An extension of the Serrin’s lower semicontinuity theorem

Michele Gori & Paolo Marcellini Dipartimento di Matematica “U. Dini” Università di Firenze Viale Morgagni 67/A 50134 Firenze, Italy

Abstract

In this paper we present a new extension of a celebrated Serrin’s lower semicontinuity theorem. We consider an integral of the calculus of variation ∫_{Ω} f (x, u, Du) dx and we prove its lower semicontinuity in W_{loc}^{1,1} (Ω) with respect to the strong L_{loc}^{1} norm topology, under the usual continuity and convexity property of the integrand f(x, s, ξ), only assuming a mild (more precisely, local) condition on the independent variable x ∈ ℝ^{n}, say local Lipschitz continuity, which - we show with a specific counterexample

- cannot be replaced, in general, by local Hölder continuity.

Keywords: Lower semicontinuity, strong convergence in L^{1}, convex functions, local Lipschitz continuity, local Hölder continuity, calculus of variations.

1991 Mathematics Subject Classification: Primary 49J45. Secondary 35D05.

1 Introduction

The aim of this paper is to determinate some new sufficient conditions for lower semicontinuity with respect to the strong convergence in L_{loc}^{1} for functionals of integral type

F (u, Ω) = ∫_{Ω} f (x, u(x), Du(x)) dx ,  (1)

where Ω is an open set of ℝ^{n}, u varies in the Sobolev class W_{loc}^{1,1}(Ω), Du denotes the gradient of u, and the function f = f(x, s, ξ), for x ∈ Ω, s ∈ ℝ, ξ ∈ ℝ^{n}, satisfies the conditions

\{ \begin{matrix}f is continuous in Ω × ℝ × ℝ^{n}, \\ f is nonnegative in Ω × ℝ × ℝ^{n}, \\ f(x, s, ξ) is convex in ξ ∈ ℝ^{n} for all (x, s) ∈ Ω × ℝ. \end{matrix}  (2)

The integral functional F is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1} if, for every u_{h}, u ∈ W_{loc}^{1,1} (Ω) such that u_{h} → u in L_{loc}^{1} (Ω), then

lim inf_{h→+∞} F (u_{h}, Ω) ≥ F (u, Ω) .

Since the example given in 1941 by Aronzjain (see Pauc [15]; in particular page 54), it is known that condition (2) alone is not sufficient for strong lower semicontinuity of the integral F in (1). Serrin published in 1961 an article [16] proposing, in addition to (2), some sufficient conditions for strong lower semicontinuity. One of the most known and celebrated Serrin’s theorem on this subject is the following one (see Theorem 12 in [16]).

Theorem 1 (Serrin) Let f satisfy, in addition to (2), one of the following conditions:

(a) f(x, s, ξ) → +∞ when |ξ| → +∞, for all (x, s) ∈ Ω × ℝ;

(b) f(x, s, ξ) is strictly convex in ξ ∈ ℝ^{n} for all (x, s) ∈ Ω × ℝ;

(c) the derivatives f_{x}(x, s, ξ), f_{ξ}(x, s, ξ) and f_{ξx}(x, s, ξ) exist and are continuous.

Then F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}.

Many attempts have been made to weaken the assumptions on the integrand f. Serrin himself gave in 1961 the following further result (see Theorem 11 in [16]).

Theorem 2 (Serrin) Let us assume that f satisfies (2) and the following (uniform) continuity condition

|f(x_{1}, s_{1}, ξ) − f(x_{2}, s_{2}, ξ)| ≤ λ (|x_{1} − x_{2}| + |s_{1} − s_{2}|) · {1 + f(x_{1}, s_{1}, ξ)} , (3)

for every (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ Ω × ℝ and for all ξ ∈ ℝ^{n}, where λ is a modulus of continuity. Then

lim inf_{h→+∞} F (u_{h}, Ω) ≥ F (u, Ω) ,

for every u_{h}, u ∈ W_{loc}^{1,1} (Ω) such that u_{h} → u in L_{loc}^{1} (Ω), assuming in addition that u ∈ C^{0}(Ω).

The aims of some further studies in the direction of Theorem 2 tempt to remove the assumption of continuity of u and to weaken the uniform continuity condition (3) on f. Dal Maso [3] in 1980 gave the following lower semicontinuity result, without continuity assumptions on the limit function u (and in fact Dal Maso was able to extend his analysis to u ∈ BV (Ω), the functional space of functions of class L^{1} (Ω) with bounded variation, also considering more generally sequences of integral functionals which Γ-converge). However Dal Maso had to introduce some coercivity and growth conditions, as follows.

Theorem 3 (Dal Maso) Let us assume that f satisfies (2), (3) and that there exist functions m, q ∈ C^{0}(Ω), with m(x) > 0 for every x ∈ Ω, and a positive constant M such that

m(x) |ξ| ≤ f(x, s, ξ) ≤ M |ξ| + q(x) ,

for every (x, s, ξ) ∈ Ω × ℝ × ℝ^{n}. Then F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}.

Let us also mention that Dal Maso himself, revisiting the already quoted example by Aronzjain [15], emphasized that the continuity of f with respect to (x, u) alone is not sufficient for lower semicontinuity of F (u, Ω) in L^{1}. See Section 4 of this paper for further details.

A recent extension of Theorem 2 is due to Fonseca and Leoni (see Theorem 1.1 in [10], where the case u ∈ BV (Ω) is considered too; see also [11]).

Theorem 4 (Fonseca - Leoni) Let f(x, s, ξ) be a Borel function, convex with respect to ξ ∈ ℝ^{n}. Let us also assume that, for every (x_{0}, s_{0}) ∈ Ω × ℝ and for every ε > 0, there exists δ > 0 such that

f(x_{0}, s_{0}, ξ) − f(x, s, ξ) ≤ ε {1 + f(x, s, ξ)} ,  (4)

for every (x, s) ∈ Ω × ℝ such that |x − x_{0}| + |s − s_{0}| ≤ δ and for all ξ ∈ ℝ^{n}. Then F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}.

Notice that assumption (4) is a kind of lower semicontinuity of f with respect to (x, s) ∈ Ω × ℝ, uniform with respect to ξ ∈ ℝ^{n}. Lower semicontinuity of integrands of the type f(x, ξ) with respect to x ∈ Ω has been pointed out by Fusco [12] in 1979, as a necessary condition for lower semicontinuity of the respective (one-dimensional) integrals with respect to the strong convergence in L^{1}(Ω), on discussing the case of linear growth, with f(x, ξ) = a(x) |ξ|.

As already said, Theorems 3 and 4 has been obtained, respectively by Dal Maso and by Fonseca and Leoni, in the more general setting of BV (Ω), the subspace of L^{1} (Ω) whose functions have bounded variation. We quoted above the particular case when u_{h}, u ∈ W_{loc}^{1,1}(Ω), for a better comparison with the other results presented in this paper.

Some researches, as in Theorem 4, had the aim to relax the assumptions on f(x, s, ξ) related to the dependence on s, starting from a result in [7] by De Giorgi, Buttazzo and Dal Maso in 1983.

Theorem 5 (De Giorgi - Buttazzo - Dal Maso) Let f = f(s, ξ) be a nonnegative Borel function, convex with respect to ξ ∈ ℝ^{n}, only measurable with respect to s ∈ ℝ, although lower semicontinuous with respect to s ∈ ℝ at ξ = 0. If

lim sup_{|ξ|→0} \begin{matrix}(f(s, 0) − f(s, ξ)) & + \\ | & ξ & | \end{matrix} ∈ L_{loc}^{1}(ℝ) ,

then F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}.

Theorem 5 has been generalized in 1987 by Ambrosio [2], and in 1990-91 by De Cicco [4], [5] to the BV (Ω) setting. In [10] Fonseca and Leoni obtained the same conclusion of Theorem 5 for integrands f(x, s, ξ), depending explicitly on the x variable too, under the assumption of uniform continuity of f with respect to x ∈ Ω, similarly to (4).

From the above exposition it should be clear that the dependence of f on (x, s) must be treated carefully in studying lower semicontinuity of the integral F (u, Ω) with respect to the strong convergence in L^{1}(Ω). Of course (x, s) dependence gives some difficulties in the proofs, which are not only technical difficulties, since the existence of a counterexample to lower semicontinuity under explicit (x, s) dependence of the integrand f. In particular measurability of f(x, s, ξ) with respect to (x, s), or only with respect to x, is not appropriate for strong lower semicontinuity. As already mentioned, Fusco [12] gave a one-dimensional example, related to the integrand f(x, ξ) = a(x) |ξ|, where lower semicontinuity with respect to x was a necessary condition; in this context we refer also to the example in [14], related to an integrand of the type f(x, ξ) = a(x) |ξ|^{2}, which may have a relaxed functional in the strong L^{1} norm topology identical equal to zero, although the coefficient a(x) is a nonnegative function, not identical equal to zero (thus the corresponding integral F (u, Ω) is not lower semicontinuous). In Section 5 of this paper we will give further details, as well as we will propose a coercivity condition also sufficient for lower semicontinuity in L_{loc}^{1} .

In this paper we consider specifically the dependence of f(x, s, ξ) with respect to x ∈ Ω. In the previous results some qualified assumptions of uniform continuity, or of uniform lower semicontinuity, of f(x, s, ξ) with respect to x have been made (in the sense made more precise in the statements). On the contrary, in this paper we propose a new simple condition, in addition to (2), sufficient for lower semicontinuity. In fact we assume that f(x, s, ξ) is Lipschitz continuous with respect to x, locally respect to (s, ξ) and not necessarily globally. That is, we do not assume that the Lipschitz constant is uniform for (s, ξ) ∈ ℝ × ℝ^{n}. This main difference allows us to obtain, as a corollary, an improvement of Serrin’s Theorem 1(c); in fact we get the lower semicontinuity of F (u, Ω) under the only assumption that the derivative f_{x}(x, s, ξ) exist and is continuous, condition which clearly implies Lipschitz continuity of f on compact subsets of Ω × ℝ × ℝ^{n}, but not necessarily Lipschitz continuity of f on the full set Ω × ℝ × ℝ^{n}.

One of the main results of this paper (an other is Example 11) is the following Theorem 6. It has as a direct consequence Corollary 7, which of course generalizes Serrin’s Theorem 1(c).

Theorem 6 Assume that f(x, s, ξ) satisfies (2) and that, for every compact set K ⊂ Ω × ℝ × ℝ^{n}, there exists a constant L = L(K) such that

|f(x_{1}, s, ξ) − f(x_{2}, s, ξ)| ≤ L |x_{1} − x_{2}| ,  (5)

for every (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ K. Then the integral F (u, Ω) in (1) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}(Ω).

The proof of Theorem 6 is divided in several steps. In one of these steps we use an approximation procedure, due to De Giorgi [6], of the integrand f by a sequence f_{j}, each f_{j} being the maximum of a finite number of functions g(x, s, ξ) = a_{0}(x, s) + ∑_{i=1}^{n} a_{i}(x, s)ξ_{i}, linear with respect to ξ, with coefficients a_{0}(x, s) and a_{i}(x, s) (i = 1, 2, …, n) as in (18), which can be reduced to have compact support in Ω×ℝ. By this reason it is enough to assume local Lipschitz continuity of f with respect to x as in (5), and not necessarily global Lipschitz continuity.

Direct consequence of the Theorem 6 is the following result.

Corollary 7 Assume that f(x, s, ξ) satisfies (2) and that the derivative f_{x}(x, s, ξ) exists and is continuous (or only locally bounded). Then F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}(Ω).

