disjoint open intervals I_{j} ⊂ [0, 2π] such that u (I_{j}) is contained either in γ^{+} or in γ^{−} for every j = 1, 2, …, N, and u (ϑ) = (0, 0) when ϑ ∉ ∪_{j=1}^{N}I_{j}. Assume, in addition, that the function

ϑ ⟶ u^{1} (ϑ) u_{ϑ}^{2} (ϑ) − u^{2} (ϑ) u_{ϑ}^{1} (ϑ)  (82)

has piecewise constant sign in [0, 2π]. Then, for every ε > 0, there exists a Lipschitz-continuous map w̃ : B_{1} → ℝ^{2} satisfying the boundary condition w̃ (1, ϑ) = u (ϑ) for every ϑ ∈ [0, 2π], and such that

∫_{B_{1}} |det Dw̃ (x)| dx < ε + \frac{1}{2} ∑_{j=1}^{N} |∫_{I_{j}} \{u^{1} (ϑ) u_{ϑ}^{2} (ϑ) − u^{2} (ϑ) u_{ϑ}^{1} (ϑ)\} dϑ| .  (83)

Proof. Fix j ∈ {1, 2, …, N} and assume that u (I_{j}) ⊂ γ^{+}. We follow the method of proof of Lemma 27, using Lemma 32 (in place of Lemma 23) and Lemma 33 (in place of Lemma 26). Setting I_{j} := (α_{j}, β_{j}), we construct a Lipschitz-continuous map w̃_{j} : S (α_{j}, β_{j}) → ℝ^{2} satisfying the boundary conditions

\{ \begin{matrix}w̃_{i} (1, ϑ) = u (ϑ) ,  ∀ ϑ ∈ [α_{j}, β_{j}] \\ w̃_{i} (ϱ, α_{j}) = ϱ · u (α_{j}) = (0, 0) , ∀ ϱ ∈ [0, 1] \\ w̃_{i} (ϱ, β_{j}) = ϱ · u (β_{j}) = (0, 0) , ∀ ϱ ∈ [0, 1] \end{matrix}  (84)

and the estimate

∫_{S(α_{j},β_{j})} |det Dw̃_{i} (x)| dx < \frac{ε}{N} + \frac{1}{2} |∫_{I_{j}} \{u^{1} (ϑ) u_{ϑ}^{2} (ϑ) − u^{2} (ϑ) u_{ϑ}^{1} (ϑ)\} dϑ | .  (85)

A similar conclusion holds if, instead, we have u (I_{j}) ⊂ γ^{−}. Then the result follows by taking w̃ : B_{1} → ℝ^{2} defined by

w̃ (ϱ, ϑ) := \{ \begin{matrix}w̃_{j} (ϱ, ϑ) , ∀ ϑ ∈ (α_{j}, β_{j}) = I_{j} \\ (0, 0) , ∀ ϑ ∉ ∪_{j=1}^{N}I_{j} \end{matrix} .

Proof of Theorem 4. Step 1 (lower bound - first part). Let v : [0, 2π] → γ = γ^{+} ∪ γ^{−} be a Lipschitz-continuous map. With u (x) := v (x/ |x|) then u ∈ L^{∞} (B_{1}; ℝ^{2}) ∩ W^{1,p} (B_{1}; ℝ^{2}) ∩ W_{loc}^{1,∞} (B_{1}\ {0}; ℝ^{2}) for every p ∈ (1, 2). By (90) (lower bound obtained in Lemma 36 in the general n−dimensional case) we have

T V (u, B_{1}) ≥ |∫_{B_{1}} det Dũ (x) dx| ,

where ũ : B_{1} → ℝ^{2} is any Lipschitz-continuous map which assumes the boundary value ũ = u on ∂B_{1} (e.g., ũ (x) = |x| v (x/ |x|) = |x| u (x) for x ∈ B_{1}\ {0} and ũ (0) = 0). By formula (61) of Lemma 28 (valid on B_{r} for every r ∈ (0, 1]), since ũ = u, ∂ũ/∂ϑ = ∂u/∂ϑ on ∂B_{1}, we have

T V (u, B_{1}) ≥ | \frac{1}{2} ∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .  (86)

As in the statement of Theorem 4, we denote by (I_{j})_{j∈ℕ} a sequence of disjoint open intervals in [0, 2π] (possibly empty) such that the image v (I_{j}) is contained either in γ^{+} or in γ^{−}, and v (ϑ) = (0, 0) when ϑ ∉ ∪_{j∈ℕ}I_{j}. Then we can write (86) equivalently

T V (u, B_{1}) ≥ \frac{1}{2} |∑_{j∈ℕ} ∫_{I_{j}} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .  (87)

Step 2 (lower bound - second part). Let δ > 0 and let {u_{h} = (u_{h}^{1}, u_{h}^{2})}_{h∈ℕ} be a sequence in W^{1,2} (B_{1}; ℝ^{2}) converging to u in the weak topology of W^{1,p} (B_{1}; ℝ^{2}), p ∈ (1, 2), and such that

T V (u, B_{1}) + δ ≥ lim_{h→+∞} ∫_{B_{1}} |det Du_{h} (x)| dx .

Let u_{h}^{+} := (|u_{h}^{1}| , u_{h}^{2}) ∈ W^{1,2} (B_{1}; ℝ^{2}), which converges to u^{+} = (|u^{1}| , u^{2}) in the weak topology of W^{1,p} (B_{1}; ℝ^{2}) as h → +∞. Since |det Du_{h}^{+} (x)| = |det Du_{h} (x)| for almost every x ∈ B_{1}, we obtain

T V (u, B_{1}) + δ ≥ lim_{h→+∞} ∫_{B_{1}} |det Du_{h} (x)| dx = lim_{h→+∞} ∫_{B_{1}} |det Du_{h}^{+} (x)| dx ≥ T V (u^{+}, B_{1}) .

The total variation of the map u^{+} : B_{1} → γ^{+} can be obtained using the formula (10), with u^{+} = (|u^{1}| , u^{2}) = (|v^{1}| , v^{2}); therefore, as δ → 0^{+} we have

T V (u, B_{1}) ≥ T V (u^{+}, B_{1}) = \frac{1}{2} |∫_{0}^{2π} \{|v^{1}| \frac{dv^{2}}{dϑ} − v^{2} \frac{d |v^{1}|}{dϑ}\} dϑ| .

Recall that v^{1} (ϑ) > 0 if ϑ ∈ I_{j}, where v (I_{j}) ⊂ γ^{+} (and analogously v^{1} (ϑ) < 0 if v (I_{j}) ⊂ γ^{−}, while v^{1} (ϑ) = 0 if ϑ ∉ ∪_{j∈ℕ}I_{j}). Again, as in the statement of Theorem 4, we denote by I_{j}^{+}, with the + sign, any interval I_{j} such that v (I_{j}) ⊂ γ^{+}, and by I_{k}^{−} any interval I_{k} such that v (I_{k}) ⊂ γ^{−}. Thus

T V (u, B_{1}) ≥ \frac{1}{2} |∑_{j∈ℕ}∫_{I_{j}^{+}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ − ∑_{k∈ℕ}∫_{I_{k}^{−}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| .  (88)

Step 3 (lower bound - conclusion). Using the results of the previous Steps 1 and 2, in particular (87) and (88), we have

T V (u, B_{1}) ≥ \frac{1}{2} max_{±} |∑_{j∈ℕ}∫_{I_{j}^{+}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ ± ∑_{k∈ℕ}∫_{I_{k}^{−}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| .

Since max_{±} |a ± b| = |a| + |b|, we finally obtain the lower bound (13).

Step 4 (upper bound). Assume first that u : [0, 2π] → γ = γ^{+}∪γ^{−} is a Lipschitz-continuous map satisfying the further assumptions of Lemma 34. In particular, we assume that there exist N disjoint open intervals I_{j} ⊂ [0, 2π] such that u (I_{j}) is contained either in γ^{+} or in γ^{−} for every j = 1, 2, …, N, and u (ϑ) = (0, 0) when ϑ ∉ ∪_{j=1}^{N}I_{j}. We also assume that the function ϑ ⟶ u^{1} (ϑ) u_{ϑ}^{2} (ϑ)−u^{2} (ϑ) u_{ϑ}^{1} (ϑ) has piecewise constant sign in [0, 2π]. Then, for every ε > 0, there exists a Lipschitz-continuous map w̃ : B_{1} → ℝ^{2} satisfying the boundary condition w̃ (1, ϑ) = u (ϑ) for ϑ ∈ [0, 2π], and (83). For every h ∈ ℕ we define

u_{h} (ϱ, ϑ) := \{ \begin{matrix}u (ϑ) ,  if 1/h ≤ ϱ ≤ 1, \\ w̃ (ϱh, ϑ) ,  if 0 ≤ ϱ ≤ 1/h \end{matrix} .

As in Step 2 of the proof of Theorem 1, u_{h} converges to u strongly in W^{1,p} (B_{1}; ℝ^{2}) for every p ∈ (1, 2), as h → +∞. Finally, by (83),

∫_{B_{1}} |det Du_{h} (x)| dx = ∫_{B_{1/h}} |h^{2} det Dw (ϱh, ϑ)| dx = ∫_{B_{1}} |det Dw̃ (ϱ, ϑ)| dx

< ε + \frac{1}{2} ∑_{j=1}^{N} |∫_{I_{j}} \{u^{1} (ϑ) u_{ϑ}^{2} (ϑ) − u^{2} (ϑ) u_{ϑ}^{1} (ϑ)\} dϑ | ,

and thus we obtain the conclusion (12) in this case, i.e.,

T V (u, B_{1}) ≤ T V^{s} (u, B_{1}) ≤ \frac{1}{2} ∑_{j=1}^{N} |∫_{I_{j}} \{u^{1}u_{ϑ}^{2} − u^{2}u_{ϑ}^{1}\} dϑ | .

Step 5 (upper bound again). Consider first the case where u : [0, 2π] → γ satisfies the conditions of the previous Step 4, with the possible exception that the function ϑ ⟶ u^{1} (ϑ) u_{ϑ}^{2} (ϑ) − u^{2} (ϑ) u_{ϑ}^{1} (ϑ) has piecewise constant sign in [0, 2π]. Assume further that there exist N disjoint open intervals I_{j} ⊂ [0, 2π] such that u (I_{j}) is contained either in γ^{+} or in γ^{−} for every j = 1, 2, …, N, and u (ϑ) = (0, 0) when ϑ ∉ ∪_{j=1}^{N}I_{j}. We proceed in a way similar to that of Step 3 of the proof of Theorem 1. We consider one of such intervals I_{j} such that u (I_{j}) ⊂ γ^{+} and, without loss of generality, we can assume that u (0) = (2, 0) ∈ γ^{+}. By Lemma 31, the map u (ϑ) can be represented in the form

u (ϑ) = 2 cos A_{w} (ϑ) (cos A_{w} (ϑ) , sin A_{w} (ϑ))

for ϑ ∈ I_{j}. As in Step 3 of the proof of Theorem 1, we may find a sequence (u_{j,k})_{k∈ℕ}, with u_{j,k} : I_{j} → ℝ^{2}, such that, as k → +∞,

\{ \begin{matrix}u_{j,k} → u in C^{0} (\overline{I_{j}}) , \\ \frac{du_{j,k}}{dϑ} → \frac{du}{dϑ} strongly in L^{q} (I_{j}) ∀ q ≥ 1 \end{matrix} .

