
Seminario di Calcolo delle
Variazioni & Equazioni alle Derivate Parziali

I seminari si tengono di norma di venerdì alle ore 14:30
nella Sala Conferenze "Franco Tricerri" del Dipartimento di
Matematica e Informatica "Ulisse Dini" (Viale Morgagni 67/A).
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A.A. 2017 / 2018
::: PROSSIMO SEMINARIO: :::
10 luglio 14:30 
James C. Robinson (Warwick UnVersity)

Unexpected behaviour for solutions of the heat equation on the whole
space


Abstract: (joint with Aníbal RodríguezBernal, Complutense Madrid)
We identify the class of (locally integrable) initial data is the
largest that allows the solution of the heat equation, posed on the
whole space, to be written in terms of the Gaussian heat kernel. A
positive solution exists on the time interval $[0,T]$ if and only if
$u_0\in L^1_{1/4T}$, where
$$
L^1_\epsilon=\{f\in L^1_{\rm loc}:\ \int_{R^d}
e^{\epsilonx^2}f(x)\,dx<\infty\}.
$$
We show that for such initial data, which can be large at infinity,
solutions can blow up in finite time, and can exhibit "wild
oscillations" as $t\to\infty$. In fact, these wild oscillations are
generic (in an appropriate sense).
:: Seminari in programma: ::
:: Seminari passati: ::
22 giugno 14:30 
Francesco Maggi (University of Texas at Austin)

Il teorema di Alexandrov rivisitato


Abstract: Si presenta una caratterizzazione delle sfere come unici insiemi di perimetro
finito a curvatura media distribuzionale costante. Lavoro in collaborazione con Matias
Delgadino (Imperial College London)
15 giugno 14:30 
Marco Caroccia (Universidade de Lisboa)

Perimeter with densities: an energy for epitaxial growth


Abstract: In this talk I will introduce a free energy, first studied by Ratz and
Vöigt, defined as the integration over the reduced boundary of a crystal
$E$ of a suitable psi(u(x)) where psi is a convex function and
u:\partial?* E \rightarrow \R_+ denotes the adatom density (the density
of atoms not attached to the crystal and free to move on the surface).
This energy leads the surface diffusion evolution in absence of the
elastic stress and for small adatom densities. We treat the variational
point of view, by analyzing the semicontinuity issues and the relaxad
functional.
8 giugno 14:30 
Cristina Trombetti (Università di Napoli Federico II)

Alcune disuguaglianze spettrali in forma ottimale


Abstract: In questo seminario racconterò alcuni risultati riguardanti stime ottimali per autovalori del Laplaciano (o del pLaplaciano) con varie condizioni al bordo.
lunedì 21 maggio 14:30 
Piotr Hajlasz (University of Pittsburgh)

Topologically nontrivial counterexamples to Sard's theorem


Abstract: We prove the following dichotomy: if n = 2, 3 and f in C^1 (S^{n+1}, S^n) is not homotopic to a constant map, then there is an open set Omega in S^{n+1} such that rank Df = n on Omega and f(Omega) is dense in S^n, while for
any n larger or equal than 4, there is a map f in C^1 (S^n+1, S^n ) that is not homotopic to
a constant map and such that rank Df is less than n everywhere. While the result
deals with mappings between spheres, the motivation comes from the
theory of mappings from Euclidean spaces into the Heisenberg groups H^n
, because the rank of the derivative of such a mapping is bounded by n.
In fact, our proofs are based on methods that have previously been
applied to study Lipschitz homotopy groups of the Heisenberg groups.
18 maggio 14:30 
Luca Bisconti (Università di Firenze)

Some recent results on a deconvolution model for the mean Boussinesq equations


Abstract: We consider a Large Eddy Simulation (LES) scheme for the approximation of the 3D Boussinesq equations. This model is based on Van Cittern's approximate deconvolution operators, applied to the filtered Boussinesq equations (Helmholtz filter). We show existence and uniqueness of regular weak solutions and prove convergence towards a weak solution of the filtered Boussinesq equations. In the 2D case, for the same model, we study the dynamics associated with regular weak solutions, proving the existence of an inertial manifold.
11 maggio 14:30 
Giovanni S. Alberti (Università di Genova)