In Sections 2 and 3 we will give the proof of Theorem 6. In Section 4 we will exhibit some examples showing that assumptions made in Theorem 6 (and in Corollary 7) are relevant for strong lower semicontinuity, in the sense that the only property of continuity of f(x, s, ξ) with respect to x ∈ Ω (of course, together with the other conditions in (2)) is not enough for strong lower semicontinuity of F (u, Ω) in L_{loc}^{1} , although sufficient for weak lower semicontinuity in W_{loc}^{1,1}(Ω).

More precisely, in Example 11 we will compare the assumptions of local Lipschitz continuity of f with respect to x with the assumption of local Hölder continuity of f (with respect to x) with exponent α < 1. In fact we will show that, for every exponent α ∈ (0, 1), there exists an n-dimensional integral F (u, Ω) (the dimension n depends on α, precisely n > 4α/(1 − α)) which is not lower semicontinuous in L_{loc}^{1} and whose integrand f(x, s, ξ) is Hölder continuous with respect to x (and of course nonnegative, continuous for (x, s, ξ) ∈ Ω × ℝ × ℝ^{n} and convex with respect to ξ ∈ ℝ^{n}).

In Section 5 we will show that lower semicontinuity results, corresponding to those of Theorem 6 and Corollary 7, do not hold in the vector-valued case, for applications u ∈ W_{loc}^{1,1}(Ω, ℝ^{m}), i.e., u : Ω ⊂ ℝ^{n} → ℝ^{m} with m > 1, not just for quasiconvex integrands, but under convexity conditions too. Finally, having in mind some relaxation formulas due to Marcellini [14] and Fusco [12], we give in Proposition 17 a coercivity condition sufficient for lower semicontinuity in L_{loc}^{1} of the integral F (u, Ω) in (1).

Acknowledgment. This research, partially done while Michele Gori was carrying out his degree thesis in Mathematics [13] at the Università di Firenze, has been supported by the Italian Ministero dell’Università e della Ricerca Scientifica e Tecnologica (MURST).

2 A preliminary lemma

In this section we give some preliminary results that will be used in the proof of Theorem 6. The first lemma is a modification of an argument introduced by Serrin (see the proof of Theorem 12 in [16]).

Lemma 8 Let us assume that f satisfies (2) and that the derivative f_{ξ}(x, s, ξ) exists and is a continuous function in Ω × ℝ × ℝ^{n}. Let us also assume that, for every compact set K ⊂ Ω × ℝ × ℝ^{n}, there exists a constant L = L (K) such that

|f_{ξ}(x_{1}, s, ξ) − f_{ξ}(x_{2}, s, ξ)| ≤ L |x_{1} − x_{2}| , ∀ (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ K, (6)

and, for every compact set K_{1} ⊂ Ω × ℝ, there exists a constant L_{1} = L_{1} (K_{1}) such that

\{ \begin{matrix}|f_{ξ}(x, s, ξ)| ≤ L_{1}, ∀ (x, s) ∈ K_{1}, ∀ ξ ∈ ℝ^{n}, \\ |f_{ξ}(x, s, ξ_{1}) − f_{ξ}(x, s, ξ_{2})| ≤ L_{1} |ξ_{1} − ξ_{2}| , ∀ (x, s) ∈ K_{1}, ∀ ξ_{1}, ξ_{2} ∈ ℝ^{n}. \end{matrix}

(7)

Then F (u, Ω) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}.

Proof. Let us consider a sequence f_{i}(x, s, ξ) = α_{i}(x, s)f(x, s, ξ), i = 1, 2, …, where {α_{i}}_{i∈ℕ} is an increasing sequence of smooth functions with compact support in Ω × ℝ, converging pointwise to 1 in Ω × ℝ. It is clear that, for all i ∈ ℕ, f_{i} still satisfies the hypothesis of Lemma 8 and also vanishes if (x, s) vary outside a compact set of Ω × ℝ. Moreover f_{i} is an increasing sequence of functions which pointwise converge to f. Thus, by the monotone convergence theorem we can go to the limit as i → +∞ and, since the supremum of a family of lower semicontinuous functionals is lower semicontinuous, it is sufficient to prove the stated lower semicontinuity for the integral functional associated to a generic integrand α_{i}f. In other words, in the proof of Lemma 8, without loss of generality, we can assume that the integrand f(x, s, ξ) vanishes if (x, s) vary outside a compact set of Ω × ℝ. For the same reason in the exposition below we will always assume that (x, s) vary on a compact set of Ω × ℝ and that f is equal to zero in the complement of this compact set.

Let u_{h}, u ∈ W_{loc}^{1,1} (Ω) such that u_{h} → u in L_{loc}^{1} (Ω). We will prove that

lim inf_{h→+∞} F (u_{h}, Ω) ≥ F (u, Ω) .  (8)

Without loss of generality, we can assume that u_{h} converges to u almost everywhere in Ω and that

lim inf_{h→+∞} F (u_{h}, Ω) = lim_{h→+∞} F (u_{h}, Ω) .

Let Ω′ be an open set whose closure is contained in Ω, such that f(x, s, ξ) = 0 when x ∈ Ω − Ω′; thus in particular F (u_{h}, Ω) = F (u_{h}, Ω′) and F (u, Ω) = F (u, Ω′). Let us denote by α_{ρ} a mollifier and by u_{ρ} = u ∗ α_{ρ} the mollified function of u with step ρ. Since u ∈ W_{loc}^{1,1}(Ω), for every ε > 0 there exists ρ > 0 such that

∫_{Ω′} |Du − Du_{ρ}| dx ≤ ε .  (9)

By Fatou lemma we can choose ρ small enough such that u_{ρ} satisfies also the condition

∫_{Ω′} f(x, u, Du_{ρ}) dx ≥ ∫_{Ω′} f(x, u, Du) dx − ε .  (10)

We estimate the difference of the integrands in (8)

f(x, u_{h}, Du_{h}) − f(x, u, Du) = f(x, u_{h}, Du_{h}) − f(x, u_{h}, Du_{ρ})  (11)
+f(x, u_{h}, Du_{ρ}) − f(x, u, Du_{ρ}) + f(x, u, Du_{ρ}) − f(x, u, Du) .

From (11), by the convexity of f(x, s, ξ) with respect to ξ, we have

f(x, u_{h}, Du_{h}) − f(x, u, Du) ≥ (f_{ξ}(x, u_{h}, Du_{ρ}), Du_{h} − Du_{ρ})
+ {f(x, u_{h}, Du_{ρ}) − f(x, u, Du_{ρ})} + {f(x, u, Du_{ρ}) − f(x, u, Du)}
= (f_{ξ}(x, u_{h}, Du_{ρ}), Du_{h}) − (f_{ξ}(x, u, Du_{ρ}), Du)
+ (f_{ξ}(x, u, Du_{ρ}), Du − Du_{ρ}) + (f_{ξ}(x, u, Du_{ρ}) − f_{ξ}(x, u_{h}, Du_{ρ}), Du_{ρ})
+ {f(x, u_{h}, Du_{ρ}) − f(x, u, Du_{ρ})} + {f(x, u, Du_{ρ}) − f(x, u, Du)} .

We observe that (x, s) → f_{ξ}(x, s, Du_{ρ}(x)) is a continuous function with compact support in Ω×ℝ and that, by (7), |f_{ξ}(x, s, Du_{ρ}(x))| ≤ L_{1} for every (x, s) ∈ Ω×ℝ and for every ρ. We obtain

f(x, u_{h}, Du_{h}) − f(x, u, Du) ≥ (f_{ξ}(x, u_{h}, Du_{ρ}), Du_{h}) − (f_{ξ}(x, u, Du_{ρ}), Du)

−L_{1} |Du − Du_{ρ}| + (f_{ξ}(x, u, Du_{ρ}) − f_{ξ}(x, u_{h}, Du_{ρ}), Du_{ρ})

+ {f (x, u_{h}, Du_{ρ}) − f (x, u, Du_{ρ})} + {f (x, u, Du_{ρ}) − f (x, u, Du)} .

We integrate both sides over Ω′. By (9) and (10) we have

∫_{Ω′} {f(x, u_{h}, Du_{h}) − f(x, u, Du)} dx  (12)
≥ ∫_{Ω′} {(f_{ξ}(x, u_{h}, Du_{ρ}), Du_{h}) − (f_{ξ}(x, u, Du_{ρ}), Du)} dx
+ ∫_{Ω′} (f_{ξ}(x, u, Du_{ρ}) − f_{ξ}(x, u_{h}, Du_{ρ}), Du_{ρ}) dx
+ ∫_{Ω′} {f(x, u_{h}, Du_{ρ}) − f(x, u, Du_{ρ})} dx − (1 + L_{1}) ε .

We can go to the limit as h → +∞. First we observe that, since (x, s) → f(x, s, Du_{ρ}(x)) and (x, s) → f_{ξ}(x, s, Du_{ρ}(x)) are bounded functions (in fact are continuous function with compact support), by the Lebesgue’s dominated convergence theorem we obtain

lim_{h→+∞} ∫_{Ω′} (f_{ξ}(x, u, Du_{ρ}) − f_{ξ}(x, u_{h}, Du_{ρ}), Du_{ρ}) dx = 0 ,

lim_{h→+∞} ∫_{Ω′} {f(x, u_{h}, Du_{ρ}) − f(x, u, Du_{ρ})} dx = 0 .

Since ε in (12) can be arbitrarily small, to obtain the conclusion (8) it remains to prove that

lim_{h→+∞} ∫_{Ω′} {(f_{ξ}(x, u_{h}, Du_{ρ}), Du_{h}) − (f_{ξ}(x, u, Du_{ρ}), Du)} dx = 0 .  (13)

With the aim to prove (13), similarly to Tonelli [17], we denote by g(x, s) the continuous vector-valued function with compact support (g : Ω × ℝ → ℝ^{n}) defined by g(x, s) = f_{ξ}(x, s, Du_{ρ}(x)), i.e., more precisely,

g(x, s) = (g^{(i)}(x, s))_{i=1}^{n} = (f_{ξ_{i}}(x, s, Du_{ρ}(x)))_{i=1}^{n} .

Let us first prove that g is Lipschitz continuous with respect to x ∈ Ω′. Recall that (x, s) vary on a compact set of Ω × ℝ (and out of this compact set f is equal to zero; in particular f(x, s, ξ) = 0 when x ∈ Ω − Ω′); also recall that ξ = Du_{ρ} is bounded. For all x_{1}, x_{2} ∈ Ω′ and s ∈ ℝ, by using the assumptions (6) and (7), we have

|g(x_{1}, s) − g(x_{2}, s)| = |f_{ξ}(x_{1}, s, Du_{ρ}(x_{1})) − f_{ξ}(x_{2}, s, Du_{ρ}(x_{2}))|

≤ |f_{ξ}(x_{1}, s, Du_{ρ}(x_{1})) − f_{ξ}(x_{2}, s, Du_{ρ}(x_{1}))|
+ |f_{ξ}(x_{2}, s, Du_{ρ}(x_{1})) − f_{ξ}(x_{2}, s, Du_{ρ}(x_{2}))|
≤ L |x_{1} − x_{2}| + L_{1} |Du_{ρ}(x_{1}) − Du_{ρ}(x_{2})| ≤ L_{2} |x_{1} − x_{2}| ,

where L_{2} = max {L; L_{1} ∥Du_{ρ}∥_{W^{1,∞}(Ω′)}}. As before, let us denote by α_{σ} a mollifier with parameter σ → 0^{+} (α_{σ} ∈ C_{c}^{∞}(ℝ^{n}), with α_{σ} ≥ 0 and ∫_{ℝ^{n}} α_{σ}(η) dη = 1); then we pose

g_{σ}(x, s) = ∫_{ℝ^{n}} α_{σ}(y) g(x − y, s) dy = (∫_{ℝ^{n}} α_{σ}(y) g^{(i)}(x − y, s) dy)_{i=1}^{n} .