Moreover u_{j,k} (ϑ) = u (ϑ) for ϑ ∈ ∂I_{j}, and u_{j,k}^{1}du_{j,k}^{2}/dϑ − u_{j,k}^{2}du_{j,k}^{1}/dϑ has piecewise constant sign in I_{j}. Then the map u_{j,k} (ϑ) satisfies all the assumptions of the previous Step 4. We define

u_{k} (ϑ) := \{ \begin{matrix}u_{j,k} (ϑ) , if ϑ ∈ I_{j} \\ (0, 0) , if ϑ ∉ ∪_{j}I_{j} \end{matrix} .

Clearly the maps u_{k} (ϑ) converge to u in the strong topology of W^{1,p} (B_{1}; ℝ^{2}) for every p ∈ [1, 2), as k → +∞, and from Step 4 we obtain the upper bound (12) under our assumptions, i.e.,

T V (u, B_{1}) ≤ T V^{s} (u, B_{1}) ≤ lim inf_{k→+∞} T V^{s} (u_{k}, B_{1})  (89)

= lim_{k→+∞} \frac{1}{2} ∑_{j=1}^{N} |∫_{I_{j}} \{u_{j,k}^{1} \frac{du_{j,k}^{2}}{dϑ} − u_{j,k}^{2} \frac{du_{j,k}^{1}}{dϑ} \} dϑ| = \frac{1}{2} ∑_{j=1}^{N} |∫_{I_{j}} \{u^{1} \frac{du^{2}}{dϑ} − u^{2} \frac{du^{1}}{dϑ} \} dϑ| .

Finally, when the intervals I_{j} are infinitely many, the upper bound (12) is deduced from the previous case of finitely many intervals I_{j} (j = 1, 2, …, N), by approximating u with

u_{N} (ϑ) := \{ \begin{matrix}u (ϑ) , if ϑ ∈ ∪_{j=1}^{N}I_{j} \\ (0, 0) , if ϑ ∉ ∪_{j=1}^{N}I_{j} \end{matrix} .

Indeed, applying (89) to each u_{N} and passing to the limit as N → +∞, we obtain

T V (u, B_{1}) ≤ T V^{s} (u, B_{1}) ≤ lim inf_{N→+∞} T V^{s} (u_{N}, B_{1})

≤ lim_{N→+∞} \frac{1}{2} ∑_{j=1}^{N} |∫_{I_{j}} \{u^{1} \frac{du^{2}}{dϑ} − u^{2} \frac{du^{1}}{dϑ} \} dϑ| = \frac{1}{2} ∑_{j=1}^{∞} |∫_{I_{j}} \{u^{1} \frac{du^{2}}{dϑ} − u^{2} \frac{du^{1}}{dϑ} \} dϑ| .

7 The n−dimensional case

In this section we prove Theorem 10. We first recall a lower semicontinuity result, valid for polyconvex integrands (and for quasiconvex integrands as well), related to the weak topology of W^{1,p} (Ω; ℝ^{n}) for some p below the critical exponent n. These may be called non-standard lower semicontinuity results, as opposed to the classical setting of lower semicontinuity results in the weak topology of W^{1,p} (Ω; ℝ^{n}) when p is equal to the growth exponent of the integrand f (see Morrey [48], Acerbi and Fusco [3], Marcellini [44]). We refer to polyconvex integrals as in Theorem 35 below, of the type

∫_{Ω} f (Du) dx , with 0 ≤ f (ξ) ≤ c (1 + |ξ|^{p}) .

In the case considered here, the integrand f (ξ) := |det ξ| ≤ n^{−n/2} |Du (x)|^{n} has growth exponent equal to n, while we need to consider the weak topology of W^{1,p} (Ω; ℝ^{n}) for some p < n.

Theorem 35 has been proved by Marcellini [44], [45] for p > n^{2}/ (n + 1) and by Dacorogna and Marcellini [19] for p > n−1 (p ≥ 1 if n = 2). A limiting case, with p = n−1, has been considered under different assumptions by Acerbi and Dal Maso [2], Celada and Dal Maso [14], Dal Maso and Sbordone [21] and by Fusco and Hutchinson [28]. The relaxation in this context has been first considered by Fonseca and Marcellini [27].

Precisely the following theorem holds (we limit ourselves to quote here the polyconvex case, related to maps u : Ω ⊂ ℝ^{n} → ℝ^{m} with m = n). Given a map u : Ω → ℝ^{n}, we denote by M (Du) the vectorvalued map

M (Du) = (Du, adj_{2}Du, …, adj_{n−1}Du, det Du) ∈ ℝ^{N},

where, for j = 2, …, n − 1, adj_{j}Du denotes the matrix of all minors j × j of Du and N = ∑_{j=1}^{n} \binom{n}{j}^{2} (in particular N = 5 if n = 2).

Theorem 35 (Lower semicontinuity below the critical exponent) Let Ω be an open set of ℝ^{n}. Let g : ℝ^{N} → ℝ be a nonnegative convex function. Then

lim inf_{h→+∞} ∫_{Ω} g (M (Du_{h})) dx ≥ ∫_{Ω} g (M (Du)) dx ,

for every sequence u_{h} which converge to u in the weak topology of W^{1,p} (Ω; ℝ^{n}) for some p > n − 1, with u, u_{h} ∈ W_{loc}^{1,n} (Ω; ℝ^{n}) for every h ∈ ℕ.

The following Lemmas 36, 37 and 38, are also used in the 2−dimensional Section 5. The first result stated in Lemma 36 gives a lower bound for the total variation. It is a variant of Lemma 5.1 (see also Lemma 2.3) by Marcellini [45], who considered the general quasiconvex case with the exponent p below the critical growth exponent n, precisely n^{2}/ (n + 1) < p < n.

Lemma 36 (Lower bound - first estimate) Let u ∈ L^{∞} (Ω; ℝ^{n})∩W^{1,p} (Ω; ℝ^{n})∩W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for some p ∈ (n − 1, n). The following estimate holds

T V (u, Ω) ≥ |∫_{Ω} det Dũ (x) dx| ,  (90)

whenever ũ : Ω → ℝ^{n} is a Lipschitz-continuous map which agrees with u on the boundary of Ω, i.e., ũ (x) = u (x) on ∂Ω.

Proof. For fixed p ∈ (n − 1, n), δ > 0, consider a sequence {u_{h}}_{h∈ℕ} in W^{1,n} (Ω; ℝ^{n}) that converges to u in the weak topology of W^{1,p} (Ω; ℝ^{n}), and such that

lim_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx ≤ T V (u, Ω) + δ .  (91)

Let M := ∥u∥_{L^{∞}(Ω;ℝ^{n})} ∈ ℝ. Truncate each u_{h} into w_{h} = (w_{h}^{1}, w_{h}^{2}, …, w_{h}^{n}) whose components are given by

w_{h}^{j} (x) := \{ \begin{matrix}−M  if u_{h}^{j} (x) ≤ −M \\ u_{h}^{j} (x)  if − M ≤ u_{h}^{j} (x) ≤ M \\ M  if u_{h}^{j} (x) ≥ M \end{matrix} , ∀ j = 0, 1, …, n.

Clearly {u_{h}}_{h∈ℕ} still converges to u, as h → +∞, in the weak topology of W^{1,p} (Ω; ℝ^{n}) and the L^{∞}−norm ∥w_{h}∥_{L^{∞}(Ω;ℝ^{n})} is uniformly bounded as h ∈ ℕ. Moreover, since

w_{h} (x) ≠ u_{h} (x) ⟹ det Dw_{h} (x) = 0 ,

we obtain |det Dw_{h} (x)| ≤ |det Du_{h} (x)| for almost every x ∈ Ω, and

lim inf_{h→+∞} ∫_{Ω} |det Dw_{h} (x)| dx ≤ lim_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx < T V (u, Ω) + δ .

Therefore, without loss of generality, passing to a subsequence, we can assume that the limit relation (91) holds, together with the uniform bound

sup_{h∈ℕ} ∥u_{h}∥_{L^{∞}(Ω;ℝ^{n})} = M < +∞ .  (92)

Let Ω_{0} be an open set compactly contained in Ω and let R := dist (Ω_{0}, ∂Ω) /2, with 0 ∈ Ω_{0}. For every k ∈ ℕ set

Ω_{i} := \{x ∈ Ω : dist (x, Ω_{0}) < \frac{iR}{k} \} , ∀ i = 1, 2, …, k.

For every i = 1, 2, …, k, consider a Lipschitz continuous cut-off scalar function φ_{i} : Ω → [0, 1], defined by

φ_{i} (x) := \{ \begin{matrix}1  if x ∈ Ω_{i−1} \\ 0 ≤ φ ≤ 1, |Dφ_{i}| ≤ \frac{k +1}{R} if x ∈ Ω_{i}\Ω_{i−1} \\ 0  if x ∈ Ω\Ω_{i} \end{matrix} .

Again, for every i = 1, 2, …, k, and for h ∈ ℕ, define

w_{h,i} (x) := (1 − φ_{i} (x)) u (x) + φ_{i} (x) u_{h} (x) .

Then w_{h,i} (x) = u (x) for every x ∈ Ω\Ω_{i}, and in particular for every x ∈ Ω\Ω_{0}. Since u (x) is a smooth map in Ω\Ω_{0} and since w_{h,i} (x) and ũ (x) are smooth maps in Ω, which coincide with u (x) on the boundary ∂Ω, using the fact that the integral of the Jacobian depends only on the trace at the boundary, we have

|∫_{Ω} det Dũ (x) dx| = |∫_{Ω} det Dw_{h,i} (x) dx| ≤ ∫_{Ω} |det Dw_{h,i} (x)| dx

= ∫_{Ω_{i−1}} |det Du_{h} (x)| dx + ∫_{Ω_{i}\Ω_{i−1}} |det Dw_{h,i} (x)| dx + ∫_{Ω\Ω_{i}} |det Du (x)| dx .

Letting h → +∞, taking into account the limit relation (91), summing up the above relation with respect to i = 1, 2, …, k, and dividing both sides by k, we obtain

|∫_{Ω} det Dũ (x) dx| ≤ T V (u, Ω) + δ  (93)

+ \frac{1}{k} lim sup_{h→+∞} ∑_{i=1}^{k} ∫_{Ω_{i}\Ω_{i−1}} |det Dw_{h,i} (x)| dx + ∫_{Ω\Ω_{0}} |det Du (x)| dx .

We estimate the second integral in the right hand side. To this aim, we recall the inequality (2.9) of Marcellini [44], valid for every quasiconvex function f (ξ) with q ∈ [1, +∞) growth: there exists a positive constant c such that, for every ξ, η,

|f (ξ) − f (η)| ≤ c (1 + |ξ|^{q−1} + |η|^{q−1}) |ξ − η| .  (94)

When applied to the quasiconvex function f (ξ) := |det ξ|, since |det ξ| ≤ n^{−n/2} |ξ|^{n}, from the previous inequality with q = n we deduce that

||det ξ| − |det η|| ≤ c (1 + |ξ|^{n−1} + |η|^{n−1}) |ξ − η| .  (95)

As Dw_{h,i} (x) = D [(1 − φ_{i} (x)) u (x) + φ_{i} (x) u_{h} (x)], in Ω_{i}\Ω_{i−1} we have

|Dw_{h,i} (x) − φ_{i} (x) Du_{h} (x)| ≤ |Dφ_{i} (x)| · |u_{h} (x) − u (x)|

+ |1 − φ_{i} (x)| · |Du (x)| ≤ \frac{k + 1}{R} |u_{h} (x) − u (x)| + |Du (x)| .