Calderón's Inverse Problem with a Finite Number of Measurements


Abstract: In this talk I will discuss how ideas from applied harmonic analysis, in particular sampling theory and compressed sensing (CS), may be applied to inverse problems in PDEs. The focus will be on inverse boundary value problems for the conductivity and the Schrodinger equations, and I will give uniqueness and stability results, both in the linearized and in the nonlinear case. These results make use of a recent general theory of infinitedimensional CS for deterministic and nonisometric operators, which will be briefly surveyed. This is joint work with Matteo Santacesaria (University of Helsinki).
20 aprile 14:30 
Serena Matucci (Università di Firenze)

Ground State Solutions to nonlinear equations with pLaplacian


Abstract: We present some recent results obtained in collaboration with Zuzana Dosla (Masaryk University of Brno) about existence of radial ground state solutions for nonlinear equations with pLaplacian operator, both for the superhomogeneous and for the subhomogeneous case, in presence of a changing sign nonlinearity describing diffusion/absorption phenomena. The existence is proved outside a fixed ball. The resulting BVP problem for a nonlinear ordinary differential equation on a noncompact interval is studied by means a fixed point approach; a key role is played by properties of solutions of some auxiliary halflinear equations, the notion of disconjugacy, principal solutions and comparison theorems. Sufficient conditions are proved for the existence of positive solutions y decaying to zero, satisfying any initial conditions y(R)=c>0, both in case of negative initial derivative and in case of zero initial derivative.
13 aprile 14:30 
Giovanni Franzina (Università di Firenze)

Prescribed mean curvature in hyperbolic space.


Abstract: We present a joint work with Adriano Pisante and Marcello Ponsiglione (Sapienza University of Rome) dealing with a variation on AllenCahn equation in hyperbolic nspace. Our contribution includes asymptotic regularity results at infinity for entire solutions. The results rely on a Liouvilletype theorem inspired by Caffarelli, Gidas, and Spruck’s analysis of the multimeron solutions of YangMills Equation. In the sharp interface limit we provide solutions of the Plateau problem at infinity for hypersurfaces of prescribed mean curvature together with quantitative information about the contact angle with the data on the visual boundary.
6 aprile 14:30 
Guido de Philippis (SISSA)

Boundary Regularity for Mass Minimising currents


Abstract: In this talk I will present a first boundary regularity result for mass minimising currents in any dimension and codimension, and some of its consequences. In particular I will show that the regular points are dense in the boundary. Note that in this general setting, it was unknown even the existence of one regular point. This is a joint work with C. De Lellis, J. Hirsch and A. Massaccesi.
23 marzo 2018 14:30 e 15:30 
Camillo de Lellis (Universitat Zurich) 14:30

REGULARITY AND SINGULARITY OF GENERALIZED NAVIERSTOKES EQUATIONS

Norisuke Ioku (Ehime University) 15:30

REMARK ON A SOBOLEV TYPE INEQUALITY IN THE UNIT BALL


Abstract De Lellis: I will discuss the classical incompressible NavierStokes equations where
we substitute the classical Laplacian with a fractional one. When the exponent is
larger than 1, I will present a suitable extension of the CaffarelliKohnNirenberg
regularity theory. When the exponent is close to 0 I will illustrate how ``convex
integration'' techniques prove illposedness of Leray solutions.
Abstract Ioku: It is well known that the sharp constant of the Sobolev inequality in the
whole space is attained by the Talenti function, but not in bounded domains since a
scale invariance is breaking. In this talk, we introduce a scale invariant form of
the Sobolev inequality in the unit ball, and show that the sharp constant is attained
by a Talenti like function.
16 marzo 2018 14:30 
Sergio Vessella (Università di Firenze)

Stime quantitative di continuazione unica e problemi inversi


Abstract: Le stime quantitative di continuazione unica sono di importanza
fondamentale nello studio della stabilità dei problemi inversi e, nello stesso tempo, hanno interesse di per sé.
In questo seminario vorrei fornire una rapida panoramica sulla problematica e presentare
alcuni recenti risultati sulle stime quantitative di continuazione unica per problemi di
trasmissione per equazioni ellittiche e paraboliche.
9 marzo 14:30 
Berardo Ruffini (Université de Montpellier)