Let us observe that, if σ is sufficiently small, then g_{σ} ∈ C_{c}^{0}(Ω′ × ℝ), because g ∈ C_{c}^{0}(Ω′ × ℝ) too. Moreover, for every x_{1}, x_{2} ∈ Ω′ and for every s ∈ ℝ, we obtain

|g_{σ}(x_{1}, s) − g_{σ}(x_{2}, s)| = |∫_{ℝ^{n}} α_{σ}(y) [g(x_{1} − y, s) − g(x_{2} − y, s)] dy|

≤ ∫_{ℝ^{n}} α_{σ}(y) |g(x_{1} − y, s) − g(x_{2} − y, s)| dy ≤ L_{2} |x_{1} − x_{2}| .

Therefore we also have

|\frac{∂g_{σ}^{(i)}}{∂x_{j}} (x, s)| ≤ L_{2} , ∀ i, j = 1, 2, … n, ∀ (x, s) ∈ Ω′ × ℝ.  (14)

For every σ > 0 we denote by G_{σ,h}(x) = (G_{σ,h}^{(i)}(x))_{i=1}^{n} the sequence of vectorvalued functions defined by

G_{σ,h}(x) = ∫_{u(x)}^{u_{h}(x)} g_{σ}(x, s) ds .

By the chain rule we can compute the trace of the n × n matrix DG_{h}; we have

trace DG_{σ,h}(x) = ∑_{i=1}^{n} \frac{∂G_{σ,h}^{(i)}}{∂x_{i}} = (g_{σ} (x, u_{h}(x)) , Du_{h}(x))  (15)

− (g_{σ} (x, u(x)) , Du(x)) + ∑_{i=1}^{n} ∫_{u(x)}^{u_{h}(x)} \frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s) ds .

Since also G_{σ,h}(x) vanishes outside Ω′, it results

∫_{Ω′} DG_{σ,h}(x) dx = 0 , ∀ h ∈ ℕ .  (16)

From (15), (16) and (14) we deduce that

|∫_{Ω′} {(g_{σ} (x, u_{h}(x)) , Du_{h}(x)) − (g_{σ} (x, u(x)) , Du(x))} dx|

= |− ∫_{Ω′} \{∑_{i=1}^{n} ∫_{u(x)}^{u_{h}(x)} \frac{∂g_{σ}^{(i)}}{∂x_{i}} (x, s) ds \} dx| ≤ nL_{2} ∫_{Ω′} |u_{h}(x) − u(x)| dx .

We go first to the limit as σ → 0^{+} (and h fixed). We obtain the same inequality when in the left hand side g_{σ} is replaced by g. Thus we also have

|∫_{Ω′} {(f_{ξ}(x, u_{h}, Du_{ρ}), Du_{h}) − (f_{ξ}(x, u, Du_{ρ}), Du)} dx|

= |∫_{Ω′} {(g (x, u_{h}(x)) , Du_{h}(x)) − (g (x, u(x)) , Du(x))} dx|
≤ nL_{2} ∫_{Ω′} |u_{h}(x) − u(x)| dx ,

which goes to zero as h → +∞, since, by assumption, u_{h} → u in L_{loc}^{1} (Ω). Therefore (13) holds and the proof of Lemma 8 is complete.

The following approximation result has been given by De Giorgi (see [6]).

Lemma 9 If f = f(x, s, ξ) satisfies (2) and vanishes outside a compact set of Ω × ℝ, there exists an increasing sequence of functions {f_{j}(x, s, ξ)}_{j∈ℕ} that converges to f(x, s, ξ) uniformly on the compact sets of Ω × ℝ × ℝ^{n} and such that, for all j ∈ ℕ, f_{j} satisfies (2) and

|f_{j}(x, s, ξ_{1}) − f_{j}(x, s, ξ_{2})| ≤ L_{j} |ξ_{1} − ξ_{2}| ,  (17)

for some constant L_{j} and for every (x, s, ξ_{1}), (x, s, ξ_{2}) ∈ Ω × ℝ × ℝ^{n}.

Proof. We do not give all the details and we refer to the original proof by De Giorgi [6]. By using the support tangent hyperplanes to the graph of f(x, s, ξ), up to a regularization procedure, De Giorgi shows that, for every j ∈ ℕ, f_{j} can be defined as the maximum between the zero function and a finite number of (affine with respect to ξ ∈ ℝ^{n}) functions of the type

g(x, s, ξ) = a_{0}(x, s) + ∑_{i=1}^{n} a_{i}(x, s)ξ_{i} .

For the use that we will make in the next section, we recall that the coefficients a_{i} (i = 0, 1, 2, …, n) are given by

\{ \begin{matrix}a_{i}(x, s) = − ∫_{ℝ^{n}} f(x, s, η)D_{i}α(η) dη , ∀ i = 1, 2, …, n, \\ a_{0}(x, s) = ∫_{ℝ^{n}} f(x, s, η) \{(n + 1)α(η) + ∑_{i=1}^{n} η_{i}D_{i}α(η)\} dη , \end{matrix}  (18)

for some mollifier α ∈ C_{c}^{∞}(ℝ^{n}), with α ≥ 0 and ∫_{ℝ^{n}} α(η) dη = 1.

3 Proof of theorem 6

In this section we will prove Theorem 6. With the same argument used at the beginning of the proof of Lemma 8, without loss of generality, we can assume that f(x, s, ξ) vanishes outside a compact set of Ω × ℝ. Therefore we can also assume that Ω is a set with finite measure (we will use this remark in the definition (20) of the integral F_{j}(u, Ω)).

Let {f_{j}(x, s, ξ)}_{j∈ℕ} be the increasing sequence that pointwise converges to f(x, s, ξ), as in Lemma 9. Let us denote by φ_{ρ} a mollifier (φ_{ρ} ∈ C_{c}^{∞}(ℝ^{n}), φ_{ρ} ≥ 0, φ_{ρ}(η) = 0 if |η| ≥ ρ, ∫_{ℝ^{n}} φ_{ρ}(η) dη = 1) and, for all j ∈ ℕ, by f_{j,ρ} = f_{j} ∗ φ_{ρ} the mollified function of f_{j} , with respect to the variable ξ ∈ ℝ^{n}, with step ρ. That is, for every j ∈ ℕ and ρ > 0, the function f_{j,ρ} : Ω × ℝ × ℝ^{n} → ℝ is defined by

f_{j,ρ}(x, s, ξ) = ∫_{ℝ^{n}} f_{j}(x, s, ξ − η)φ_{ρ}(η) dη .

By the Lipschitz continuity (17) of f_{j} with respect to ξ ∈ ℝ^{n}, we have

|f_{j,ρ}(x, s, ξ) − f_{j}(x, s, ξ)| ≤ ∫_{ℝ^{n}} |f_{j}(x, s, ξ − η) − f_{j}(x, s, ξ)| φ_{ρ}(η) dη

≤ ∫_{supp φ_{ρ}} L_{j} |η| φ_{ρ}(η) dη ≤ L_{j}ρ .

Thus we can choose ρ = ρ_{j} =: 1/ (jL_{j}) → 0 so that

f_{j}(x, s, ξ) − \frac{2}{j} ≤ f_{j,ρ_{j}}(x, s, ξ) − \frac{1}{j} ≤ f_{j}(x, s, ξ) ≤ f(x, s, ξ) ,  (19)

for every (x, s, ξ) ∈ Ω × ℝ × ℝ^{n}. By the monotone convergence theorem we have

lim_{j→+∞} ∫_{Ω} f_{j} (x, u(x), Du(x)) dx = ∫_{Ω} f (x, u(x), Du(x)) dx .

Thus, if we consider the sequence of integrals

F_{j}(u, Ω) = ∫_{Ω} \{f_{j,ρ_{j}} (x, u(x), Du(x)) − \frac{1}{j}\} dx ,  (20)

by (19) we obtain that F_{j}(u, Ω) converges, as j → +∞, to the main integral F (u, Ω) = ∫_{Ω} f (x, u(x), Du(x)) dx, and at the same time F_{j}(u, Ω) ≤ F (u, Ω) for every j ∈ ℕ. Therefore F (u, Ω), being the supremum of the family of functionals {F_{j}(u, Ω)}_{j∈ℕ}, will be lower semicontinuous if every of such F_{j}(u, Ω) is lower semicontinuous.

Thus we must prove that, for every fixed j ∈ ℕ, the integral functional F_{j} in (20) is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}. To this aim we apply Lemma 8. Of course f_{j,ρ_{j}}(x, s, ξ) satisfies (2) and (17); thus it satisfies also the bound for the derivative in (7). It remains to verify that f_{j,ρ_{j}}(x, s, ξ) also satisfies the second assumption in (7) and (6). We first compute the n partial derivatives of f_{j,ρ_{j}} with respect to the gradient variable ξ, i.e., we compute the vector field ∂f_{j,ρ_{j}}/∂ξ:

\frac{∂f_{j,ρ_{j}}}{∂ξ} (x, s, ξ) = ∫_{ℝ^{n}} f_{j}(x, s, ξ − η)\frac{∂φ_{ρ_{j}}}{∂ξ} (η) dη .

Then from (17) we deduce that

|\frac{∂f_{j,ρ_{j}}}{∂ξ} (x, s, ξ_{1}) − \frac{∂f_{j,ρ_{j}}}{∂ξ} (x, s, ξ_{2})| ≤ M_{j}L_{j} |ξ_{1} − ξ_{2}| ,

for every (x, s, ξ_{1}), (x, s, ξ_{2}) ∈ Ω × ℝ × ℝ^{n}, where

M_{j} = ∫_{ℝ^{n}} |\frac{∂φ_{ρ_{j}}}{∂ξ} (η)| dη .  (21)

Therefore the second assumption in (7) is satisfied. To prove (6), we recall that f_{j}(x, s, ξ) is the maximum between the zero function and a finite number of affine functions, with respect to ξ ∈ ℝ^{n}, of the type

g(x, s, ξ) = a_{0}(x, s) + ∑_{i=1}^{n} a_{i}(x, s)ξ_{i} ,  (22)

where the coefficients a_{i}(x, s) (i = 0, 1, 2, …, n) are given in (18). From assumption (5), of local Lipschitz continuity of f with respect to x, for every compact set K ⊂ Ω × ℝ × ℝ^{n} and for every (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ K we have

|f(x_{1}, s, ξ) − f(x_{2}, s, ξ)| ≤ L |x_{1} − x_{2}| ,

for some constant L = L(K). Then the coefficients a_{i} (i = 0, 1, 2, …, n) in (18) are locally Lipschitz continuous with respect to x too; in fact, for example, for every i = 1, 2, …, n,

|a_{i}(x_{1}, s) − a_{i}(x_{2}, s)| = |∫_{ℝ^{n}} {f(x_{1}, s, η) − f(x_{2}, s, η)} D_{i}α(η) dη|
≤ m_{i}L (K) |x_{1} − x_{2}| ,

for every (x_{1}, s), (x_{2}, s) which vary on a compact set K_{0} of Ω × ℝ (in fact the two points (x_{1}, s, η), (x_{2}, s, η) vary in the compact set K = K_{0} × supp α) and m_{i} is given by

m_{i} = ∫_{ℝ^{n}} |\frac{∂α}{∂ξ_{i}} (η)| dη .