From (95) with ξ := Dw_{h,i} (x) and η := φ_{i} (x) Du_{h} (x) we obtain

| |det Dw_{h,i} (x)| − |det φ_{i} (x) Du_{h} (x)| |

≤ c (1 + |Dw_{h,i} (x)|^{n−1} + |φ_{i} (x) Du_{h} (x)|^{n−1}) |Dw_{h,i} (x) − φ_{i} (x) Du_{h} (x)|

≤ c (1 + |Dw_{h,i} (x)|^{n−1} + |Du_{h} (x)|^{n−1}) [ \frac{k + 1}{R} |u_{h} (x) − u (x)| + |Du (x)|] .

Set M_{1} := ∥Du∥_{L^{∞}(Ω\Ω_{0};ℝ^{n×n})} ∈ ℝ. Then, since p > n − 1, for the second integral in the right hand side of (93) we have the following bound

∫_{Ω_{i}\Ω_{i−1}} |det Dw_{h,i} (x)| dx ≤ ∫_{Ω_{i}\Ω_{i−1}} |det φ_{i} (x) Du_{h} (x)| dx

+c ∫_{Ω_{i}\Ω_{i−1}} \{(1 + |Dw_{h,i} (x)|^{n−1} + |Du_{h} (x)|^{n−1}) · [ \frac{k + 1}{R} |u_{h} (x) − u (x)| + M_{1}]\} dx

≤ ∫_{Ω_{i}\Ω_{i−1}} |det φ_{i} (x) Du_{h} (x)| dx + c \{∫_{Ω_{i}\Ω_{i−1}} (1 + |Dw_{h,i} (x)|^{n−1} + |Du_{h} (x)|^{n−1})^{\frac{p}{n−1}} dx\}^{\frac{n−1}{p}} ·

· \{∫_{Ω_{i}\Ω_{i−1}} [ \frac{k}{(R − r)} |u_{h} (x) − u (x)| + M_{1}]^{\frac{p}{p−(n−1)}} dx\}^{\frac{p−(n−1)}{p}}

The sequences {u_{h}}_{h∈ℕ} and {w_{h}}_{h∈ℕ} converge to u in the weak topology of W^{1,p} (Ω; ℝ^{n}) and the L^{p}−norm of their gradients remains bounded. Up to a subsequence, as h → +∞, the difference {u_{h} (x) − u (x)}_{h∈ℕ} converges almost everywhere to zero. By taking into account the uniform bound (92), we can go to the limit as h → +∞ and we obtain

lim sup_{h→+∞} ∑_{i=1}^{k} ∫_{Ω_{i}\Ω_{i−1}} |det Dw_{h,i} (x)| dx

≤ lim sup_{h→+∞} ∑_{i=1}^{k} ∫_{Ω_{i}\Ω_{i−1}} |det φ_{i} (x) Du_{h} (x)| dx + c_{1} · M_{1} |Ω_{i}\Ω_{i−1}|^{\frac{p−(n−1)}{p}}

≤ lim sup_{h→+∞} ∑_{i=1}^{k} ∫_{Ω_{i}\Ω_{i−1}} |φ_{i} (x)|^{n} · |det Du_{h} (x)| dx + c_{1}M_{1} |Ω_{i}\Ω_{i−1}|^{\frac{p−(n−1)}{p}}  (96)

≤ lim sup_{h→+∞} ∫_{Ω\Ω_{0}} |det Du_{h} (x)| dx + c_{1}kM_{1} |Ω\Ω_{0}|^{\frac{p−(n−1)}{p}} = T V (u, Ω) + δ + c_{1}kM_{1} |Ω\Ω_{0}|^{\frac{p−(n−1)}{p}} .

From (93) and (96) we deduce that

|∫_{Ω} det Dũ (x) dx| ≤ T V (u, Ω) + δ

+ \frac{1}{k} \{T V (u, Ω) + δ + c_{1}kM_{1} |Ω\Ω_{0}|^{\frac{p−(n−1)}{p}} \} + ∫_{Ω\Ω_{0}} |det Du (x)| dx .

Letting k → +∞, Ω_{0} → Ω and δ → 0^{+}, we conclude

|∫_{Ω} det Dũ (x) dx| ≤ T V (u, Ω) .

Lemma 37 (Lower bound - second estimate) Let u be a function of class L^{∞} (Ω; ℝ^{n})∩W^{1,p} (Ω; ℝ^{n})∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for some p ∈ (n − 1, n). For every r > 0 such that B_{r} ⊂ Ω the following estimate holds

T V (u, Ω) ≥ ∫_{Ω\B_{r}} |det Du (x)| dx + |∫_{B_{r}} det Dũ (x) dx| ,  (97)

where ũ : B_{r} → ℝ^{n} is any Lipschitz-continuous map which coincides with u on the boundary of B_{r}, i.e., ũ (x) = u (x) on ∂B_{r}.

Proof. Fix δ > 0 and consider a sequence {u_{h}}_{h∈ℕ} in W^{1,n} (Ω; ℝ^{n}) which converges to u in the weak topology of W^{1,p} (Ω; ℝ^{n}) for some p ∈ (n − 1, n) and such that (91) holds. For every r > 0 such that B_{r} ⊂ Ω we have

T V (u, Ω) + δ ≥ lim_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx ≥ lim inf_{h→+∞} ∫_{Ω\B_{r}} |det Du_{h} (x)| dx

+ lim inf_{h→+∞} ∫_{B_{r}} |det Du_{h} (x)| dx ≥ lim inf_{h→+∞} ∫_{Ω\B_{r}} |det Du_{h} (x)| dx + T V (u, B_{r}) .

We estimate the term T V (u, B_{r}) with (90). Moreover, since u, u_{h} belong to W^{1,n} (Ω\B_{r}; ℝ^{n}) for every h ∈ N (and u_{h} converge to u in the weak topology of W^{1,p} (Ω\B_{r}; ℝ^{n}) for p > n−1), we can apply the lower semicontinuity result below the critical exponent stated in Theorem 35. We reach the conclusion (97) as δ → 0^{+}.

Lemma 38 Let u ∈ W^{1,p} (B_{1}; ℝ^{n}) ∩ W_{loc}^{1,∞} (B_{1}\ {0}; ℝ^{n}) for some p ∈ [1, n). If

\frac{1}{ϱ^{n−p}} ∫_{B_{ϱ}} |D_{τ} u|^{p} dx ≤ M_{0}

for every ϱ ∈ (0, 1) and for some positive constant M_{0}, then there exists a constant c (n, p) and a sequence ϱ_{j} → 0 such that

\frac{1}{ϱ_{j}^{n−p−1}} ∫_{∂B_{ϱ_{j}}} |D_{τ} u|^{p} dH^{n−1} ≤ c (n, p) M_{0} .

Proof. For every j ≥ 2 we have

∫_{1/(2j)}^{1/j} dϱ ∫_{∂B_{ϱ}} |D_{τ} u|^{p} dH^{n−1} ≤ ∫_{B_{1/j}} |D_{τ} u|^{p} dx ≤ \frac{M_{0}}{j^{n−p}} .  (98)

Therefore there exist ϱ_{j} ∈ (\frac{1}{2j} , \frac{1}{j}) such that

∫_{∂B_{ϱ_{j}}} |D_{τ} u|^{p} dH^{n−1} ≤ \frac{3M_{0}}{j^{n−p−1}} ;  (99)

in fact, if (99) does not hold, then for every ϱ ∈ (\frac{1}{2j} , \frac{1}{j}) we should have

∫_{∂B_{ϱ}} |D_{τ} u|^{p} dH^{n−1} ≥ \frac{3M_{0}}{j^{n−p−1}}

and thus

∫_{1/(2j)}^{1/j} dϱ ∫_{∂B_{ϱ}} |D_{τ} u|^{p} dH^{n−1} ≥ \frac{3M_{0}}{j^{n−p−1}} · \frac{1}{2j} > \frac{M_{0}}{j^{n−p}} ,

which is in contradiction with (98). Since \frac{1}{2j} < ϱ_{j} < \frac{1}{j}, we deduce that ϱ_{j} → 0, and that (\frac{1}{j})^{n−p−1} ≤

ϱ_{j}^{n−p−1} if p ≥ n − 1, while (\frac{1}{j})^{n−p−1} ≤ (2ϱ_{j})^{n−p−1} if p < n − 1. From (99) we finally have

∫_{∂B_{ϱ_{j}}} |D_{τ} u|^{p} dH^{n−1} ≤ \frac{3M_{0}}{j^{n−p−1}} ≤ c (n, p) M_{0} ϱ_{j}^{n−p−1} ,

where c (n, p) = 3 if p ∈ [n − 1, n), c (n, p) = 3 · 2^{n−p−1} if p ∈ [1, n − 1).

We denote a generic element of the surface of the unit ball ∂B_{1} = S^{n−1} by ω. Let ω_{0} ∈ S^{n−1} be fixed. For every j ∈ {1, 2, …, n − 1} let τ_{j} : S^{n−1} − {ω_{0}} → S^{n−1} by a vector field of class C^{1} such that, for every x ∈ S^{n−1} − {ω_{0}}, the set of vectors {τ_{1} (ω) , τ_{2} (ω) , …, τ_{n−1} (ω)} is an orthonormal basis for the tangent plane to the surface S^{n−1} at the point ω. Without loss of generality (up to a change of sign to one of the vectors) we can assume that τ_{1} (ω) , τ_{2} (ω) , …, τ_{n−1} (ω) have the property that, if we denote by ν (ω) the exterior normal unit vector to S^{n−1} at ω, then the system of vectors {ν (ω) , τ_{1} (ω) , …, τ_{n−1} (ω)} is a positively oriented basis of ℝ^{n}. I.e.,

ν (ω) ∧ τ_{1} (ω) ∧ … ∧ τ_{n−1} (ω) = e_{1} ∧ e_{2} ∧ … ∧ e_{n}

or, equivalently, that the determinant of the matrix whose column vectors are the components of ν (ω) , τ_{1} (ω) , …, τ_{n−1} (ω) with respect to e_{1}, e_{2}, …, e_{n}, is equal to 1.

If v : S^{n−1} → ℝ^{n}, v ∈ W^{1,∞} (S^{n−1}; ℝ^{n}), v = (v^{1}, v^{2}, …, v^{n}), is a Lipschitz-continuous map, we denote by D_{τ} v the vector of ℝ^{n−1} whose components are D_{τ_{1}}v, D_{τ_{2}}v, …, D_{τ_{n−1}}v.