Some results on an isoperimetric model for charged liquid drops


Abstract: We consider an isoperimetric model, originally proposed by Lord
Rayleigh, aimed to describe the (lack of) equilibria of a liquid conducting
drop in presence of a charge on its surface. The resulting functional
contains an attracting term, usually modeled by the perimeter of the drop,
and a repulsiveterm depending on the amount of charge considered and the
electric capacity of the drop. We show that, quite surprisingly, the
resulting variational problem is ill posed. We then consider several
modification of it and we investigate existence, uniqueness and stability
issues about those problem. The talk is based on works in collaborations
with M. Goldman, C. Muratov and M. Novaga
23 febbraio 14:30 
Iwona Skrzypczak (IMPAN at Polish Academy of Sciences & MIMUW at University of
Warsaw)

Approximable solutions to measure data and L^1data elliptic problems in the Orlicz setting without growth restrictions


Abstract: We study approximable solutions to a general nonlinear elliptic Dirichlet
equation $div A(x,\nabla u)= \mu$ on a bounded Lipschitz domain in $\rn$ in the
OrliczSobolev space avoiding growth restrictions. The growth of~the~monotone vector
field $A$ is controlled by an $N$function $B$. We do not require any particular type
of growth condition of $B$ or its conjugate $\widetilde{B}$, so we deal with the problem
in a nonreflexive space.
When the problem involves measure data, we prove existence. For $L^1$data problems we
infer also uniqueness and regularity in the OrliczMarcinkiewicztype spaces. Joint work
with Anna ZatorskaGoldstein (MIMUW at University of Warsaw).
GIOVEDI 8 febbraio 14:30 
Sunra Mosconi (Università Catania)

Fine boundary regularity for nonlocal problems


Abstract: It is a classical theorem that functions having bounded
$p$Laplacian in a smooth domain are $C^{1,\alpha}$ up to the
boundary. We will discuss the possibility of deriving a similar
result for solutions of nonlocal nonlinear equations with bounded
righthand side. In this case reflection methods are bound to fail,
and actually we will show through examples that the boundary behavior
is necessarily worse than the interior one. We will next describe the
optimal regularity to be expected and give a brief sketch of its proof.
2 febbraio 14:30 
Jan Kristensen (University of Oxford)

On regularity of BV minimizers


Abstract:
We discuss a partial regularity result for minimizers of quasiconvex variational integrals
defined on the space of maps of bounded variation. The talk is based on joint work with Franz Gmeineder (Bonn).
15 dicembre 14:30 
Edouard Oudet (Université Grenoble Alpes and Centro di Ricerca Matematica E De Giorgi)

Numerical study of 1D optimal structure


Abstract: We focus our attention on shape optimization problems in which one dimensional connected objects are involved. Very old and classical problems in calculus of variation are of this kind: euclidean Steiner’s tree problem, optimal irrigation networks, cracks propagation, etc.
In a first part we quickly recall some previous work in collaboration with F. Santambrogio related to the functional relaxation of the irrigation cost. We establish a Γconvergence of Modica and Mortola’s type and illustrate its efficiency from a numerical point of view by computing optimal networks associated to simple sources/sinks configurations. We also present more evolved situations with non Dirac sinks in which a fractal behavior of the optimal network is expected.
In the second part of the talk we restrict our study to the euclidean Steiner’s tree problem. We recall recent numerical approach which have been developed the last five years to approximate optimal trees: partitioning formulation, relaxation with geodesic distance terms and energetic constraints. We describe the first results obtained in collaboration with A. Massaccesi and B. Velichkov to certify the optimality of a given tree. With our discrete parametrization of generalized calibration, we are able to recover the theoretical optimal matrix fields which certify the optimality of simple trees associated to the vertices of regular polygons.
Finally, we focus on the delicate problem of the identification of the optimal structure. Based ona recent approach obtained in collaboration with G. Orlandi and M. Bonafini, we describe the first convexification framework associated to the Euclidean Steiner tree problem which provide relevant tools from a numerical point of view
1 dicembre 14:30 
Maria Giovanna Mora (Università di Pavia)