Therefore the affine functions g(x, s, ξ) in (22), since ξ vary on a bounded set, are local Lipschitz continuous with respect to x, for (x, s, ξ) ∈ K. Finally f_{j}(x, s, ξ), being the maximum between the zero function and a finite number of affine functions of the type of g(x, s, ξ) in (22), is local Lipschitz continuous with respect to x, i.e., for every compact set K ⊂ Ω × ℝ × ℝ^{n}, there exists a constant L′_{j} = L′_{j}(K) such that

|f_{j}(x_{1}, s, ξ) − f_{j}(x_{2}, s, ξ)| ≤ L′_{j}(K) |x_{1} − x_{2}| ,

for every (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ K. For the same values of (x_{i}, s, ξ), i = 1, 2, we deduce that

|\frac{∂f_{j,ρ_{j}}}{∂ξ} (x_{1}, s, ξ) − \frac{∂f_{j,ρ_{j}}}{∂ξ} (x_{2}, s, ξ)|

≤ ∫_{ℝ^{n}} |f_{j}(x_{1}, s, ξ − η) − f_{j}(x_{2}, s, ξ − η)| · |Dφ_{ρ_{j}}(η)| dη
≤ M_{j}L′_{j}(K′) |x_{1} − x_{2}| ,

where M_{j} is the constant in (21) and K′ is a suitable compact subset of Ω × ℝ × ℝ^{n} containing K. Therefore also the assumption (6) is satisfied and the proof of Theorem 6 is complete.

4 Aronzjain-Dal Maso’s example revisited

In this section, and in the next one, we collect some examples, partially known, partially new or revisited, modified and adapted to a more general context, with the aim to introduce some parameters which will allow us to test more carefully the assumptions. More precisely, we will show that the assumption of continuity alone of f(x, s, ξ) with respect to x ∈ Ω (together with (2)) is not sufficient for the lower semicontinuity of the integral functional F (u, Ω) with respect to the strong convergence in L_{loc}^{1}(Ω) (see Example 10). Then, with a more precise analysis of some parameters and with the study of the n-dimensional context, we compare in Example 11 assumptions of Lipschitz continuity of f with respect to x with the weaker assumption of Hölder continuity.

The first example that we propose in this section has been inspired by an old example by Aronzjain in 1941 (see Pauc [15], starting from page 54), more recently exploited by Dal Maso (see Section 4 in [3]). In the new version proposed here we consider, in particular, a simplified sequence u_{h}; as already said, this simplification will allow us to compare Lipschitz continuity versus Hölder continuity of f with respect to x. Notice also that the original example by Aronzjain is related to a multiple integral, i.e., with n = 2, although Aronzjain’s integrand f(x, ξ) does not explicitly depends on s. A one-dimensional version of Aronzjain’s example was known to Dal Maso, who gave us some handwritten notes on the subject.

In this and in the following section we propose several examples; for completeness we also mention the case considered by Acerbi, Buttazzo and Fusco [1], with the main difference that their example is posed in the vector-valued setting of polyconvex integrands, but similar to the next two examples, at least for two aspects: the fact that the integrand f(x, s, ξ) = |a(x, s) ξ − 1| has not minimum at ξ = 0 and the L^{∞} convergence of the sequence u_{h}. Other similarities seem to exist, and we hope to come back to the vectorial setting in the future.

Example 10 Let Ω be the open interval (0, 2π). Let u_{h} be the sequence (converging to u ≡ 0 in L^{∞}(Ω), but not in the weak topology of W^{1,1}(Ω)) defined by

u_{h}(x) = \frac{1}{2^{h}} (1 − \frac{1}{4} cos (4^{h}x)) .

Then there exists a function a(x, s), bounded and uniformly continuous for (x, s) ∈ Ω × ℝ, such that, if we define

f(x, s, ξ) = |a(x, s) ξ − 1| , x ∈ Ω ⊂ ℝ, s ∈ ℝ, ξ ∈ ℝ,

then

lim_{h→+∞} F (u_{h}, Ω) = lim_{h→+∞} ∫_{Ω} f (x, u_{h}, u′_{h}) dx = 0 ,

while of course

F (u, Ω) = ∫_{Ω} f (x, 0, 0) dx = 2π .

Thus in this case the integral F is not lower semicontinuous with respect to the strong convergence in L^{∞}(Ω), although f(x, s, ξ) is a continuous nonnegative function, convex with respect to ξ, i.e., f satisfies (2).

Proof. Let us observe that, for every h ∈ ℕ,

min \{u_{h}(x) : x ∈ [0, 2π]\} = \frac{1}{2^{h}} (1 − \frac{1}{4})

> \frac{1}{2^{h+1}} (1 + \frac{1}{4}) = max \{u_{h+1}(x) : x ∈ [0, 2π]\};

thus the graph G_{h} = {(x, s) ∈ Ω × ℝ : s = u_{h}(x)} of u_{h} is disjoint from the graph of u_{h+1} , more precisely, they have positive distance each other. We will first define the function a(x, s) on a subset of the union ⋃_{h∈ℕ} G_{h}.

With the aim to define this subset, we notice that, for periodicity reasons, for every positive λ the measure of the set {x ∈ [0, 2π] : |sin(mx)| < λ} does not depend on the integer m. Moreover, since for x > 0 sufficiently small sin x > x/2, then for the same x values (i.e., for λ > 0 sufficiently small) the following inclusion holds {x : sin x < λ} ⊂ {x : x/2 < λ}. This implies that the measure of the set {x > 0 close to zero such that sin x < λ} is less than 2λ. Taking into account the three zeroes x = 0, π, 2π of the sinus function in the interval [0, 2π], finally we have

meas {x ∈ [0, 2π] : |sin(mx)| < λ} < 8λ, ∀ m ∈ ℕ.  (23)

Let us denote by E_{h} the open subset of [0, 2π] given by

E_{h} = {x ∈ [0, 2π] : |sin (4^{h}x)| < h2^{−h}} .

By (23) we have meas (E_{h}) < h2^{−h+3}. For every h ∈ ℕ we compute the derivative u′_{h}(x) = 2^{h−2} sin (4^{h}x). We define the function a(x, s) on the following subset of the union {G_{h}}_{h∈ℕ}: if x ∈ [0, 2π] − E_{h} and s = u_{h}(x), then

a(x, s) = \frac{1}{u′_{h}(x)} = \frac{1}{2^{h−2} sin (4^{h}x)} .

Since h by h the graphs G_{h} are disjoint sets of \overline{Ω} × ℝ, then the above definition is consistent. Here we use the relevant fact that the derivative u′_{h} of u_{h}, as h → +∞, diverges (in absolute value) for x ∈ [0, 2π] − E_{h} (otherwise we should expect lower semicontinuity of the integral F); in fact 2^{h−2} |sin (4^{h}x)| ≥ h/4 for every x ∈ [0, 2π] − E_{h}. Therefore 1/u′_{h}(x) converges to zero as h → +∞ and we can also define by continuity

a(x, s) = 0, if x ∈ [0, 2π] and s = 0.

At this stage the function a(x, s) has been defined as a continuous function on a compact subset of \overline{Ω} × ℝ. Then it can be extended to the full \overline{Ω} × ℝ remaining uniformly continuous (and bounded) on \overline{Ω} × ℝ.

By definition, for every h ∈ ℕ and for x ∈ [0, 2π] − E_{h} , we have

f (x, u_{h}(x), u′_{h}(x)) = |a(x, u_{h}(x)) u′_{h}(x) − 1| = 0 .

Thus, since |u′_{h}(x)| = 2^{h−2} |sin (4^{h}x)| ≤ 2^{h−2}h2^{−h} = h/4 for x ∈ E_{h}, and meas (E_{h}) < h2^{−h+3}, if we denote by M > 0 a bound for a(x, s) in \overline{Ω} × ℝ, we obtain

∫_{Ω} f (x, u_{h}, u′_{h}) dx = ∫_{E_{h}} f (x, u_{h}, u′_{h}) dx

= ∫_{E_{h}} |a(x, u_{h}(x)) u′_{h}(x) − 1| dx

≤ ∫_{E_{h}} \{M |u′_{h}(x)| + 1\} dx ≤ (\frac{Mh}{4} + 1) h2^{−h+3},

which converges to zero as h → +∞.

Let us go back to the main Theorem 6, in particular to the assumption that f(x, s, ξ) is Lipschitz continuous with respect to x, locally respect to (x, s, ξ), i.e., for every compact set K ⊂ Ω × ℝ × ℝ^{n}, there exists a constant L = L(K) such that

|f(x_{1}, s, ξ) − f(x_{2}, s, ξ)| ≤ L |x_{1} − x_{2}| ,

for every (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ K. We may ask if we can assume a less restrictive local continuity assumption of f with respect to x. For example, we may ask if Theorem 6 holds under the only assumption that f(x, s, ξ) satisfies (2) and is Hölder continuous with respect to x, locally respect to (x, s, ξ), i.e., there exists a real number α ∈ (0, 1) with the property that, for every compact set K ⊂ Ω × ℝ × ℝ^{n}, there exists a constant L = L(K) such that

|f(x_{1}, s, ξ) − f(x_{2}, s, ξ)| ≤ L |x_{1} − x_{2}|^{α} ,  (24)

for every (x_{1}, s, ξ) , (x_{2}, s, ξ) ∈ K.

By the next example we give an answer to this question; in fact we will prove that, for every exponent α ∈ (0, 1), it is possible to find a nonnegative continuous integrand f = f_{α}(x, s, ξ) (depending on α too), convex with respect to ξ ∈ ℝ^{n}, satisfying the Hölder continuity property (24), but whose corresponding integral is not lower semicontinuous, even in C^{∞}(\overline{Ω}), with respect to the strong L^{∞}(Ω) convergence.

We emphasize that, in the next example, we do not consider an arbitrary independent dimension n ≥ 1, but we impose the constraint

n > \frac{4α}{1 − α}  (25)

on the dimension, or equivalently the constraint

α < \frac{n}{n + 4}  (26)

on the Hölder exponent α. The less restrictive constraint α < 1/3 is assumed when n = 1. Thus it remains open the interesting question to know if, for every n ∈ ℕ, there exists a critical exponent α(n) such that the integral (1) is lower semicontinuous with respect to the strong convergence in L_{loc}^{1} under the usual condition (2) and the Hölder continuity property (24) for some exponent α such that α(n) ≤ α < 1. In particular we may ask if, for example, the integral (1) is lower semicontinuous in the one-dimensional case n = 1, when the integrand f is Hölder continuous with exponent α ≥ 1/3.