Lemma 39 Let v ∈ W^{1,∞} (S^{n−1}; ℝ^{n}), η ∈ C^{1} ([0, 1]), with η (0) = 0 and let w (x) = η (|x|) v (\frac{x}{|x|}). For almost every x ∈ B_{1} we have

|Dw (x)|^{2} = |η′ (|x|) v (\frac{x}{|x|})|^{2} + \frac{η^{2} (|x|)}{|x|^{2}} |D_{τ} v (\frac{x}{|x|})|^{2} ;  (100)

det Dw (x) = \frac{η′ (|x|) η^{n−1} (|x|)}{|x|^{n−1}} ∑_{i=1}^{n} (−1)^{i−1} v^{i} (\frac{x}{|x|}) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (\frac{x}{|x|}) . (101)

Moreover, if η (t) = t for every t ∈ [0, 1], then

∫_{B_{1}} det Dw (x) dx = \frac{1}{n} ∫_{∂B_{1}} ∑_{i=1}^{n} (−1)^{i−1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) dH^{n−1} .  (102)

Proof. Since v : S^{n−1} → ℝ^{n} is a Lipschitz-continuous map, then D_{τ} v (ω) exists (in the classical sense) H^{n−1} almost everywhere on S^{n−1} and the map x → v (x/ |x|) is classically differentiable for almost every x ∈ B_{1}. Let x ≠ 0 be a point of B_{1} where v (x/ |x|) is differentiable; since the vectors

ν = ν (\frac{x}{|x|}) , τ_{1} = τ_{1} (\frac{x}{|x|}) , …, τ_{n−1} = τ_{n−1} (\frac{x}{|x|}) ,

form a basis of ℝ^{n}, for every i = 1, 2, …, n we have

Dw^{i} (x) = \frac{∂w^{i} (x)}{∂ν}ν + ∑_{j=1}^{n−1} \frac{∂w^{i} (x)}{∂τ_{j}}τ_{j} = η′ (|x|) v^{i} (\frac{x}{|x|}) ν + \frac{η (|x|)}{|x|} ∑_{j=1}^{n−1} \frac{∂v^{i} (x/ |x|)}{∂τ_{j}}τ_{j} ,

and thus we obtain (100). Moreover, Dw (x) is equal to the matrix {Dw^{1} (x) , Dw^{2} (x) , …, Dw^{n} (x)}. If we express each column of Dw (x) as linear combination of the elements of the basis {ν, τ_{1}, …, τ_{n−1}}, since w (x) = η (|x|) v (\frac{x}{|x|}), we obtain the matrix

Dw (x) = ( \begin{matrix}η′ (|x|) v^{1} (\frac{x}{|x|}) & η′ (|x|) v^{2} (\frac{x}{|x|}) & … & η′ (|x|) v^{n} (\frac{x}{|x|}) \\ \frac{η(|x|)}{|x|} \frac{∂v^{1}}{∂τ_{1}} (\frac{x}{|x|}) & \frac{η(|x|)}{|x|} \frac{∂v^{2}}{∂τ_{1}} (\frac{x}{|x|}) & … & \frac{η(|x|)}{|x|} \frac{∂v^{n}}{∂τ_{1}} (\frac{x}{|x|}) \\ … & … & … & … \\ \frac{η(|x|)}{|x|} \frac{∂v^{1}}{∂τ_{n−1}} (\frac{x}{|x|}) & \frac{η(|x|)}{|x|} \frac{∂v^{2}}{∂τ_{n−1}} (\frac{x}{|x|}) & … & \frac{η(|x|)}{|x|} \frac{∂v^{n}}{∂τ_{n−1}} (\frac{x}{|x|}) \end{matrix} ) .

Thus the determinant of the matrix Dw (x), computed by developing the first row, is given by (101). By integrating over B_{1} both sides of (101), with η (t) = t for every t ∈ [0, 1], since \frac{η′(|x|)η^{n−1}(|x|)}{|x|^{n−1}} = 1, we obtain

∫_{B_{1}} det Dw (x) dx = ∫_{B_{1}} ∑_{i=1}^{n} (−1)^{i−1} v^{i} (\frac{x}{|x|}) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (\frac{x}{|x|}) dx

= ∫_{0}^{1} dϱ ∫_{∂B_{ϱ}} ∑_{i=1}^{n} (−1)^{i−1} v^{i} (\frac{x}{|x|}) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (\frac{x}{|x|}) dH^{n−1}

= ∫_{0}^{1} ϱ^{n−1}dϱ ∫_{∂B_{1}} ∑_{i=1}^{n} (−1)^{i−1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) dH^{n−1}

= \frac{1}{n} ∫_{∂B_{1}} ∑_{i=1}^{n} (−1)^{i−1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) dH^{n−1} .

Lemma 40 Let Ω be an open set containing the origin. Let us assume that, for some p ∈ (n − 1, n), u ∈ W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) satisfies

sup_{ϱ>0} \frac{1}{ϱ^{n−p}} ∫_{B_{ϱ}} |D_{τ} u|^{p} dx ≤ M_{0}

for a positive constant M_{0}. Let v ∈ W^{1,∞} (S^{n−1}; ℝ^{n}) be such that

lim_{ϱ→0^{+}} max {|u (ϱω) − v (ω)| : ω ∈ ∂B_{1}} = 0 .

Then there exists a sequence ϱ_{j} → 0 such that

lim_{j→+∞} ∫_{B_{1}} det Dw_{j} (x) dx = ∫_{B_{1}} det Dw (x) dx ,  (103)

where w_{j} (x) := |x| u (ϱ_{j} \frac{x}{|x|}) and w (x) := |x| v (\frac{x}{|x|}).

Proof. Let ϱ_{j} be the real sequence converging to zero of Lemma 38. By assumption w_{j} (x) := |x| u (ϱ_{j} \frac{x}{|x|}) converges to w (x) := |x| v (\frac{x}{|x|}) uniformly in B_{1}. Let us prove that w_{j} weakly converge in W^{1,p} (B_{1}; ℝ^{n}) to w. In fact, by (100) of Lemma 39 we have

|Dw_{j} (x)|^{2} = |u (ϱ_{j} \frac{x}{|x|})|^{2} + ϱ_{j}^{2} |D_{τ} u (ϱ_{j} \frac{x}{|x|})|^{2}

and thus the L^{p} norm of Dw_{j} remains bounded. In fact, by Lemma 38,

∫_{B_{1}} |Dw_{j} (x)|^{p} dx ≤ c_{1} + c_{2}ϱ_{j}^{p} ∫_{B_{1}} |D_{τ} u (ϱ_{j} \frac{x}{|x|})|^{p} dx = c_{1} + c_{2}ϱ_{j}^{p} ∫_{0}^{1} dr ∫_{∂B_{r}} |D_{τ} u (ϱ_{j} \frac{x}{|x|})|^{p} dH^{n−1}

= c_{1} + c_{2}ϱ_{j}^{p} ∫_{0}^{1} \frac{r^{n−1}}{ϱ_{j}^{n−1}} dr ∫_{∂B_{ϱ_{j}}} |D_{τ} u (y)|^{p} dH_{y}^{n−1}

= c_{1} + \frac{c_{2}}{nϱ_{j}^{n−p−1}} ∫_{∂B_{ϱ_{j}}} |D_{τ} u (y)|^{p} dH_{y}^{n−1} = c_{1} + \frac{c_{2}}{n} c (n, p) M_{0} .

By (101) we also have, with α = p/ (n − 1),

∫_{B_{1}} |det Dw_{j} (x)|^{α} dx ≤ c_{3}ϱ_{j}^{α(n−1)} ∫_{B_{1}} |u (ϱ_{j} \frac{x}{|x|})|^{α} |D_{τ} u (ϱ_{j} \frac{x}{|x|})|^{α(n−1)} dx  (104)

≤ c_{4}ϱ_{j}^{α(n−1)}ϱ_{j}^{1−n} ∫_{∂B_{ϱ_{j}}} |D_{τ} u (y)|^{p} dH_{y}^{n−1} = c_{4} \frac{1}{ϱ_{j}^{n−1−p}} ∫_{∂B_{ϱ_{j}}} |D_{τ} u (y)|^{p} dH_{y}^{n−1},

which is bounded, again by Lemma 38. Therefore, since α > 1, to obtain the conclusion (103) it is sufficient to prove that

lim_{j→+∞} ∫_{B_{1}} φ det Dw_{j} (x) dx = ∫_{B_{1}} φ det Dw (x) dx ,  ∀ φ ∈ C_{0}^{1} (B_{1}) .  (105)

Since p > n−1, we apply Reshetnyak’s [56] weak continuity result on the matrix adj_{n−1}Dw_{j} of minors (n − 1) × (n − 1) of Dw_{j}, which weakly converge in L^{\frac{p}{n−1}} to the corresponding matrix adj_{n−1}Dw of minors of Dw (see 44). By the uniform convergence of w_{j} to w, for every φ ∈ C_{0}^{1} (B_{1}) we get the conclusion

lim_{j→+∞} ∫_{B_{1}} φ det Dw_{j} dx = lim_{j→+∞} − ∫_{B_{1}} w_{j}^{1} \frac{∂ (φ, w_{j}^{2}, …, w_{j}^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx

= − ∫_{B_{1}} w^{1} \frac{∂ (φ, w^{2}, …, w^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx = ∫_{B_{1}} φ det Dw dx .

Proof of Theorem 7. Let u ∈ L_{loc}^{∞} (Ω; ℝ^{2}) ∩ W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for some p ∈ (n − 1, n). Let {ε_{h}}_{h∈ℕ} be a sequence converging to zero and consider the convolution u_{h} := u∗η_{ε_{h}} of u with a smooth mollifier η_{ε_{h}}. For every h ∈ ℕ, u_{h} ∈ C^{1} (Ω_{h}; ℝ^{n}), where Ω_{h} = {x ∈ Ω : dist (x, ∂Ω) > ε_{h}}. Moreover, for every Ω′ ⊂⊂ Ω\ {0}, u_{h} → u uniformly in Ω′, Du_{h} (x) → Du (x) for every x ∈ Ω\E, where E is a Borel set of zero measure, and the sequence {u_{h}}_{h∈ℕ} is Lipschitz-continuos in Ω′, with a Lipschitz constant independent of h. Denote by N_{0} the set of real numbers given by

N_{0} := {ϱ > 0 : H^{n−1} (∂B_{ϱ} ∩ E) > 0} .

If B_{r} ⊂⊂ Ω then we have

0 = |E ∩ B_{r}| = ∫_{0}^{r} H^{n−1} (∂B_{ϱ} ∩ E) dϱ ,

and thus the one-dimensional Lebesgue measure of N_{0} is equal to zero. We can repeat the proof of Lemma 40 to reach the same conclusion for a sequence {ϱ_{j}}_{j∈ℕ} ⊂ (0, r), {ϱ_{j}}_{j∈ℕ} ∩ N_{0} = ∅, ϱ_{j} → 0. Since u_{h} → u uniformly on B_{ϱ_{j}}, D_{τ} u_{h} (x) → D_{τ} u (x) H^{n−1}−almost everywhere on B_{ϱ_{j}}, and the sequence {u_{h}}_{h∈ℕ} is Lipschitz-continuos on B_{ϱ_{j}} with a Lipschitz constant independent of h, then D_{τ} u_{h} → D_{τ} u in L^{q} (∂B_{ϱ_{j}}) for every q ≥ 1. Fixed φ ∈ C_{0}^{1} (Ω) and denoting by ν = ν (x) = (ν^{1}, ν^{2}, …, ν^{n}) the exterior normal unit vector to ∂B_{ϱ_{j}}, we have

∫_{Ω\B_{ϱ_{j}}} u^{1} \frac{∂ (φ, u^{2}, …, u^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx = lim_{h→+∞} ∫_{Ω\B_{ϱ_{j}}} u_{h}^{1} \frac{∂ (φ, u_{h}^{2}, …, u_{h}^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx

= − lim_{h→+∞} \{∫_{Ω\B_{ϱ_{j}}} φ \frac{∂ (u_{h}^{1}, u_{h}^{2}, …, u_{h}^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx

+ ∫_{∂B_{ϱ_{j}}} u_{h}^{1} ∑_{i=1}^{n} (−1)^{i−1} φ\frac{∂ (u_{h}^{2}, u_{h}^{3}, …, u_{h}^{n}) ν^{i}}{∂ (x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n})} dH^{n−1}\}

= − ∫_{Ω\B_{ϱ_{j}}} φ det Du dx − ∫_{∂B_{ϱ_{j}}} u^{1} ∑_{i=1}^{n} (−1)^{i−1} φ\frac{∂ (u^{2}, u^{3}, …, u^{n}) ν^{i}}{∂ (x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n})} dH^{n−1} .