The equilibrium measure for a nonlocal dislocation energy


Abstract: In this talk I will discuss the minimization problem for a nonlocal energy,
that describes
the interaction of a large number of dislocations in the plane. The interaction kernel is
given by
the sum of the Coulomb potential and of an anisotropic term, that makes the potential
nonradially
symmetric. The purely logarithmic potential has been studied in a variety of contexts
(GinzburgLandau vortices, Coulomb gases, random matrices, Fekete sets) and in this case
it is
well known that the equilibrium measure is given by the circle law. I will show that the
presence of
the anisotropy in the kernel changes dramatically the nature of the equilibrium measure,
which
turns out to be supported on the vertical axis and distributed according to Wigner?s
semicircle law.
This result is one of the few examples where the minimizer of a nonlocal energy is
explicitly computed
and it gives a positive answer to the conjecture that positive dislocations tend to
arrange themselves
in vertical walls. A few extension of this result will also be discussed.
24 novembre 14:30 
Enrico Valdinoci (Università di Milano)

Longrange phase transitions and minimal surfaces


Abstract: We discuss some recent results on nonlocal minimal surfaces and discuss
their connections with nonlocal phase transitions.
In particular, we will consider the "genuinely nonlocal regime" in which
the diffusion operator is of order less than 1 and present some rigidity
and symmetry results.
We also discuss the connections between the problems presented and related
models in crystal dislocation theory and water waves.
17 novembre 14:30 
Antoine Henrot (Université de Lorraine)

On two functionals invoving the maximum of the torsion function


Abstract: In this talk we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function.
More precisely , we consider T(D)/(M(D)D) and M(D)L_1(D), where D is a bounded
open set of R^N with finite Lebesgue measure D, M(D) denotes the maximum of the torsion function,
T(D) the torsion, and L_1(D) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.
3 novembre 14:30 E 15:30 
14h30 Guy Bouchitté (Université de Toulon)

Convex relaxation for a class
of free boundary problems
 15h30 Giuseppe Buttazzo (Università di Pisa)

Shape optimization under
uncertainty


Abstract: TBA
27 ottobre 14:30 
Eugenia Malinnikova (Norwegian University of Science and Technology (NTNU), Trondheim)

On propagation of smallness for solutions of elliptic PDEs


Abstract: We consider a solution of a second order uniformly elliptic equation in some domain. We show that if this solution is small on some set of positive measure than it is small on each compact subset of the domain. The result generalizes the three ball theorem, similar results for sets of positive measure were obtained earlier independently by Nadirashvili and by Vessella. We improve the propagation of smallness estimate and also consider sets of zero measure but codimension smaller than one. The final estimate depends on the measure (or Hausdorff content) of the initial set but not on its geometry. Similar inequalities are obtained for the gradients of solutions. The talk is based on a joint work with A. Logunov.
13 ottobre 14:30 E 15:30 
14H30 Piermarco Cannarsa (Università degli Studi di Roma "Tor Vergata")
Control of degenerate equations of evolution 
15H30 Cristina Pignotti (Università degli Studi dell'Aquila)
Stability estimates for evolution equations with time delay 

Abstract Cannarsa: The null controllability properties of degenerate parabolic operators in low space dimension, via either boundary or locally distributed controls, are by now fairly well understood if degeneracy occurs at the boundary of the space domain. The same is true for certain classes of parabolic equations associated with hypoelliptic operators on Cartesian products. Such results will be surveyed in this talk, in which special attention will be devoted to estimates for the blowup of the cost of control as degeneracy approaches the controllability threshold.
Abstract Pignotti [30 min]: We consider abstract semilinear evolution equations with a time lag.
We show that, if the C_0semigroup describing the linear (undelayed)
part of the model is exponentially stable, then the whole system keeps this
property when an appropriate smallness condition on the time delay feedback
or on the initial data is satisfied. Some examples illustrating our abstract
approach are presented.
18 settembre 14:30 
Carlos Perez (Basque Center for Applied Mathematics)

Singular Integrals and Ap weights: an expository
lecture


Abstract
Organizzatori: Chiara Bianchini, Matteo Focardi.
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