Example 11 We use notations similar to the previous Example 10, with some parameters. Thus let Ω be the open hyper-rectangle (0, 2π)^{n} ⊂ ℝ^{n}. For h ∈ ℕ let u_{h}(x) = u_{h}(x_{1}, x_{2}, …, x_{n}) be defined by

u_{h}(x) = ∑_{i=1}^{n} \frac{1}{a_{h}} (1 − \frac{1}{4} cos (b_{h}x_{i})) ,  (27)

with {a_{h}}, {b_{h}} sequences of positive real numbers diverging to +∞ as h → +∞. Thus u_{h} converges to u ≡ 0 in L^{∞}(Ω). Then, for every α ∈ (0, 1), if

n > \frac{4α}{1 − α} ,  (28)

there exist \{a_{h}\}, \{b_{h}\} and a vector-valued function a_{α} : \overline{Ω} × ℝ ⊂ ℝ^{n} × ℝ → ℝ^{n} with the properties:

(i) a_{α}(x, s) is bounded and uniformly continuous for (x, s) ∈ \overline{Ω} × ℝ; (ii) for every s ∈ ℝ, a_{α}(x, s) is Hölder continuous (of exponent α) with

respect to x ∈ \overline{Ω}; more precisely, there exists a constant L such that

|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)| ≤ L |x_{1} − x_{2}|^{α} , ∀ x_{1}, x_{2} ∈ \overline{Ω} ⊂ ℝ^{n}, ∀ s ∈ ℝ;

(iii) if we denote by (·, ·) the scalar product in ℝ^{n} and we define

f_{α}(x, s, ξ) = |(a_{α}(x, s), ξ) − 1| , x ∈ \overline{Ω}, s ∈ ℝ, ξ ∈ ℝ^{n},

then f_{α}(x, s, ξ) satisfies (2), the Hölder continuity property (24) and

lim inf_{h→+∞} ∫_{Ω} f_{α} (x, u_{h}, Du_{h}) dx = 0 ; ∫_{Ω} f_{α} (x, 0, 0) dx = (2π)^{n} .  (29)

The same construction for the one-dimensional case n = 1 has all the stated properties, under the less restrictive assumption (instead of (28)) that α < 1/3.

Proof. Step 1 (definition of a_{α}): passing to a subsequence if necessary (this is the reason to have in (29) the limit inferior, instead of the limit, that, however, can be easily reduced to became a limit), we can assume that the graphs of the functions u_{h} are disjoint; more precisely that

min \{u_{h}(x) : x ∈ [0, 2π]^{n}\} = \frac{3n}{4a_{h}} > \frac{5n}{4a_{h+1}} = max \{u_{h+1}(x) : x ∈ [0, 2π]^{n}\}

for every h ∈ ℕ. With similar notations as in the previous example we define

E_{h}^{i} = {x_{i} ∈ [0, 2π] : |sin (b_{h}x_{i})| < λ_{h}} , i = 1, 2, …, n,

E_{h} = {x = (x_{1}, x_{2}, …, x_{n}) ∈ [0, 2π]^{n} : ∑_{i=1}^{n} sin^{2} (b_{h}x_{i}) < λ_{h}^{2}} ,

where {λ_{h}} is a sequence of positive real numbers converging to 0 as h → +∞. We have

meas (E_{h}^{i}) < 8λ_{h} , ∀ h ∈ ℕ, ∀ i = 1, 2, …, n,  (30)

and, since E_{h} ⊂ E_{h}^{1} × E_{h}^{2} × … × E_{h}^{n},

meas (E_{h}) < 8^{n} (λ_{h})^{n} , ∀ h ∈ ℕ.  (31)

The partial derivatives of u_{h}(x) are equal to

\frac{∂u_{h}}{∂x_{i}} = \frac{b_{h}}{4a_{h}} sin (b_{h}x_{i}) , i = 1, 2, …, n,

and, for x ∈ [0, 2π]^{n} − E_{h}, we have

|Du_{h}(x)| = \frac{b_{h}}{4a_{h}} \{∑_{i=1}^{n} sin^{2} (b_{h}x_{i})\}^{1/2} ≥ \frac{b_{h}}{4a_{h}} λ_{h} .  (32)

Similarly to the previous example, we define the vector-valued function a_{α}(x, s) on a part of the graphs of the functions u_{h}. Precisely, for every h ∈ ℕ and for x ∈ [0, 2π]^{n} − E_{h} and s = u_{h}(x), we define

a_{α}(x, s) = \frac{Du_{h}}{|Du_{h}|^{2}} ,  (33)

i.e., with the notation a_{α}(x, s) = (a_{α}^{i}(x, s))_{i=1}^{n}, we define

a_{α}^{i}(x, s) = \frac{∂u_{h}}{∂x_{i}} · \frac{1}{|Du_{h}|^{2}} = \frac{4a_{h} sin (b_{h}x_{i})}{b_{h} ∑_{j=1}^{n} sin^{2} (b_{h}x_{j})} .

We have

|a_{α}(x, s)| = \frac{4a_{h}}{b_{h} \{∑_{j=1}^{n} sin^{2} (b_{h}x_{j})\}^{1/2}} .  (34)

We also define a_{α}(x, 0) by continuity

a_{α}(x, s) = 0, if x ∈ [0, 2π]^{n} and s = 0;

to this aim, since u_{h}(x) → 0 as h → +∞, we impose the condition

max \{|a_{α}(x, s)| : x ∈ [0, 2π]^{n} − E_{h} , s = u_{h}(x)\} = \frac{4a_{h}}{b_{h}λ_{h}} → 0 , as h → +∞.

(35)

At this stage the vector-valued function a_{α}(x, s) has been defined as a continuous function on a closed subset of \overline{Ω} × ℝ. In Step 2 we will extend it to the full \overline{Ω} × ℝ.

Step 2 (extension of a_{α} to \overline{Ω}×ℝ): fixed h ∈ ℕ, the vector-valued function a_{α}(x, s) has been defined in Step 1 at the points (x, s) ∈ \overline{Ω} × ℝ related by the conditions x ∈ [0, 2π]^{n} − E_{h} and s = u_{h}(x) (the function a_{α}(x, s) has been also defined for s = 0 with the zero value). In fact, for every fixed h ∈ ℕ, a_{α}(x, s) has been defined at the points (x, s) of the set R_{h} ⊂ \overline{Ω} × ℝ (a subset of the graph of u_{h}) given by

R_{h} = {(x, s) : x ∈ [0, 2π]^{n} − E_{h}, s = u_{h}(x)} .

Recalling the analytic expression of u_{h} in (27), the set R_{h} is contained in the hyper-rectangle

\{(x, s) : x ∈ [0, 2π]^{n} , \frac{3}{4} · \frac{n}{a_{h}} < s < \frac{5}{4} · \frac{n}{a_{h}} \} .

We will extend a_{α}(x, s) to the larger hyper-rectangle R′_{h} ⊂ \overline{Ω} × ℝ, given by

R′_{h} = \{(x, s) : x ∈ [0, 2π]^{n} , \frac{1}{2} · \frac{n}{a_{h}} ≤ s ≤ \frac{3}{2} · \frac{n}{a_{h}} \} .

First we define a_{α}(x, s) = 0 when s = \frac{n}{2a_{h}} and s = \frac{3n}{2a_{h}} , so that a_{α} will be continuously defined passing from an hyper-rectangle to an other; in fact we extend a_{α} also equal to zero out of the union ∪_{h}R′_{h}. Note that, passing possibly to a subsequence, we can assume that R′_{h} ∩ R′_{k} = if h ≠ k.

In order to estimate the oscillation |a_{α}(x_{1}, s_{1}) − a_{α}(x_{2}, s_{2})| when (x_{1}, s_{1}), (x_{2}, s_{2}) vary in R′_{h}, we first consider (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ R_{h} and we prove the following Lipschitz estimate (with constant depending on h)

|a_{α}(x_{1}, s_{1}) − a_{α}(x_{2}, s_{2})| ≤ \frac{16n · a_{h}}{λ_{h}^{4}} · |x_{1} − x_{2}| .  (36)

In fact, under the conditions x_{1}, x_{2} ∈ [0, 2π]^{n} − E_{h}, s_{1} = u_{h}(x_{1}), s_{2} = u_{h}(x_{2}) and with the notations a_{α}(x, s) = (a_{α}^{i}(x, s))_{i=1}^{n} , x_{1} = (x_{1}^{i})_{i=1}^{n}, x_{2} = (x_{2}^{i})_{i=1}^{n}, for every i = 1, 2, …, n we have

|a_{α}^{i}(x_{1}, s_{1}) − a_{α}^{i}(x_{2}, s_{2})| = \frac{4a_{h}}{b_{h}} |\begin{matrix}sin & (b_{h}x_{1}^{i}) \\ ∑ & {n \atop j=1} & sin & 2 & (b_{h}x_{1}^{j}) \end{matrix} − \begin{matrix}sin & (b_{h}x_{2}^{i}) \\ ∑ & {n \atop j=1} & sin & 2 & (b_{h}x_{2}^{j}) \end{matrix}|

≤ \frac{4a_{h}}{b_{h} · ∑_{j=1}^{n} sin^{2} (b_{h}x_{1}^{j})} |sin (b_{h}x_{1}^{i}) − sin (b_{h}x_{2}^{i})|

+\frac{4a_{h} · |sin (b_{h}x_{2}^{i})|}{b_{h} ∑_{j=1}^{n} sin^{2} (b_{h}x_{1}^{j}) · ∑_{j=1}^{n} sin^{2} (b_{h}x_{2}^{j})} |∑_{j=1}^{n} \{sin^{2} (b_{h}x_{1}^{j}) − sin^{2} (b_{h}x_{2}^{j})\}| .

Since x_{1}, x_{2} ∉ E_{h}, we also have ∑_{j=1}^{n} sin^{2}(b_{h}x_{1}^{j}) ≥ λ_{h}^{2} and ∑_{j=1}^{n} sin^{2}(b_{h}x_{2}^{j}) ≥ λ_{h}^{2}; therefore, by the Lipschitz continuity of the sinus function, we obtain

|a_{α}^{i}(x_{1}, s_{1}) − a_{α}^{i}(x_{2}, s_{2})| ≤ \frac{4a_{h}}{b_{h}λ_{h}^{2}} · b_{h} |x_{1}^{i} − x_{2}^{i}| + \frac{4a_{h}}{b_{h}λ_{h}^{4}} · 2b_{h} ∑_{j=1}^{n} |x_{1}^{j} − x_{2}^{j}| .

Since this estimate holds for every i = 1, 2, …, n, for the modulus of the vector field a_{α}(x_{1}, s) − a_{α}(x_{2}, s) we obtain

|a_{α}(x_{1}, s_{1}) − a_{α}(x_{2}, s_{2})| ≤ \frac{4a_{h}}{λ_{h}^{2}} (1 + \frac{2n}{λ_{h}^{2}} ) |x_{1} − x_{2}| .

The sequence λ_{h} converges to zero as h → +∞; therefore, as h is sufficiently large, we get the proof of (36). Of course (36) also gives

|a_{α}(x_{1}, s_{1}) − a_{α}(x_{2}, s_{2})| ≤ \frac{16 n · a_{h}}{λ_{h}^{4}} · (|x_{1} − x_{2}| + |s_{1} − s_{2}|) ,  (37)

for every h ∈ ℕ and for every (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ R_{h} with s_{1} = u_{h}(x_{1}), s_{2} = u_{h}(x_{2}).

Recalling (34), we have the bound

max \{|a_{α}(x, s)| : (x, s) ∈ R_{h}\} = \frac{4a_{h}}{b_{h}λ_{h}} ;

therefore, if (x_{1}, s_{1}) ∈ R_{h} and (x_{2}, s_{2}) ∈ R′_{h} with either s = \frac{n}{2a_{h}} or s = \frac{3n}{2a_{h}} , we deduce that

|a_{α}(x_{1}, s_{1}) − a_{α}(x_{2}, s_{2})| = |a_{α}(x_{1}, s_{1})| ≤ \frac{4a_{h}}{b_{h}λ_{h}}

and, since |s_{1} − s_{2}| ≥ \frac{1}{4} · \frac{n}{a_{h}} ,

|a_{α}(x_{1}, s_{1}) − a_{α}(x_{2}, s_{2})| ≤ \frac{16 a_{h}^{2}}{nb_{h}λ_{h}} · |s_{1} − s_{2}| ≤ \frac{16 a_{h}^{2}}{nb_{h}λ_{h}} · (|x_{1} − x_{2}| + |s_{1} − s_{2}|) .