By the analytic expression (114) of ν, together with (iii) of Lemma 41, with the notation w_{j} (x) := |x| u (ϱ_{j} \frac{x}{|x|}), we obtain

∫_{Ω\B_{ϱ_{j}}} u^{1} \frac{∂ (φ, u^{2}, …, u^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx = − ∫_{Ω\B_{ϱ_{j}}} φ det Du dx − ∫_{∂B_{ϱ_{j}}} u^{1}φ\frac{∂ (u^{2}, u^{3}, …, u^{n})}{∂ (τ_{1}, τ_{2}, …, , τ_{n−1})} dH^{n−1}

= − ∫_{Ω\B_{ϱ_{j}}} φ det Du dx − ∫_{∂B_{1}} w_{j}^{1}φ (ϱ_{j}ω) \frac{∂ (w_{j}^{2}, w_{j}^{3}, …, w_{j}^{n})}{∂ (τ_{1}, τ_{2}, …, , τ_{n−1})} dH^{n−1}  (106)

= − ∫_{Ω\B_{ϱ_{j}}} φ det Du dx − ∫_{∂B_{1}} w_{j}^{1}φ (ϱ_{j}ω) ∑_{i=1}^{n} (−1)^{i−1} \frac{∂ (w_{j}^{2}, w_{j}^{3}, …, w_{j}^{n}) ν^{i}}{∂ (x_{1}, …, x_{i−1}, x_{i+1}, …, x_{n})} dH^{n−1}

= − ∫_{Ω\B_{ϱ_{j}}} φ det Du dx − ∫_{B_{1}} \frac{∂ (w_{j}^{1}φ (ϱ_{j} \frac{x}{|x|}) , w_{j}^{2}, …, w_{j}^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx

= − ∫_{Ω\B_{ϱ_{j}}} φ det Du dx − ∫_{B_{1}} φ (ϱ_{j} \frac{x}{|x|}) det Dw_{j} dx − ∫_{B_{1}} w_{j}^{1} \frac{∂ (φ (ϱ_{j} \frac{x}{|x|}) , w_{j}^{2}, …, w_{j}^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx .

As j → +∞ the quantity φ (ϱ_{j} \frac{x}{|x|}) converges to φ (0) uniformly in B_{1}. Then, by the bound (104) and by (105), we obtain

lim_{j→+∞} ∫_{B_{1}} φ (ϱ_{j} \frac{x}{|x|}) det Dw_{j} (x) dx = ∫_{B_{1}} φ (0) det Dw (x) dx .

Moreover, as in the proof of Lemma 40, the sequence {|Dw_{j}|}_{j∈ℕ} is bounded in L^{p} (B_{1}) and

|∫_{B_{1}} w_{j}^{1} \frac{∂ (φ (ϱ_{j} \frac{x}{|x|}) , w_{j}^{2}, …, w_{j}^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx| ≤ c_{1}ϱ_{j} ∫_{B_{1}} \frac{|w_{j}| |Dw_{j}|^{n−1}}{|x|} dx ≤ c_{2}ϱ_{j} ,

which converges to zero as j → +∞. Therefore, since det Du ∈ L^{1} (Ω), letting j → +∞ in (106) we obtain

∫_{Ω} u^{1} \frac{∂ (φ, u^{2}, …, u^{n})}{∂ (x_{1}, x_{2}, …, x_{n})} dx = − ∫_{Ω} φ det Du dx − φ (0) ∫_{B_{1}} det Dw dx ,

with w (x) := |x| v (\frac{x}{|x|}); i.e.,

Det Du = det Du + m_{0}δ_{0} , where m_{0} = ∫_{B_{1}} det Dw dx .

Then, the total variation |Det Du| (Ω) of Det Du is equal to

|Det Du| (Ω) = ∫_{Ω} |det Du| dx + |m_{0}| ,

which agrees with the conclusion (20).

Proof of Theorem 10. Step 1 (lower bound). We first notice that, by virtue of (17), there exists r > 0 such that u ∈ L^{∞} (B_{r}; ℝ^{n}). Let p ∈ (n − 1, n). Let ϱ_{j} → 0 be the sequence of the Lemma 38, and consider j ∈ ℕ sufficiently large so that B_{ϱ_{j}} ⊂ B_{r} ⊂ Ω. By the estimate (97) of Lemma 37 we have

T V (u, Ω) ≥ ∫_{Ω\B_{ϱ_{j}}} |det Du (x)| dx + |∫_{B_{ϱ_{j}}} det Dũ (x) dx| ,  (107)

where ũ : B_{ϱ_{j}} → ℝ^{n} is any Lipschitz-continuous map which assumes the boundary value ũ (x) = u (x)

on ∂B_{ϱ_{j}}. In particular, we consider the extension ũ = w̃_{j} given by w̃_{j} (x) := \frac{|x|}{ϱ_{j}} u (ϱ_{j} \frac{x}{|x|}), and, using a change of variables, we have

∫_{B_{ϱ_{j}}} det Dw̃_{j} (x) dx = ∫_{B_{1}} det Dw_{j} (x) dx ,

where w_{j} (x) := |x| u (ϱ_{j} \frac{x}{|x|}). Letting j → +∞ in (107), by Lemma 40 we get

T V (u, Ω) ≥ lim inf_{j→+∞} ∫_{Ω\B_{ϱ_{j}}} |det Du (x)| dx + lim_{j→+∞} |∫_{B_{ϱ_{j}}} det Dw̃_{j} (x) dx|

= ∫_{Ω} |det Du (x)| dx + lim_{j→+∞} |∫_{B_{1}} det Dw_{j} (x) dx| = ∫_{Ω} |det Du (x)| dx + |∫_{B_{1}} det Dw (x) dx| ,

where w (x) := |x| v (\frac{x}{|x|}). We represent det Dw (x) using (102) of Lemma 39, and we obtain the lower bound

T V (u, Ω) ≥ ∫_{Ω} |det Du (x)| dx + |∫_{B_{1}} det Dw (x) dx| = ∫_{Ω} |det Du (x)| dx

+\frac{1}{n} |∫_{∂B_{1}} ∑_{i=1}^{n} (−1)^{i−1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) dH^{n−1}| .

Step 2 (upper bound in the radially symmetric case). Here we assume that u (x) := v (x/ |x|). Let ϱ_{h} be a sequence of positive numbers converging to zero as h → +∞ and let h ∈ ℕ be sufficiently large so that B_{ϱ_{h}} ⊂ Ω. As before, we use the notation w (x) := |x| v (x/ |x|), and we define

u_{h} (x) := \{ \begin{matrix}\frac{|x|}{ϱ_{h}} v (\frac{x}{|x|}) = \frac{1}{ϱ_{h}} w (x) = w ( \frac{x}{ϱ_{h}} ) , if x ∈ B_{ϱ_{h}}, \\ u (x) = v (\frac{x}{|x|}) , if x ∈ Ω\B_{ϱ_{h}} \end{matrix} .

Then {u_{h}}_{h∈ℕ} converges to u in in the strong norm topology of W^{1,p} (Ω; ℝ^{n}). Therefore we can use the definition (26) of T V^{s} (u, Ω) and, since det Du (x) = 0 in Ω\B_{ϱ_{h}} we have

T V^{s} (u, Ω) ≤ lim inf_{h→+∞} ∫_{B_{ϱ_{h}}} | \frac{1}{ϱ_{h}^{n}} det Dw ( \frac{x}{ϱ_{h}} )| dx = ∫_{B_{1}} |det Dw (x)| dx = |∫_{B_{1}} det Dw (x) dx| ,

(108)

where the last equality follows from the fact that, by assumption, det Dw (x) has constant sign in B_{1}. In fact, by (101) of Lemma 39, with η (|x|) = |x|, we have

det Dw (x) = ∑_{i=1}^{n} (−1)^{i−1} v^{i} (\frac{x}{|x|}) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (\frac{x}{|x|}) ,

and thus, by the sign condition (22), the left hand side has constant sign as well as the right hand side. Therefore, from Step 1 and from (108), when u (x) := v (x/ |x|) we get

T V (u, Ω) = T V^{s} (u, Ω) = T V (v, B_{1})

= \frac{1}{n} |∫_{∂B_{1}} ∑_{i=1}^{n} (−1)^{i+1} v^{i} (ω) \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})} (ω) dH^{n−1}| .

We explicitly observe that, as a consequence of what we have shown in Steps 1 and 2, we have achieved the proof of Theorem 12 in the radially symmetric case; moreover, the representation formula for T V (v, Ω) is independent of the open set Ω containing the origin.

Step 3 (upper bound in the general case). By Lemma 38 there exists a sequence (ϱ_{h})_{h∈ℕ}, converging to zero as h → +∞, and such that

\frac{1}{ϱ_{h}^{n−p−1}} ∫_{∂B_{ϱ_{h}}} |D_{τ} u|^{p} dH^{n−1} ≤ c (n, p) M_{0} .  (109)

For every h ∈ ℕ, we denote by σ_{h} a real sequence in (0, 1) to be chosen later (see (112)). For every h = 1, 2, …, let η_{h} (ϱ) be a cut-off function such that η_{h} (ϱ) = 1 if 0 ≤ ϱ ≤ ϱ_{h} (1 − σ_{h}), η_{h} (ϱ) = 0 if ϱ_{h} ≤ ϱ ≤ 1, η_{h} (ϱ) is linear in the interval [ϱ_{h} (1 − σ_{h}) , ϱ_{h}]. Fix ε > 0. From Step 2 there exists a Lipschitz-continuous map w : B_{1} → ℝ^{n} such that w (x) := v (\frac{x}{|x|}) on a neighborhood of ∂B_{1} and

∫_{B_{1}} |det Dw (x)| dx < T V (v, B_{1}) + ε .  (110)

Then, with the notation ω := x/ |x|, we define

u_{h} (x) := \{ \begin{matrix}w (\frac{x}{ϱ_{h}(1−σ_{h})}) ,  if 0 ≤ |x| ≤ ϱ_{h} (1 − σ_{h}) \\ η_{h} (|x|) v (ω) + [1 − η_{h} (|x|)] u (ϱ_{h}ω) , if ϱ_{h} (1 − σ_{h}) < |x| < ϱ_{h} \\ u (x) ,  if x ∈ Ω\B_{ϱ_{h}} \end{matrix} .  (111)

We first prove that {u_{h}}_{h∈ℕ} converges to u in the strong topology of W^{1,p} (Ω; ℝ^{n}). In fact

∫_{Ω} |u_{h} − u|^{p} dx = ∫_{B_{ϱ_{h}}} |u_{h} − u|^{p} dx ≤ c ∫_{B_{ϱ_{h}(1−σ_{h})}} |w (\frac{x}{ϱ_{h} (1 − σ_{h})})|^{p} dx

+c ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} \{|v (\frac{x}{|x|})|^{p} + |u (ϱ_{h} \frac{x}{|x|})|^{p}\} dx + c ∫_{B_{ϱ_{h}}} |u (x)|^{p} dx