By (35) we have \frac{a_{h}}{b_{h}λ_{h}} → 0 as h → +∞; therefore \frac{a_{h}}{b_{h}} λ_{h}^{3} → 0 too, which implies

\frac{a_{h}^{2}}{nb_{h}λ_{h}} ≤ \frac{n · a_{h}}{λ_{h}^{4}}

for every h sufficiently large. This proves that the Lipschitz estimate (37) (with constant depending on h) holds at every (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ R′_{h} where a_{α}(x, s) has been already defined.

By using Kirszbraun theorem (see Theorem 2.10.43 in Federer [9]) for the vector-valued function a_{α}, or in a simpler way by applying Mac Shane lemma to each component of a_{α}(x, s) = (a_{α}^{i}(x, s))_{i=1}^{n}, we can extend it to the hyperrectangle R′_{h} with the same Lipschitz constant as in (37), or, in case of extension of every component separately, with the same Lipschitz constant times \sqrt{n}. That is we have

|a_{α}(x_{1}, s_{1}) − a_{α}(x_{2}, s_{2})| ≤ \frac{16 n\sqrt{n} a_{h}}{λ_{h}^{4}} · (|x_{1} − x_{2}| + |s_{1} − s_{2}|) , (38)

for every h ∈ ℕ and for every (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ R′_{h}. Moreover, by truncating each component a_{α}^{i}, we can assume that the following bound holds

max \{|a_{α}(x, s)| : (x, s) ∈ R′_{h}\} = \frac{4\sqrt{n} a_{h}}{b_{h}λ_{h}} ,  (39)

for every h ∈ ℕ and for every (x, s) ∈ R′_{h}.

Step 3 (Hölder continuity of a_{α}): now the parameter α enters. To test Hölder continuity of a_{α}(x, s) with respect to x we fix h ∈ ℕ and \frac{n}{2a_{h}} ≤ s ≤ \frac{3n}{2a_{h}} and we estimate

sup \{\frac{|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h}\} .  (40)

Let t > 0 be a new real parameter that we will choose later. We estimate the supremum in (40) separately for |x_{1} − x_{2}| ≥ t and for |x_{1} − x_{2}| ≤ t.

Under the further condition |x_{1} − x_{2}| ≥ t, the supremum in (40) can be estimate by computing separately the maximum value of the numerator and the minimum value of the denominator. By (39) we have

max {|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)| : (x_{1}, s), (x_{2}, s) ∈ R′_{h}}

≤ 2 max \{|a_{α}(x, s)| : (x, s) ∈ R′_{h}\} = \frac{8\sqrt{n} a_{h}}{b_{h}λ_{h}} .

For the same s-values, since |x_{1} − x_{2}| ≥ t, we obtain

sup \{\frac{|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h} |x_{1} − x_{2}| ≥ t\}

≤ \frac{8\sqrt{n} a_{h}}{b_{h}λ_{h} t^{α}} .  (41)

While, if |x_{1} − x_{2}| ≤ t, we use the Lipschitz estimate (38) (with constant depending on h) with s_{1} = s_{2} ≡ s

|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)| ≤ \frac{16 n\sqrt{n} a_{h}}{λ_{h}^{4}} · |x_{1} − x_{2}|

and we obtain

sup \{\frac{|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h} |x_{1} − x_{2}| ≤ t\}

≤ \frac{16 n\sqrt{n} a_{h}}{λ_{h}^{4}} · t^{1−α} .  (42)

From (41) and (42) we deduce that

sup \{\frac{|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : (x_{1}, s), (x_{2}, s) ∈ R′_{h}\}

≤ \frac{16 n\sqrt{n} a_{h}}{λ_{h}} · max \{ \frac{1}{b_{h} t^{α}} ; \frac{t^{1−α}}{λ_{h}^{3}} \} .  (43)

The above inequality is valid for every t > 0. We consider the minimum of the right hand side with respect to t > 0, which is assumed when \frac{1}{b_{h} t^{α}} = \frac{t^{1−α}}{λ_{h}^{3}} , i.e., when t = \frac{λ_{h}^{3}}{b_{h}}.We obtain that the Hölder quotient in the left hand side of (43) is less than or equal to

\frac{16 n\sqrt{n} a_{h}}{λ_{h}b_{h} t^{α}} = \frac{16 n\sqrt{n} a_{h}}{(b_{h})^{1−α} (λ_{h})^{1+3α}} .  (44)

Previously we estimated the Hölder continuity with respect to x of a_{α}(x, s) in R′_{h}, for every fixed h ∈ ℕ. Thus, to obtain the Hölder continuity of a_{α}(x, s) with respect to x, with (x, s) ∈ \overline{Ω} × ℝ, we impose the further condition that the sequence in (44) remains bounded, i.e., that there exists L > 0 such that

\frac{a_{h}}{(b_{h})^{1−α} (λ_{h})^{1+3α}} ≤ L , ∀ h ∈ ℕ.  (45)

Step 4 (lower semicontinuity test): let us prove that

lim_{h→+∞} ∫_{Ω} f (x, u_{h}, Du_{h}) dx = 0 .  (46)

By the definition f_{α}(x, s, ξ) = |(a_{α}(x, s), ξ) − 1| and by (33), for every h ∈ ℕ we obtain

f (x, u_{h}, Du_{h}) = |(a_{α}(x, u_{h}), Du_{h}) − 1| = 0 , ∀ x ∈ [0, 2π]^{n} − E_{h} .

Thus, since |Du_{h}| < \frac{b_{h}λ_{h}}{4a_{h}} for x ∈ E_{h} (see (32)) and meas (E_{h}) < 8^{n} (λ_{h})^{n} (see (31)), we obtain (46); in fact, if we denote by M > 0 a bound for a_{α}(x, s) in \overline{Ω} × ℝ, we have

∫_{Ω} f (x, u_{h}, Du_{h}) dx = ∫_{E_{h}} f (x, u_{h}, Du_{h}) dx

≤ ∫_{E_{h}} \{M |Du_{h}| + 1\} dx ≤ \{\frac{M}{4} \frac{b_{h}λ_{h}}{a_{h}} + 1\} 8^{n} (λ_{h})^{n} ,  (47)

which converges to zero as h → +∞, if we assume that \frac{b_{h}}{a_{h}} (λ_{h})^{n+1} → 0 as h → +∞.

Step 5 (compatibility conditions; i.e., necessary conditions): looking above, we required the following limit relations (see in particular (35), (45) and (47))

\{ \begin{matrix}a_{h} → +∞ , b_{h} → +∞ , λ_{h} → 0 , \\ \frac{a_{h}}{b_{h}λ_{h}} → 0 , \frac{b_{h}}{a_{h}} (λ_{h})^{n+1} → 0 , \\ \frac{a_{h}}{(b_{h})^{1−α}(λ_{h})^{1+3α}} ≤ L , ∀ h ∈ ℕ , \end{matrix}

which, since

\frac{a_{h}}{(b_{h})^{1−α} (λ_{h})^{1+3α}} = \frac{a_{h}}{b_{h}λ_{h}} · (b_{h})^{α} · (λ_{h})^{−3α} ,

can be reduced to

\{ \begin{matrix}a_{h} → +∞ , b_{h} → +∞ , λ_{h} → 0 , \\ \frac{b_{h}}{a_{h}} (λ_{h})^{n+1} → 0 , \\ \frac{a_{h}}{(b_{h})^{1−α}(λ_{h})^{1+3α}} ≤ L , ∀ h ∈ ℕ . \end{matrix}  (48)

Then, from the identity

\frac{a_{h}}{(b_{h})^{1−α} (λ_{h})^{1+3α}} = \{\frac{b_{h}}{a_{h}} (λ_{h})^{n+1}\}^{α−1} · (a_{h})^{α} (λ_{h})^{n(1−α)−4α} ,

we find out the following compatibility condition

\{ \begin{matrix}\{\frac{b_{h}}{a_{h}} (λ_{h})^{n+1}\}^{α−1} → +∞ \\ (a_{h})^{α} → +∞ \\ \frac{a_{h}}{(b_{h})^{1−α}(λ_{h})^{1+3α}} ≤ L \end{matrix} ⟹ (λ_{h})^{n(1−α)−4α} → 0 ,

and, since λ_{h} → 0, we must have that n(1 − α) − 4α > 0, i.e., (28) is a necessary condition to let this construction work.

Step 6 (sufficient conditions): it remains to exhibit sequences of real parameters which satisfy the limit relations (48). Fixed α ∈ (0, 1) we consider n > \frac{4α}{1−α} so that α < \frac{n}{n+4} . Then there exists β ∈ ℕ (β > 1) large enough so that

α < \frac{n(β − 1)}{(n + 4) β − 3} .  (49)

By a simple computation we can see that inequality (49) is equivalent to

\frac{β − 1}{n + 1} < \frac{β (1 − α) − 1}{1 + 3α} .  (50)

We are ready to choose

a_{h} = 2^{h}, b_{h} = 2^{βh}, λ_{h} = 2^{−γh},

with \frac{β−1}{n+1} < γ ≤ \frac{β(1−α)−1}{1+3α} , a possibility that we can take since (50) holds. Then we can verify that all the conditions in (48) are satisfied. In fact a_{h}, b_{h} → +∞, λ_{h} → 0, and

\frac{b_{h}}{a_{h}} (λ_{h})^{n+1} = 2^{h[β−1−γ(n+1)]} → 0 ,

since γ > \frac{β−1}{n+1}. Finally

\frac{a_{h}}{(b_{h})^{1−α} (λ_{h})^{1+3α}} = 2^{h[1−β(1−α)+γ(1+3α)]} ≤ 1, ∀ h ∈ ℕ ,

since γ ≤ \frac{β(1−α)−1}{1+3α} .

Step 7 (the case n = 1): in the one-dimensional case we can go faster. In fact to obtain Hölder continuity of a_{α}(x, s) with respect to x we can estimate the Hölder quotient and we can compute separately the maximum value of the numerator and the minimum value of the denominator, as follows:

max {|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)| : x_{1}, x_{2} ∈ [0, 2π] − E_{h} , s = u_{h}(x_{1}) = u_{h}(x_{2})}

≤ 2 max \{|a_{α}(x, s)| : x ∈ [0, 2π] − E_{h} , s = u_{h}(x)\} = \frac{8a_{h}}{b_{h}λ_{h}} .

The minimum value of |x_{1} − x_{2}| is equal to the measure of E_{h}^{1} divided by the number of connected components of E_{h}^{1}, i.e., |x_{1} − x_{2}| = meas (E_{h}^{1}) /(2b_{h}) . By (30) we obtain

min {|x_{1} − x_{2}| : x_{1}, x_{2} ∈ [0, 2π] − E_{h} , s = u_{h}(x_{1}) = u_{h}(x_{2})}

= \frac{meas (E_{h}^{1})}{2b_{h}} < \frac{4λ_{h}}{b_{h}} .

Therefore, when n = 1, we have

sup \{\frac{|a_{α}(x_{1}, s) − a_{α}(x_{2}, s)|}{|x_{1} − x_{2}|^{α}} : x_{1}, x_{2} ∈ [0, 2π] − E_{h} , s = u_{h}(x_{1}) = u_{h}(x_{2})\}

≤ \frac{8a_{h}}{b_{h}λ_{h}} · (\frac{4λ_{h}}{b_{h}} )^{−α} = \frac{2^{3−2α}a_{h}}{(b_{h})^{1−α} (λ_{h})^{1+α}} .

In order to obtain the Hölder continuity of a_{α}(x, s) with respect to x, we impose the further condition that there exists L > 0 such that

\frac{a_{h}}{(b_{h})^{1−α} (λ_{h})^{1+α}} ≤ L , ∀ h ∈ ℕ.

Following this estimate, with similar computations as in the step 2 above, we can extend the function a_{α}(x, s) to \overline{Ω} × ℝ (or, in a simpler way, we could extend it linearly). As in Step 5, we obtain the compatibility conditions

\{ \begin{matrix}a_{h} → +∞ , b_{h} → +∞ , λ_{h} → 0 , \\ \frac{a_{h}}{b_{h}λ_{h}} → 0 , \frac{b_{h}}{a_{h}} (λ_{h})^{2} → 0 , \\ \frac{a_{h}}{(b_{h})^{1−α}(λ_{h})^{1+α}} ≤ L , ∀ h ∈ ℕ , \end{matrix}

(note the exponent in (λ_{h})^{1+α} instead of (λ_{h})^{1+3α}), which gives, as in Steps 5 and 6, the constraint α < \frac{1}{3} for n = 1.

5 Some other examples

In this section first we show in Example 12 that the local Lipschitz continuity of f(x, s, ξ) with respect to x ∈ Ω, although sufficient for the lower semicontinuity of F (u, Ω) in L_{loc}^{1} in the scalar case, as proved by our Theorem 6, it is not sufficient in the vector-valued case, i.e., for applications u : Ω ⊂ ℝ^{n} → ℝ^{m}, when m > 1. With Example 13 we show that also Lemma 8 cannot be extended to the vectorial setting. By Example 16 we emphasize the role of lower semicontinuity of f with respect to x ∈ Ω in the case of linear growth of f(x, s, ξ) as |ξ| → +∞, while in Example 14 we show that, however, neither continuity, nor even lower semicontinuity, of f(x, s, ξ) with respect to x ∈ Ω are necessary in the case of superlinear growth, also if the usual coercivity condition f(x, s, ξ) ≥ const |ξ|^{p}, for some p > 1, is not satisfied. We will start from this example to formulate (see Proposition 17) at the end of this section a sufficient condition for lower semicontinuity of F (u, Ω) in L_{loc}^{1} .

We can give all the examples below in the one dimensional case n = 1.

Eisen [8] showed with an example that Theorem 1(c) is false in the vectorial case. The same example shows that also Theorem 6 and Corollary 7 do not hold in the vectorial case. We recall this example, related to an integrand f independent of the variable x.

Example 12 Let Ω be the open interval (0, 1). Let us consider the function f : Ω × ℝ^{2}×ℝ^{2}→ ℝ defined by f(x, s_{1}, s_{2}, ξ_{1}, ξ_{2}) = (s_{1}ξ_{2})^{2}; thus, and for all u = (u_{1}, u_{2}) ∈ W^{1,1} (Ω, ℝ^{2}), the functional F is given by

F (u, Ω) = ∫_{Ω} (u_{1} · u′_{2})^{2} dx .

Then there exists a sequence u_{h} = (u_{1,h}, u_{2,h}) : Ω → ℝ^{2} which converges to a function u ∈ W^{1,1} (Ω, ℝ^{2}) in the strong topology of L^{1} (Ω, ℝ^{2}), such that

F (u_{h}, Ω) = 0, ∀ h ∈ ℕ; F (u, Ω) = 1.

Proof. Let u_{h} : Ω → ℝ^{2} be the sequence defined by

u_{1,h}(x) = \{ \begin{matrix}0 if x ∈ (\frac{m}{h}, \frac{m}{h} + 2^{−h}] \\ 2^{h+2} (x − \frac{m}{h} − 2^{−h}) if x ∈ (\frac{m}{h} + 2^{−h}, \frac{m}{h} + 2^{−h} + 2^{−h−2}] \\ 1 if x ∈ (\frac{m}{h} + 2^{−h} + 2^{−h−2}, \frac{m+1}{h} − 2^{−h−2}] \\ 1 − 2^{h+2} (x − \frac{m+1}{h} + 2^{−h−2}) if x ∈ (\frac{m+1}{h} − 2^{−h−2}, \frac{m+1}{h}] \end{matrix} ,

u_{2,h}(x) = \{ \begin{matrix}\frac{m}{h} + \frac{2^{h}}{h} (x − \frac{m}{h}) if x ∈ (\frac{m}{h}, \frac{m}{h} + 2^{−h}] \\ \frac{m+1}{h} if x ∈ (\frac{m}{h} + 2^{−h}, \frac{m+1}{h}] \end{matrix} ,

where m = 0, …, (h−1). Then u_{h} is Lipschitz continuous in (0, 1) for all h ∈ ℕ. With a simple calculation we get

u′_{2,h}(x) = \{ \begin{matrix}\frac{2^{h}}{h} if x ∈ (\frac{m}{h}, \frac{m}{h} + 2^{−h}) \\ 0 if x ∈ (\frac{m}{h} + 2^{−h}, \frac{m+1}{h}) \end{matrix} ,

thus u_{1,h}(x) · u′_{2,h}(x) = 0 for almost every x ∈ (0, 1). If we denote by u_{1}(x) = 1, u_{2}(x) = x, we have u_{h} → u = (1, x) in L^{1} ((0, 1), ℝ^{2}); in fact

∫_{0}^{1} |u_{1,h}(x) − 1| dx = \frac{h − 1}{2^{h−2}} → 0 , sup_{x∈(0,1)} \{|x − u_{2,h}(x)|\} ≤ \frac{1}{h} − \frac{1}{2^{h}} → 0 .

Finally the lower semicontinuity of F does not hold, since F (u_{h}, Ω) = 0 for every h ∈ ℕ, while F (u, Ω) = 1.

With a similar computation as in the previous example, following Eisen [8], we can show that also Lemma 8 cannot be extended to the vector-valued setting.

Example 13 Let Ω be the open interval (0, 1). Let us consider the function f : Ω × ℝ^{2}×ℝ^{2}→ ℝ defined by

f(x, s_{1}, s_{2}, ξ_{1}, ξ_{2}) = a(x) · b (\sqrt{s_{1}^{2} + s_{2}^{2}}) s_{1}^{2} · c(ξ_{2}) ,

where a(x) is a Lipschitz continuous function with compact support in (0, 1), not identically equal to zero and such that 0 ≤ a(x) ≤ 1; b : ℝ → [0, 1] is a Lipschitz continuous function with compact support and such that b(t) = 1 for every t ∈ [1, 2]; finally c : ℝ → ℝ is defined by

c(t) = \{ \begin{matrix}\frac{1}{2}t^{2}  if |t| ≤ 1 \\ |t| − \frac{1}{2}  if |t| > 1 \end{matrix} .

Then, the function f satisfies the assumption of Lemma 8 while, on the same sequence u_{h} : Ω → ℝ^{2} of the previous example, the integral ∫_{Ω} f(x, u_{1}, u_{2}, u′_{1}, u′_{2}) dx is not lower semicontinuous.

If there exists p > 1 such that f(x, s, ξ) ≥ const |ξ|^{p}, for some positive constant (i.e., a coercivity condition holds for f), then it is clear that the lower semicontinuity in W_{loc}^{1,p}(Ω) of the integral functional F (u, Ω) = ∫_{Ω} f (x, u, Du) dx with respect to the strong convergence in L_{loc}^{1}(Ω) is equivalent to the weak-W_{loc}^{1,p}(Ω) lower semicontinuity of F (u, Ω). Therefore, in this case the lower semicontinuity in L_{loc}^{1}(Ω) holds under the only assumption that f(x, s, ξ) is a Carathéodory function, i.e., f is measurable with respect to x ∈ Ω and continuous in (s, ξ) ∈ ℝ×ℝ^{n}, and of course f is also convex with respect to ξ ∈ ℝ^{n}. By the next examples 14 and 15, following [14], we will show that neither continuity nor even lower semicontinuity of f(x, s, ξ) with respect to x ∈ Ω are necessary in the case of superlinear growth p > 1; this fact may happen also if the usual coercivity condition is not satisfied.

Example 14 Let Ω be the open interval (0, 1). Let f(x, ξ) = a(x) |ξ|^{p} for some p > 1, where a(x) is a bounded measurable function in (0, 1), with a(x) ≥ 0 for almost every x ∈ (0, 1). Then the maximum lower semicontinuous (in the strong norm topology of L_{loc}^{1}(Ω)) functional \overline{F}_{p} (u, Ω), less than or equal to F (u, Ω) = ∫_{Ω} a(x) |u′|^{p} dx, is given by

\overline{F}_{p} (u, Ω) = ∫_{Ω} b_{p}(x) |u′|^{p} dx ,  (51)

for every u ∈ W_{loc}^{1,p}(Ω), where b_{p} is the bounded measurable function defined in (0, 1) by

b_{p}(x) = lim inf_{ε→0^{+}} \{\frac{1}{2ε} ∫_{x−ε}^{x+ε} a(t)^{− \frac{1}{p−1}} dt\}^{−(p−1)} .  (52)

Moreover, for every p > 1, b_{p} satisfies the estimates

0 ≤ \overline{a}(x) ≤ b_{p}(x) ≤ a(x), a.e.x ∈ Ω,  (53)

where \overline{a} is the maximum lower semicontinuous function less than or equal to a in Ω. Thus in particular two sufficient conditions so that the integral

F (u, Ω) = ∫_{Ω} a(x) |u′|^{p} dx

is lower semicontinuous in W_{loc}^{1,p}(Ω) with respect to the strong norm topology of L_{loc}^{1}(Ω), are: (i) the coefficient a(x) is lower semicontinuous in Ω; (ii) a^{−1/(p−1)} ∈ L_{loc}^{1}(Ω).

Proof. The representation formulas (51), (52) have been established by Marcellini in [14], in the case p = 2. The proof for general p ∈ (1, +∞) is similar.

Let us prove (53). Since \overline{F}_{p} (u, Ω) ≤ F (u, Ω) for every u ∈ W_{loc}^{1,p}(Ω), then b_{p}(x) ≤ a(x) for almost every x ∈ Ω. If c(x) is a lower semicontinuous function less than or equal to a in Ω, then, for every x ∈ Ω and every μ > 0 there exists δ > 0 such that

c(x) ≤ c(t) + μ ≤ a(t) + μ , a.e.t ∈ Ω ∩ (x − ε, x + ε) , ∀ ε ≤ δ.

Again, for 0 < ε ≤ δ, we deduce that

∫_{x−ε}^{x+ε} a(t)^{− \frac{1}{p−1}} dt ≤ 2ε (c(x) − μ)^{− \frac{1}{p−1}}

and, from the definition (52) of b_{p},

b_{p}(x) = lim inf_{ε→0^{+}} \{\frac{1}{2ε} ∫_{x−ε}^{x+ε} a(t)^{− \frac{1}{p−1}} dt\}^{−(p−1)} ≥ c(x) − μ .

Thus b_{p}(x) ≥ c(x) and also b_{p}(x) ≥ \overline{a}(x) for almost every x ∈ Ω.

From the estimates in (53) we obtain the conclusion (i), i.e., that the integral F (u, Ω) = ∫_{Ω} a(x) |u′|^{p} dx, being equal to \overline{F}_{p} (u, Ω), is lower semicontinuous in L_{loc}^{1} if the coefficient a is lower semicontinuous. Finally, if a^{− \frac{1}{p−1}} ∈ L_{loc}^{1}(Ω), then, by using the Lebesgue points of this function, we have b(x) = a(x) for almost every x ∈ (0, 1); thus again F = \overline{F}_{p}, which proves (ii).