≤ cϱ_{h}^{n} {∥w∥_{L^{∞}(B_{1})}^{p} + ∥v∥_{L^{∞}(∂B_{1})}^{p} + ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})}^{p}} + c ∫_{B_{ϱ_{h}}} |u (x)|^{p} dx,

which goes to zero as h → +∞, since ϱ_{h} → 0 and ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})}^{p} → 0. Moreover, by (100) of Lemma 39, we have

∫_{Ω} |Du_{h} − Du|^{p} dx ≤ c_{1} ∫_{B_{ϱ_{h}(1−σ_{h})}} |Dw (\frac{x}{ϱ_{h} (1 − σ_{h})})|^{p} dx + c_{1} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} \begin{matrix}|D_{τ} v| & p \\ |x| & p \end{matrix} dx

+c_{1}∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |D_{τ} u (ϱ_{h} \frac{x}{|x|})|^{p} dx+ \frac{c_{1}}{ϱ_{h}^{p}σ_{h}^{p}} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |u(ϱ_{h} \frac{x}{|x|})−v(\frac{x}{|x|})|^{p} dx+c_{1}∫_{B_{ϱ_{h}}} |Du|^{p} dx

≤ c_{2}ϱ_{h}^{n−p} (1 − σ_{h})^{n−p} ∫_{B_{1}} |Dw (x)|^{p} dx + c_{2} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} \frac{1}{|x|^{p}} dx

+c_{2} ∫_{ϱ_{h}(1−σ_{h})}^{ϱ_{h}} dr ∫_{∂B_{r}} |D_{τ} u (ϱ_{h} \frac{x}{|x|})|^{p} dH^{n−1} + c_{2} \frac{ϱ_{h}^{n−p}}{σ_{h}^{p−1}} ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})}^{p} + c_{1} ∫_{B_{ϱ_{h}}} |Du|^{p} dx

≤ c_{3}ϱ_{h}^{n−p} + \frac{c_{3}}{ϱ_{h}^{n−1}} ϱ_{h}^{n}σ_{h} ∫_{∂B_{ϱ_{h}}} |D_{τ} u (y)|^{p} dH_{y}^{n−1} + c_{3} \frac{ϱ_{h}^{n−p}}{σ_{h}^{p−1}} ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})}^{p} + c_{1} ∫_{B_{ϱ_{h}}} |Du|^{p} dx .

By the bound (109) we obtain

∫_{Ω} |Du_{h} − Du|^{p} dx ≤ c (w, v, M_{0}) ϱ_{h}^{n−p} + c \frac{ϱ_{h}^{n−p}}{σ_{h}^{p−1}} ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})}^{p} + c_{1} ∫_{B_{ϱ_{h}}} |Du|^{p} dx

and this quantity goes to zero as h → +∞ if we assume that

σ_{h} := ϱh^{\frac{n−p}{p−1}}  (112)

(we use here the fact that p < n). Therefore, as h → +∞, u_{h} converges to u in the strong norm topology of W^{1,p} (Ω; ℝ^{n}). Thus, by (110) and by the lower semicontinuity of T V^{s} (u, Ω) with respect to the strong convergence in W^{1,p} (Ω; ℝ^{n}), we have

T V (u, Ω) ≤ T V^{s} (u, Ω) ≤ lim inf_{h→+∞} ∫_{Ω} |det Du_{h} (x)| dx

≤ ∫_{B_{1}} |det Dw (x)| dx + ∫_{Ω} |det Du (x)| dx + lim inf_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx

≤ T V (v, B_{1}) + ε + ∫_{Ω} |det Du (x)| dx + lim inf_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx .

If we prove that

lim_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx = 0 ,  (113)

then, letting ε → 0^{+} we reach the upper bound

T V (u, Ω) ≤ T V^{s} (u, Ω) ≤ T V (v, B_{1}) + ∫_{Ω} |det Du (x)| dx

which, together to the lower bound in Step 1, yields the conclusion

T V (u, Ω) = T V^{s} (u, Ω) = T V (v, B_{1}) + ∫_{Ω} |det Du (x)| dx .

Therefore it remains to prove (113). To this aim, arguing as in the proof of (101), we can evaluate det Du_{h} (x) by taking first the derivative of u_{h} with respect to the radial direction, and then the tangential derivatives. We get

∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx ≤ \frac{c_{1}}{ϱ_{h}σ_{h}} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} \{|u (ϱ_{h} \frac{x}{|x|}) − v (\frac{x}{|x|})| ·

· [|D_{τ} v (\frac{x}{|x|})|^{n−1} + |D_{τ} u (ϱ_{h} \frac{x}{|x|})|^{n−1}]\} dx

≤ \frac{c_{1}}{ϱ_{h}σ_{h}} ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})} · \{c_{2} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} \frac{1}{|x|^{n−1}} dx + \frac{ϱ_{h}^{n}σ_{h}}{ϱ_{h}^{n−1}} ∫_{∂B_{ϱ_{h}}} |D_{τ} u|^{n−1} dH^{n−1}\}

≤ c_{3} ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})} {c_{2} + ∫_{∂B_{ϱ_{h}}} |D_{τ} u|^{n−1} dH^{n−1}} .

Finally, since by (109) we also have

∫_{∂B_{ϱ_{h}}} |D_{τ} u|^{n−1} dH^{n−1} ≤ c_{4} \{\frac{1}{ϱ_{h}^{n−p−1}} ∫_{∂B_{ϱ_{h}}} |D_{τ} u|^{p} dH^{n−1}\}^{\frac{n−1}{p}} ≤ c_{5} ,

then, from the above inequality, we deduce that

∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx ≤ c_{6} ∥u (ϱ_{h}ω) − v (ω)∥_{L^{∞}(∂B_{1})} ,

which converges to zero as h → +∞. Thus (113) is proved.

We conclude this section with some algebraic results used in the paper. We introduce some notations. Denote M^{m×n} by the family of m × n matrices. If A is an n × n matrix (A ∈ M^{n×n}), X_{i,j} (A) is the matrix obtained from A by deleting the i−th row and the j−th column of A. If S is an (n − 1) × n matrix (S ∈ M^{(n−1)×n}), X_{,j} (S) stands for the matrix obtained from S by deleting the j−th column of S. If T is an n × (n − 1) matrix (T ∈ M^{n×(n−1)}), then X_{i,} (T) is the matrix obtained from T by deleting the i−th row of T.

The properties stated in the next two lemmas are known and we do not give their proofs. We refer the reader for instance to the book by Cartan [13].

Lemma 41 (Algebraic lemma) The following properties hold:

(i) Let ξ, η ∈ ℝ^{n} and let B ∈ M^{n×n}. If A_{ij} = ξ_{i}η_{j} and A = (A_{ij}) ∈ M^{n×n}, then

det (A + B) = ∑_{i,j=1}^{n} (−1)^{i+j} ξ_{i}η_{j} det (X_{i,j} (B)) + det (B) .

(ii) Let T ∈ M^{n×(n−1)} be a matrix whose column vectors {τ_{1}, τ_{2}, …, τ_{n−1}} form an orthonormal basis of ℝ^{n}. Then

∑_{i=1}^{n} [det (X_{i,} (T))]^{2} = 1 .

(iii) Let S ∈ M^{(n−1)×n} and T ∈ M^{n×(n−1)}. Then

det (S · T) = ∑_{i=1}^{n} det (X_{,i} (S)) · det (X_{i,} (T)) .

As in Section 2, fixed ω_{0} ∈ ∂B_{1}, for every j ∈ {1, 2, …, n − 1} we consider a vector field τ_{j} : ∂B_{1}\ {ω_{0}} → ℝ^{n} of class C^{1} such that, for every x ∈ ∂B_{1}\ {ω_{0}}, the set of vectors {τ_{1} (x) , τ_{2} (x) , …, τ_{n−1} (x)} is an orthonormal basis for the tangent plane to the surface ∂B_{1} at the point x. For every x ∈ ∂B_{1}\ {ω_{0}} we denote by T (x) the n × (n − 1) matrix whose columns are given by the vectors {τ_{1} (x) , τ_{2} (x) , …, τ_{n−1} (x)}. Consider the vector

ν (x) := ∑_{i=1}^{n} (−1)^{i+1} det (X_{i,} (T (x))) e_{i} .  (114)

Up to a change of sign to one of the vectors τ_{1} (x) , τ_{2} (x) , …, τ_{n−1} (x), we can assume that, at every x ∈ ∂B_{1}\ {ω_{0}}, ν (x) represent the exterior normal unit vector to ∂B_{1}. That ν (x) is a normal unit vector to the surface ∂B_{1} follows from the following result.

Lemma 42 (On the normal unit vector) For every x ∈ ∂B_{1}\ {ω_{0}} the vector ν (x) has norm equal to 1 and it is orthogonal to the vectors τ_{1} (x) , τ_{2} (x) , …, τ_{n−1} (x); i.e.,

\{ \begin{matrix}|ν (x)| = 1,  ∀ x ∈ ∂B_{1}\ \{ω_{0}\}; \\ ⟨ν (x) , τ_{i} (x)⟩ = 0, ∀ x ∈ ∂B_{1}\ \{ω_{0}\} , ∀ i = 1, 2, …, n − 1. \end{matrix}

8 Relaxation in the general polyconvex case

As mentioned in Section 4, the characterization of T V (u, Ω) may be viewed within a broader context, namely as part of a program to search for the description and identification of the defect measure obtained through relaxation of energies when there is a gap between the space of coercivity and the space guaranteeing apriori continuity. Indeed, T V (u, Ω) is a particular case of a functional of the type F_{p,q}(u, Ω) in (41).

Here formally we may consider

\overline{F} (u, Ω) := inf \{lim inf_{h→+∞} ∫_{Ω} g (M (Du_{h} (x))) dx :  (115)
u_{h} ⇀ u weakly in W^{1,p} (Ω; ℝ^{n}) , u_{h} ∈ W^{1,n} (Ω; ℝ^{n})} .

Then \overline{F} (u, Ω) is the relaxed functional of the integral functional

F (u, Ω) := ∫_{Ω} g (M (Du)) dx ,

where u : Ω → ℝ^{n}. The vector-valued map M (Du) of minors of Du is given by

M (Du) := (Du, adj_{2}Du, …, adj_{n−1}Du, det Du) ∈ ℝ^{N},

where, for j = 2, …, n − 1, adj_{j}Du denotes the matrix of all minors j × j of Du and N = ∑_{j=1}^{n} \binom{n}{j}^{2} (in particular N = 5 if n = 2). Finally g : ℝ^{N} → [0, +∞) is a convex function satisfying the growth conditions

g_{∞} |det ξ| ≤ g (M (ξ)) ≤ L (1 + |ξ|^{p}) + g_{∞} |det ξ| ,  (116)

for some constants L ≥ 0, g_{∞} > 0, for all matrices ξ ∈ ℝ^{n×n} and for some exponent p ∈ [1, n). A particularly important case of F (u, Ω) is the area integral

A (u, Ω) := ∫_{Ω} \sqrt{1 + |M (Du)|^{2}} dx ,  (117)

which in the 2 − d setting reduces to

A (u, Ω) = ∫_{Ω} \sqrt{1 + |Du (x)|^{2} + |det Du (x)|^{2}} dx .  (118)

It has been shown in [8] that, if p > n − 1, then \overline{F} (u, ·) is a Radon measure and, for every open set A ⊂ Ω,

\overline{F} (u, A) = g (M (Du)) L^{n}⌊A + μ_{s} (A) ,

where μ_{s} is a finite Radon measure, singular with respect to the Lebesgue measure L^{n}. A longtime question has been to identify the singular measure μ_{s}. In Theorem 43 we achieve this for the class of maps u ∈ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) considered in Section 2. Precisely, using Theorem 1 in 2−d and Theorem 10 for the general n−d case, we can prove the following relaxation result.