As in [14], we give below an explicit application of Example 14. In particular, given p > 1 and the integral F (u, Ω) = ∫_{Ω} a_{p,s}(x) |u′|^{p} dx, where the nonnegative measurable function a_{p,s} is defined below in (54), we show that there exist some values of the real parameter s such that F (u, Ω) is not lower semicontinuous in the strong norm topology of L_{loc}^{1}(Ω).

Example 15 Let Ω be the open interval (0, 1). Let us denote by {x_{i}}_{i∈ℕ} the set of rational numbers in (0, 1) ordered in a sequence and let s be a real parameter. Let f(x, ξ) = a_{p,s}(x) |ξ|^{p} for some p > 1, where a_{p,s}(x) is the bounded measurable nonnegative function in (0, 1) defined by

a_{p,s}(x) = \frac{1}{(1 + ∑_{i=1}^{∞} 2^{−i} |x − x_{i}|^{−s})^{p−1}} ,  x ∈ (0, 1)  (54)

if the denominator is finite, otherwise we pose a_{p,s}(x) = 0. Then, for every s ∈ ℝ, the measurable function a_{p,s}(x) is not identically equal to zero (more precisely, the set {x ∈ Ω : a_{p,s}(x) ≠ 0} has positive measure), while the integral F (u, Ω) = ∫_{Ω} a_{p,s}(x) |u′|^{p} dx is lower semicontinuous in W_{loc}^{1,p}(Ω) with respect to the strong norm topology of L_{loc}^{1}(Ω) if and only if s < 1.

Proof. With reference to the representation formulas (51), (52), we will prove that

b_{p,s}(x) = lim inf_{ε→0^{+}} \{\frac{1}{2ε} ∫_{x−ε}^{x+ε} a_{p,s}(t)^{− \frac{1}{p−1}} dt\}^{−(p−1)} = \{ \begin{matrix}a_{p,s}(x) if s < 1 \\ 0 if s ≥ 1 \end{matrix} .

In fact (for simplicity of notations we integrate over the whole interval (0, 1))

∫_{0}^{1} a_{p,s}(x)^{− \frac{1}{p−1}} dx = ∫_{0}^{1} \{1 + ∑_{i=1}^{∞} 2^{−i} |x − x_{i}|^{−s}\} dx

= 1 + ∑_{i=1}^{∞} 2^{−i} ∫_{0}^{1} |x − x_{i}|^{−s} dx ≤ 1 + \frac{1}{1 − s} < +∞

if s < 1 (and, in this case b_{p,s}(x) = a_{p,s}(x) for almost every x in (0, 1)), otherwise the integral is equal to +∞, when computed on any subinterval (x_{0} − ε, x_{0} + ε) too (and, in this case b_{p,s}(x) = 0 for almost every x in (0, 1)).

It remains to show that the set {x ∈ Ω : a_{p,s}(x) ≠ 0} has positive measure. To this aim we observe that, if s < 1, by the above condition

∫_{0}^{1} a_{p,s}(x)^{− \frac{1}{p−1}} dx < +∞

we deduce that a_{p,s}(x)^{− \frac{1}{p−1}} is finite almost everywhere in Ω; therefore a_{p,s}(x) is different from zero almost everywhere in Ω. Otherwise, if s ≥ 1, we compute similarly

∫_{0}^{1} a_{p,s}(x)^{− \frac{1}{2s(p−1)}} dx = ∫_{0}^{1} \{1 + ∑_{i=1}^{∞} 2^{−i} |x − x_{i}|^{−s}\}^{\frac{1}{2s}} dx

≤ ∫_{0}^{1} \{1 + ∑_{i=1}^{∞} (2^{−i} |x − x_{i}|^{−s})^{\frac{1}{2s}} \} dx ≤ 1 + ∑_{i=1}^{∞} 2^{\frac{−i}{2s}} ∫_{0}^{1} |x − x_{i}|^{− \frac{1}{2}} dx < +∞;

thus again a_{p,s}(x)^{− \frac{1}{2s(p−1)}} is finite almost everywhere and a_{p,s}(x) is different from zero almost everywhere in Ω.

In the next example we consider the limit case p = 1. Example 16 is due to Fusco [12]. We emphasize here that Example 16 can be considered as a passage to the limit from the case p > 1 in the formulas of Example 14. In fact, analogously to the well known limit relation lim_{r→+∞} ∥v∥_{L^{r}(Ω)} = ∥v∥_{L^{∞}(Ω)} , when we replace v by 1/v , as well known we also have

lim_{r→+∞} ∥v^{−1}∥_{L^{r}(Ω)}^{−1} = inf {|v(x)| : x ∈ Ω} .

Therefore, from (52), we obtain the following representation formula for the maximum lower semicontinuous function \overline{a} less than or equal to a in Ω

\overline{a}(x) = lim_{ε→0^{+}} inf \{a(t) : t ∈ (x − ε, x + ε)\}  (55)

= lim_{ε→0^{+}} lim_{p→1^{+}} \{∫_{x−ε}^{x+ε} a(t)^{− \frac{1}{p−1}} dt\}^{−(p−1)}

= lim_{ε→0^{+}} lim_{p→1^{+}} \{\frac{1}{2ε} ∫_{x−ε}^{x+ε} a(t)^{− \frac{1}{p−1}} dt\}^{−(p−1)} .

As already said, the proof of the statement of the following Example 16 can be found in Fusco [12].

Example 16 Let Ω be the open interval (0, 1). Let f(x, ξ) = a(x) |ξ|, where a(x) is a bounded measurable function in (0, 1), with a(x) ≥ 0 for almost every x ∈ (0, 1). Then the maximum lower semicontinuous (in L_{loc}^{1}) functional \overline{F}_{1} (u, Ω), less than or equal to F (u, Ω) = ∫_{Ω} a(x) |u′| dx, is given by

\overline{F}_{1} (u, Ω) = ∫_{Ω} \overline{a}(x) |u′| dx ,  (56)

for every u ∈ W_{loc}^{1,1}(Ω), where \overline{a} is the maximum lower semicontinuous function less than or equal to a in Ω. Thus in particular the integral F (u, Ω) = ∫_{Ω} a(x) |u′| dx is lower semicontinuous in L_{loc}^{1}(Ω) if and only if the coefficient a is lower semicontinuous in Ω (that is, if the measurable function a is almost everywhere equal to a lower semicontinuous function in Ω).

Having in mind Examples 14 and 15, we give the following sufficient condition for lower semicontinuity of F (u, Ω) in L_{loc}^{1} . Here we go back to the general n-dimensional case, under the assumption that f(x, s, ξ) is a Carathéodory function, i.e., that f is measurable with respect to x ∈ Ω ⊂ ℝ^{n} and continuous with respect to (s, ξ) ∈ ℝ × ℝ^{n}. We mention explicitly that the following result holds in the vector-valued setting too.

We thank Giovanni Leoni, who pointed out to us an improvement of a previous version of the following Proposition 17.

Proposition 17 Assume that f(x, s, ξ) is a Carathéodory function, convex with respect to ξ, which satisfies the coercivity condition

f(x, s, ξ) ≥ a(x) |ξ|^{p} , a.e. x ∈ Ω, ∀ (s, ξ) ∈ ℝ × ℝ^{n},  (57)

for some p > 1, where a(x) is a measurable function in an open set Ω ⊂ ℝ^{n}, a(x) ≥ 0 for almost every x ∈ Ω, and such that

a^{− \frac{1}{p−1}} ∈ L_{loc}^{1}(Ω).  (58)

Then the integral F (u, Ω) = ∫_{Ω} f (x, u, Du) dx is lower semicontinuous in W_{loc}^{1,1}(Ω) with respect to the strong convergence in L_{loc}^{1}(Ω).

Proof. By Hölder inequality and by the coercivity assumption (57), for every open set Ω′ compactly contained in Ω, we have

∫_{Ω′} |Du| dx = ∫_{Ω′} a(x)^{1/p} |Du| · a(x)^{−1/p} dx

≤ \{∫_{Ω′} a(x) |Du|^{p} dx\}^{1/p} · \{∫_{Ω′} a(x)^{− \frac{1}{p} · \frac{p}{p−1}} dx\}^{(p−1)/p}

≤ \{∫_{Ω′} f (x, u, Du) dx\}^{1/p} · \{∫_{Ω′} a(x)^{− \frac{1}{p−1}} dx\}^{(p−1)/p} .

Let u_{h}, u ∈ W_{loc}^{1,1} (Ω) such that u_{h} → u in L_{loc}^{1} (Ω). Let us also assume that

lim inf_{h→+∞} F (u_{h}, Ω) = lim_{h→+∞} F (u_{h}, Ω) = C < +∞ .

Under such conditions Du_{h} is a sequence locally equi-integrable in Ω and so u_{h} weakly converges to u in W_{loc}^{1,1} (Ω); in fact a satisfies (58) and we have

∫_{Ω′} |Du_{h}| dx ≤ C^{1/p} · \{∫_{Ω′} a(x)^{− \frac{1}{p−1}} dx\}^{(p−1)/p} .

Therefore we can apply the original lower semicontinuity theorem by De Giorgi [6].

References

[1] Acerbi E., Buttazzo G., Fusco N.: Semicontinuity and relaxation for integrals depending on vector valued functions, Journal Math. Pures et Appl., 62 (1983), 371–387.

[2] Ambrosio L.: New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XL 11 (1987), 1-42.

[3] Dal Maso G.: Integral representation on BV (Ω) of Γ-limits of variational integrals, Manuscripta Math., 30 (1980), 387-416.

[4] De Cicco V.: A lower semicontinuity result for functionals defined on BV (Ω), Ricerche Mat., 39 (1990), 293-325.

[5] De Cicco V.: Lower semicontinuity for certain integral functionals on BV (Ω), Boll. Un. Mat. Ital., 5-B (1991), 291-313.

[6] De Giorgi E.: Teoremi di semicontinuità nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Roma, 1968-1969.

[7] De Giorgi E., Buttazzo G., Dal Maso G.: On the lower semicontinuity of certain integral functions. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur., Rend., 74 (1983), 274-282.

[8] Eisen G.: A counterexample for Some Lower Semicontinuity Results. Math. Z, 162 (1978), 241-243.

[9] Federer H., Geometric measure theory, Springer-Verlag, Berlin, 1969.

[10] Fonseca I., Leoni G.: On lower semicontinuity and relaxation, to appear in Proc. Roy. Soc. Edinburgh.

[11] Fonseca I., Leoni G.: Some remarks on lower semicontinuity, Indiana Univ. Math. J., 49 (2000), 617-635.

[12] Fusco N.: Dualità e semicontinuità per integrali del tipo dell’area, Rend. Accad. Sci. Fis. Mat. Napoli, 46 (1979), 81-90.

[13] Gori M.: I teoremi di semicontinuità di Serrin e loro estensioni, Degree Thesis in Mathematics, University of Firenze, July 2000.

[14] Marcellini P.: Some problems of semicontinuity and of convergence for integrals of the calculus of variations, Recent Methods in Nonlinear Analisys, De Giorgi et al. editors, Rome, Pitagora Ed., 1978, 205-221.

[15] Pauc C. Y.: La méthode métrique en calcul des variations, Paris, Hermann, 1941.

[16] Serrin J.: On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139-167.

[17] Tonelli L.: Sur la semi-continuité des integrales doubles du calcul des variations, Acta Math., 53 (1929), 325-346.