Theorem 43 (Relaxation in n−d) Let Ω be an open set of ℝ^{n}, n ≥ 2, containing the origin. Let u ∈ W^{1,p} (Ω; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for some p ∈ (n − 1, n), such that, for a positive constant M_{0},

sup_{ϱ>0} \frac{1}{ϱ^{n−p}} ∫_{B_{ϱ}} |Du|^{p} dx ≤ M_{0} .

Let v : ∂B_{1} = S^{n−1} → ℝ^{n}, v ∈ W^{1,∞} (S^{n−1}; ℝ^{n}), be a Lipschitz-continuous map such that

lim_{ϱ→0^{+}} max \{|u (ϱ\frac{x}{|x|}) − v (\frac{x}{|x|})| : x ∈ B_{1}\ \{0\}\} = 0 .

Moreover, if n = 2 we assume that the map v has values in the set Γ defined in (6); while, if n ≥ 3, then we assume that the quantity

∑_{i=1}^{n} (−1)^{i+1} v^{i} \frac{∂ (v^{1}, …, v^{i−1}, v^{i+1}, …, v^{n})}{∂ (τ_{1}, τ_{2}, …, τ_{n−1})}

has constant sign H^{n−1}−almost everywhere on ∂B_{1}. Then the relaxed functional \overline{F} (u, Ω), defined in (115) with g : ℝ^{N} → [0, +∞) satisfying (116), is given by

\overline{F} (u, Ω) = ∫_{Ω} g (M (Du (x))) dx + g_{∞}T V (v, B_{1}) ,

where the total variation T V (v, B_{1}) of v is given in (24).

Proof. Step 1 (lower bound). Consider a sequence {u_{h}}_{h∈ℕ} of class W^{1,n} (Ω; ℝ^{n}) converging to u in the weak topology of W^{1,p} (Ω; ℝ^{n}), as h → +∞. Let ϱ ∈ (0, 1) be fixed. By Theorem 35, on the lower semicontinuity below the critical exponent, using the bound on the left hand side of (116), we have

lim inf_{h→+∞} F (u_{h}, Ω) ≥ lim inf_{h→+∞} ∫_{Ω\B_{ϱ}} g (M (Du_{h} (x))) dx + lim inf_{h→+∞} g_{∞} ∫_{B_{ϱ}} |det Du_{h} (x)| dx

≥ ∫_{Ω\B_{ϱ}} g (M (Du (x))) dx + g_{∞}T V (v, B_{1}) .

Letting ϱ → 0 we deduce the lower bound

\overline{F} (u, Ω) ≥ ∫_{Ω} g (M (Du (x))) dx + g_{∞}T V (v, B_{1}) .

Step 2 (upper bound). For every ε > 0 there exists a Lipschitz-continuous map w : B_{1} → ℝ^{n} satisfying

∫_{B_{1}} |det Dw (x)| dx < ε + T V (v, B_{1})  (119)

and such that w = v on ∂B_{1}. Indeed, if n = 2 we use (67), while if n ≥ 3 we use (110). By Lemma 38 there exists a sequence (ϱ_{h})_{h∈ℕ}, converging to zero as h → +∞, and such that

\frac{1}{ϱ_{h}^{n−p−1}} ∫_{∂B_{ϱ_{h}}} |D_{τ} u|^{p} dH^{n−1} ≤ c (n, p) M_{0} .

For every h ∈ ℕ we set σ_{h} := ϱh^{\frac{n−p}{p−1}} , and we define u_{h} (x) as in (111). As in Step 3 of the proof of Theorem 10, we can show that

lim_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |Du_{h}|^{p} dx = lim_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} |det Du_{h} (x)| dx = 0  (120)

and, by also using the inequality on the right hand side of (116), we can prove the upper bound

\overline{F} (u, Ω) ≤ lim inf_{h→+∞} F (u_{h}, Ω) ≤ ∫_{Ω} g (M (Du (x))) dx

+ lim inf_{h→+∞} ∫_{B_{ϱ_{h}}\B_{ϱ_{h}(1−σ_{h})}} {L (1 + |Du_{h}|^{p}) + g_{∞} |det Du_{h}|} dx + g_{∞} ∫_{B_{1}} |det Dw (x)| dx .

By (119) and (120), letting ε go to zero, we conclude that

\overline{F} (u, Ω) ≤ ∫_{Ω} g (M (Du (x))) dx + g_{∞}T V (v, B_{1}) .

9 A relevant n−dimensional class of maps

The singular map u : ℝ^{n}\ {0} → ℝ^{n}, defined for x ≠ 0 by

u (x) = \frac{x}{|x|} ,  (121)

belongs to the class W^{1,p} (B_{1}; ℝ^{n}) ∩ W_{loc}^{1,∞} (Ω\ {0}; ℝ^{n}) for every p ∈ [1, n), but u ∉ W^{1,n} (B_{1}; ℝ^{n}). In this case a formula for the total variation T V (u, Ω) was already known. Indeed, (122) below has been first given in 1986 by Marcellini [45] (see also Fonseca and Marcellini [27]). In this section we generalize the formula to more general maps.

To deduce (122) using the tools developed in this work, write u (x) = v (x/ |x|), where the map v : ∂B_{1} → ℝ^{n} is the identity on ∂B_{1} = S^{n−1}. The map ṽ(x) = |x| · v (x/ |x|) = x is the smooth extension of u according with Corollary 13. Clearly Dṽ(x) = Id is the identity matrix and det Dṽ(x) = 1. Therefore, if Ω is any open set of ℝ^{n} containing the origin, Corollary 13 gives

T V (\frac{x}{|x|}, Ω) = |∫_{B_{1}} det Dṽ(x) dx| = ∫_{B_{1}} dx = |B_{1}| = ω_{n} .  (122)

Next we generalize the structure (121) and we consider a class of maps recently studied by Jerrard and Soner [40]. Consider a function w ∈ C^{1} (Ω; ℝ^{n}) (or, more generally, a locally Lipschitz-continuous map w : Ω → ℝ^{n} classically differentiable at x = 0) such that det Dw (0) ≠ 0. Let Ω be an open set containing the origin and define u : Ω\ {0} → ℝ^{n} by

u (x) := \frac{w (x) − w (0)}{|w (x) − w (0)|} .  (123)

Note that the condition det Dw (0) ≠ 0 ensures the existence of r > 0 such that w (x) ≠ w (0) for every x ∈ B_{r}\ {0}, and in the sequel we limit ourselves to open sets Ω ⊂ B_{r} containing the origin.

First we show that, without loss of generality, we may assume that Dw (0) = Id is the identity matrix. Indeed, by assumption, the gradient Dw (0) of w at x = 0 is a nonsingular matrix n × n; let us denote by A := Dw (0) this matrix, and by A^{−1} its inverse matrix. Define on Ω\ {0}

z (x) := u (A^{−1}x) = \frac{w (A^{−1}x) − w (0)}{|w (A^{−1}x) − w (0)|} , ∀ x ∈ Ω\ \{0\} .

Let {u_{h}}_{h∈ℕ} be a sequence in W^{1,n} (Ω; ℝ^{n}) which converges, as h → +∞, to u weakly in W^{1,p} (Ω; ℝ^{n}). Then z_{h} (x) := u_{h} (A^{−1}x) converges weakly in W^{1,p} (Ω; ℝ^{n}) to z (x) = u (A^{−1}x). Since

∫_{A(Ω)} |det Dz_{h} (x)| dx = ∫_{A(Ω)} |det Du_{h}(A^{−1}x)| · |det A^{−1}| dx = ∫_{Ω} |det Du_{h} (x)| dx ,

we deduce that T V (z, A (Ω)) = T V (u, Ω). We also have [Dw (A^{−1}x)]_{x=0} = Dw (0)·A^{−1} = Id, where Id is the identity matrix. Therefore, the above computations show that, without loss of generality, to evaluate the total variation T V (u, Ω) of the Jacobian determinant we may assume that

A = Dw (0) = Id .  (124)

Under (124), with u given in (123), we define v : ∂B_{1} → ℝ^{n} by v (y) := y, for every y ∈ ∂B_{1}. We have

lim_{ϱ→0} max {|u (ϱy) − v (y)| : x ∈ B_{1}\ {0}} = 0 .  (125)

Indeed, since w is differentiable at x = 0, we obtain

u (ϱy) − v (y) = \frac{w (ϱy) − w (0)}{|w (ϱy) − w (0)|} − y = \frac{ϱy + o (ϱ)}{|ϱy + o (ϱ)|} − y = \frac{y + \frac{o(ϱ)}{ϱ}}{|y + \frac{o(ϱ)}{ϱ} |} − y ,

which converges to zero as ϱ → 0. Thus assertion (125) is proved.

Moreover, for every x ∈ B_{ϱ}\ {0} with B_{ϱ} compactly contained in Ω, if we denote by L the Lipschitz constant of w in B_{ϱ}, we have

|Du (x)| ≤ c_{1} \frac{|Dw (x)|}{|w (x) − w (0)|} ≤ c_{1} \frac{L}{|w (x) − w (0)|} ,

for a constant c_{1}. Since A = Dw (0) = Id, then

|w (x) − w (0)| = |Dw (0) · x + o (|x|)| = |x + o (|x|)| ≥ \frac{1}{2} |x|

for every x ∈ B_{ϱ_{0}} with ϱ_{0} sufficiently small; thus

|Du (x)| ≤ c_{1} \frac{L}{|w (x) − w (0)|} ≤ \frac{2c_{1}L}{|x|} .

Also, for every p < n, we have

sup_{0<ϱ≤ϱ_{0}} \frac{1}{ϱ^{n−p}} ∫_{B_{ϱ}} |Du|^{p} dx ≤ sup_{0<ϱ≤ϱ_{0}} \frac{c_{2}}{ϱ^{n−p}} ∫_{B_{ϱ}} \frac{1}{|x|^{p}} dx

≤ sup_{0<ϱ≤ϱ_{0}} \frac{c_{2} · ω_{n}}{ϱ^{n−p}} ∫_{0}^{ϱ} r^{n−1−p} dr = \frac{c_{2} · ω_{n}}{n − p} .

Therefore the assumptions (17), (18) are satisfied, and we can apply Theorem 10, when v : S^{n−1} → S^{n−1} is the identity map. Since |u (x)| = 1 for every x ∈ Ω\ {0}, then det Du (x) = 0 in Ω\ {0}, and hence, by (122) we finally get

T V (u, Ω) = T V (\frac{x}{|x|}, Ω) = ω_{n} , with u (x) := \frac{w (x) − w (0)}{|w (x) − w (0)|} .

10 Some 2− and 3−dimensional examples

We start with a simple application of the general 2−d result of Theorem 1.

Example 44 Let u (x) := v (x/ |x|), where v : [0, 2π] → S^{1} is the map v (ϑ) = (cos g (ϑ) , sin g (ϑ)), with g : [0, 2π] → ℝ Lipschitz-continuous function such that g (2π) = g (0) + 2kπ, for some k ∈ ℤ. Since v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ) = g′ (ϑ) , by Theorem 1 we obtain

T V (u, B_{1}) = \frac{1}{2} |∫_{0}^{2π} g′ (ϑ) dϑ| = |k| π .

Note that here g is not necessarily a monotone function and that T V (u, B_{1}) = \frac{1}{2} |∫_{0}^{2π} g′ (ϑ) dϑ|, with the absolute value sign outside the integral sign, and not inside as could have been expected. On the other hand, if w (x) = |x| u (x) is the radially linear Lipschitz-continuous extension of v, we have instead T V (w, B_{1}) = \frac{1}{2} ∫_{0}^{2π} |g′ (ϑ)| dϑ.

We propose below some examples related to the “eight” curve, i.e., to the union γ of the two circles γ^{+}, γ^{−} of radius 1 with centers at (1, 0) and at (−1, 0) respectively. We consider a Lipschitz-continuous closed curve v : [0, 2π] → γ, with parametric representation v (ϑ) = (v^{1} (ϑ) , v^{2} (ϑ)) and with v (0) = v (2π). As in Section 2, we denote by {I_{j}^{+}}_{j} and by {I_{k}^{−}}_{k} sequences of disjoint open intervals of

[0, 2π] such that v (I_{j}) ⊂ γ^{+} and v (I_{k}) ⊂ γ^{−} (and v (ϑ) = (0, 0) when ϑ ∉ (∪_{j}I_{j}^{+}) ∪ (∪_{k}I_{k}^{−})). With u (x) := v (x/ |x|), we stated in Theorem 4 the following upper and lower estimates

T V (u, B_{1}) ≤ \frac{1}{2} ∑_{j∈ℕ} |∫_{I_{j}} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ | ;  (126)

T V (u, B_{1}) ≥ \frac{1}{2} \{|∑_{j∈ℕ}∫_{I_{j}^{+}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| + |∑_{k∈ℕ}∫_{I_{k}^{−}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ|\} .  (127)

We notice that, if the curve v : [0, 2π] → γ = γ^{+} ∪ γ^{−} admits only two intervals I_{1}^{+} and I_{2}^{−} where v (I_{1}^{+}) ⊂ γ^{+}, v (I_{2}^{−}) ⊂ γ^{−} respectively, then the above estimates for T V (u, B_{1}) are in fact equalities, and

T V (u, B_{1}) = \frac{1}{2} \{|∫_{I_{1}^{+}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ| + |∫_{I_{2}^{−}} \{v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1}\} dϑ|\} .  (128)

Moreover, the total variation of the distributional determinant |Det Du| (B_{1}) is given by

|Det Du| (B_{1}) = \frac{1}{2} |∫_{0}^{2π} \{v^{1} (ϑ) v_{ϑ}^{2} (ϑ) − v^{2} (ϑ) v_{ϑ}^{1} (ϑ)\} dϑ| .  (129)

Using these estimates we can now address the next examples.

Example 45 Let h, k ∈ ℤ, and let v : [0, 2π] → γ be the curve whose image turns |h| times in γ^{−} and |k| times in γ^{+}, according to the parametric representation

v(ϑ) := \{ \begin{matrix}(−1, 0) + (cos 2hϑ, sin 2hϑ) & if 0 ≤ ϑ ≤ π \\ (1, 0) − (cos 2kϑ, sin 2kϑ) & if π ≤ ϑ ≤ 2π \end{matrix} .

Since

v^{1}v_{ϑ}^{2} − v^{2}v_{ϑ}^{1} = \{ \begin{matrix}2h (1 − cos 2hϑ) , if 0 < ϑ < π \\ 2k (cos 2kϑ − 1) , if π < ϑ < 2π \end{matrix} ,

then, with u (x) := v (x/ |x|), by the representation formulas (128), (129) we have

\{ \begin{matrix}T V (u, B_{1}) = (|h| + |k|) π \\ |Det Du| (B_{1}) = |h − k| π \end{matrix} , ∀ h, k ∈ ℤ.  (130)

Example 46 We consider the map

v(ϑ) := \{ \begin{matrix}(−1, 0) + (cos 2ϑ, sin 2ϑ) & if 0 ≤ ϑ ≤ π \\ (1, 0) ± (− cos 2ϑ, sin 2ϑ) & if π ≤ ϑ ≤ 2π \end{matrix} ,  (131)

and we extend it by periodicity from [0, 2π] to ℝ. Then we define v_{h}(ϑ) := v(hϑ), for a given parameter h ∈ ℤ. The image of v_{h} is contained in γ^{+} and γ^{−} in correspondence with two sets of disjoint open intervals of [0, 2π] which, with the notations introduced above, we denote by I_{j}^{+} and I_{k}^{−} respectively. Then v (I_{j}) ⊂ γ^{+} and v (I_{k}) ⊂ γ^{−}. With u_{h} (x) := v_{h} (x/ |x|), by (126) and (127) we obtain

T V (u_{h}, B_{1}) = \frac{1}{2} |∑_{j} ∫_{I_{j}^{+}} \{v_{h}^{1} \frac{∂v_{h}^{2}}{∂ϑ} − v_{h}^{2} \frac{∂v_{h}^{1}}{∂ϑ} \} dϑ| + \frac{1}{2} |∑_{k} ∫_{I_{k}^{−}} \{v_{h}^{1} \frac{∂v_{h}^{2}}{∂ϑ} − v_{h}^{2} \frac{∂v_{h}^{1}}{∂ϑ} \} dϑ|

= \frac{1}{2} |∑_{j} ∫_{I_{j}^{+}} 2h (1 − cos 2hϑ) dϑ| + \frac{1}{2} |∑_{k} ∫_{I_{k}^{−}} 2h (cos 2hϑ − 1) dϑ| = 2 |h| π .

In this situation we have

\{ \begin{matrix}T V (u_{h}, B_{1}) = 2 |h| π \\ |Det Du_{h}| (B_{1}) = 0 \end{matrix} , ∀ h ∈ ℤ.

Example 47 The map v : [0, 2π] → γ defined by

v(ϑ) := \{ \begin{matrix}(−1, 0) + (cos 4ϑ, sin 4ϑ), if 0 ≤ ϑ ≤ π/2 \\ (1, 0) + (− cos 4ϑ, sin 4ϑ), if π/2 ≤ ϑ ≤ π \\ (−1, 0) + (cos 4ϑ, sin 4ϑ), if π ≤ ϑ ≤ 3π/2 \\ (1, 0) + (− cos 4ϑ, sin 4ϑ), if 3π/2 ≤ ϑ ≤ 2π \end{matrix}  (132)

spans γ^{−} twice counter-clockwise, and γ^{+} twice clockwise. It is a particular case of the previous Example 46 and, with the usual notation u (x) := v (x/ |x|), we have T V (u, B_{1}) = 4π and |Det Du| (B_{1}) = 0.

Consider now the map \overline{v} : [0, 2π] → γ defined by

\overline{v}(ϑ) := \{ \begin{matrix}(−1, 0) + (cos 4ϑ, sin 4ϑ),  if 0 ≤ ϑ ≤ π/2 \\ (1, 0) + (− cos 4ϑ, sin 4ϑ),  if π/2 ≤ ϑ ≤ π \\ (−1, 0) + (cos 4ϑ, − sin 4ϑ), if π ≤ ϑ ≤ 3π/2 \\ (1, 0) + (− cos 4ϑ, − sin 4ϑ), if 3π/2 ≤ ϑ ≤ 2π \end{matrix} ,  (133)

which spans γ^{−} twice, the first time counter-clockwise and the second time clockwise; while γ^{+} is spanned first clockwise and then counter-clockwise. Then again, with \overline{u} (x) := \overline{v} (x/ |x|), the estimate (126) yields T V (\overline{u}, B_{1}) ≤ 4π, while (127) gives T V (\overline{u}, B_{1}) ≥ 0 = |Det D\overline{u}| (B_{1}). Therefore, this is an example where there is a gap between the estimates (126) and (127).

The last example related to \overline{v} was already considered by Malý [41] and by Giaquinta, Modica and Souček [35], who proved that the graph of \overline{u} cannot be approximated in area by the graphs of smooth maps.

We finally consider a 3−dimensional example.

Example 48 Let us consider the map v : S^{2} → S^{2} ⊂ ℝ^{3} defined, in spherical coordinates, by

v (ϑ, ψ) := \{ \begin{matrix}v^{1} = cos g (ϑ) sin ψ \\ v^{2} = sin g (ϑ) sin ψ \\ v^{3} = cos ψ \end{matrix} ,

for ϑ ∈ [0, 2π], ψ ∈ [0, π], where g : [0, 2π] → [0, 2π] is a Lipschitz-continuous function such that

g (2π) − g (0) = 2kπ

for some k ∈ ℤ. By formula (101) we can see that, if ω is a generic point of S^{2}, represented in the form ω = (cos ϑ sin ψ, sin ϑ sin ψ, cos ψ), then we have

v^{1} (ω) \frac{∂ (v^{2}, v^{3})}{∂ (τ_{1}, τ_{2})} (ω) − v^{2} (ω) \frac{∂ (v^{1}, v^{3})}{∂ (τ_{1}, τ_{2})} (ω) + v^{3} (ω) \frac{∂ (v^{2}, v^{1})}{∂ (τ_{1}, τ_{2})} (ω) = g′ (ϑ) .

Thus, if the function g is monotone, then the sign assumption (22) is satisfied and, by Theorem 10, we obtain

T V (v, B_{1}) = \frac{2}{3} |g (2π) − g (0)| = \frac{4}{3}π |k| ,  (134)

which, as expected, is equal to the absolute value |k| of the topological degree of the map times the volume ω_{3} = \frac{4}{3}π of the unit ball in ℝ^{3}.

However, formula (134) also holds if the function g is not monotone, i.e., if the sign assumption (22) is not satisfied. To assert this fact (that we do not want to prove in all details), we can follow the argument used in Section 5 to prove Theorem 1. In particular, if for some α, β, with 0 ≤ α < β ≤ 2π, we have g (α) = g (β), then for every ε > 0 we can construct a Lipschitz-continuous map w : S_{α,β} → ℝ^{3} such that w (x) := |x| v (\frac{x}{|x|}) if x ∈ ∂S_{α,β} and

∫_{S_{α,β}} |det Dw (x)| dx < ε ,

where S_{α,β} is the subset of B_{1} of points x = (ϱ cos ϑ sin ψ, ϱ sin ϑ sin ψ, ϱ cos ψ), with 0 ≤ ϱ ≤ 1, α ≤ ϑ ≤ β, 0 ≤ ψ ≤ π. The map w can be defined similarly to the one used in the proof of the “umbrella” Lemma 23, setting w (ϱ, ϑ, ψ) := ϱ (cos φ (ϱ, ϑ) sin ψ, sin φ (ϱ, ϑ) sin ψ, cos ψ), where φ (ϱ, ϑ) := ϱ^{h}g (ϑ) + (1 − ϱ^{h}) g (α), with h sufficiently large.

Acknowledgment. The research of Irene Fonseca was partially supported by the National Science Foundation under Grant No. DMS-9731957 and through Center for Nonlinear Analysis (CNA). The research of Nicola Fusco was partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR). The research of Paolo Marcellini was partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) and through the contribution of the Italian Ministero degli Affari Esteri, Direzione Generale per la Promozione e la Cooperazione Culturale (MAE).

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