Frank Merle (Université de Cergy-Pontoise )
Asymptotics for critical nonlinear dispersive equations and Universality properties

We consider various examples of critical nonlinear partial differential equations which have the following common features: they are Hamiltonian, of dispersive nature, have a conservation law invariant by scaling, and have solutions of nonlinear type (their asymptotic behavior in time differs from the behavior of solutions of linear equations). The main questions concern the possible behaviors one can expect asymptotically in time. Are there many possibilities, or on the contrary very few universal behaviors depending on the type of initial data? We shall see that the asymptotic behavior of solutions starting with general or constrained initial data is related to very few special solutions of the equation. This will be illustrated through different examples related to classical problems.

Maria Giovanna Mora (Università di Pavia)
Quasistatic evolution models for thin plates in perfect plasticity

In this talk I shall discuss the rigorous derivation of a quasistatic evolution model for a thin plate in the framework of Prandtl-Reuss plasticity via Gamma-convergence techniques. The limiting model has a genuinely three-dimensional nature, unless specific data are prescribed. In particular, the stretching and bending components of the stress decouple only in the equilibrium condition, while the whole stress is involved in the stress constraint and in the flow rule. I shall also present an ad hoc notion of stress-strain duality, which allows one to write a strong formulation of the flow rule in the limit problem. This is based on a joint work with Elisa Davoli (Carnegie Mellon University).

David Arcoya (Universidad de Granada)
Nonlinear problems with natural growth on the gradient and lack of an a priori $L^\infty$ estimates

The starting point is a paper by L. Boccardo, F. Murat, J.P. Puel, where it is considered the zero Dirichlet boundary value problems associated to nonlinear elliptic equaltions with quadratic dependence on the gradient and whose model may be: $$ -\Delta u - \lambda c(x) u - \mu(x)|\nabla u|^2 = h(x), \, x\in \Omega, $$ with $c, h \in L^p(\Omega) \, (p > N/2), \, c\ge c_0 >0$ and $0<\mu_1\le \mu(x) \le \mu_2$. If $\lambda < 0$, the existence of an a priori $L^\infty$ estimate for the solutions of a family of approximated problems becomes the key to prove the existence of solution. As it was proved by V. Ferone, F. Murat (and also in a paper by Porretta), this a priori estimate may fail for $\lambda = 0$. Our aim is to go beyond the case $\lambda \le 0$. Indeed, we improve jointly with L. Jeanjean, C. De Cosner and K. Tanaka the previous multiplicity result of L. Jeanjean, B. Sirakov (see also a paper by B. Abdellaoui, A. Dall'Aglio, I. Peral) for the case that $\lambda \ge 0$ is small and \mu is constant. In contrast with the constant case where it is possible to apply a suitable change of variables to reduce to a variational semilinear problem, when $\mu(x)$ is not constant, we study the bifurcation form infinity of solutions of the problem to solve it. Continuing previous works for the case $\lambda = 0$, we also see in collaboration with L. Moreno that the above multiplicity result fails when a singular term at $u=0$ is added to the term $\mu(x) |\nabla u|^2$.

Miguel Escobedo (Universidad del País Vasco)
Some remarks on the fully parabolic Keller-Segel system in the plane

We will describe recent results on the doubly parabolic Keller-Segel system in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. We analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. We are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term the solutions behave like self-similar solutions, while in presence of the degradation term global solutions behave like the heat kernel

Vladimir Maz'ya (University of Liverpool)
Criteria for the Poincare-Hardy inequalities

This is a survey of old and new necessary and sufficient conditions for validity of various integral inequalities containing arbitrary weights (measures and distributions). These results have direct applications to the spectral theory of elliptic partial differential operators.

Corrado Lattanzio (Università di L'Aquila)
Relative entropy methods and relaxation limits

We shall analyze relative entropy methods in connections with singular limits, both for the hyperbolic-to-hyperbolic and for the hyperbolic-to-parabolic relaxations. For the former case, we recall in particular the results proved in [Lattanzio, Tzavaras, ARMA 2006] for the hyperbolic stress relaxation limit for multidimensional elasticity. Moreover, for diffusive relaxations, we obtain a relative entropy estimate in the general case of a large friction theory converging toward gradient flow equations. The framework considered applies in particular to the Euler equations with large friction converging toward the porous medium equation [Lattanzio, Tzavaras, SIMA 2013], and to the diffusive relaxation limits for Keller-Segel type models and for Euler-Korteweg equations with large friction.

Josè Carmona (Univérsidad de Almeria)
Quasilinear elliptic problems with quadratic gradient term. Comparison principle and Gelfand problems

We review some recent results concerning with the existence of positive solutions of the following problem $$ \begin{cases} \displaystyle -\Delta u +H(x,u,\nabla u)= \lambda f(u) \quad & \mbox{in}\,\, \Omega, \\ u=0 &\mbox{on} \, \partial \Omega, \end{cases} $$ where $f$ is a continuous nonnegative function in $[0,+\infty)$ with $f(0)>0$ and $H$ is a Carathédory function defined on $\Omega\times [0,+\infty)\times \mathbb{R}^{N}$. Specifically, we show how this problem provides a general framework to study Gelfand type problems. We give sufficient conditions to prove that there exists $\lambda^*>0$ such that the problem has a minimal solution $u_\lambda$ provided that $0\leq \lambda<\lambda^*$ and no solution if $\lambda >\lambda^*$. We pay special interest in the extension of the classical stability condition as well as a general comparison principle.

Josè Mazòn (Univérsitat de Valencia)
Some mass transport problems as limits of p-Laplacian problems

Following the method introduced by Evans y Gangbo to solve the classical Monge-Kantorovich mass transport problem, in this lecture we present two mass transport problems obtained as limit when $p\to \infty$ of the solutions of some problem related with the $p$-Laplacian operator. The first one is an optimal matching problem that consists in transporting two commodities to a prescribed location, the target set in such a way that they match there and the total cost of the operation, measured in terms of the Euclidean distance that the commodities are transported, is minimized. We show that such a problem has a solution with matching measure concentrated on the boundary of the target set. Furthermore we perform a method to approximate the solution of the problem taking limit as $p\to \infty$ in a system of PDE's of $p$-Laplacian type. The second problem consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger or equal than a fixed one to fulfill a demand also larger or equal than a fixed one, with the obligation of paying an extra cost of $-g_1(x)$ for extra production of one unit at location x and an extra cost of $g_2(y)$ for creating one unit of demand at $y$. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as $p\to\infty$ to a double obstacle problem (with obstacles $g_1$, $g_2$) for the $p$-Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved.

Emanuele Spadaro (MPI, Leipzig)
Unique tangent cones for 2-d almost minimal currents

In this talk I will discuss some new results on the infinitesimal behavior of 2 dimensional almost minimal surfaces (relevant examples are semi-calibrated currents and section of 3 dimensional minimizing cones). In particular I will show that such surfaces posses a unique tangent cone at every point and I will comment on perspective results concerning their full regularity. This is a joint work with C. De Lellis and L. Spolaor (Zurich).

Juan Casado Diaz (Universidad de Sevilla)
On the determination of finitely many parameters in some elliptic equations and systems from boundary measurements

We consider the classical problem in optimal design consisting in mixing two materials (electric or thermic) with given proportions in order to maximize or minimize the energy (compliance and energy problem). We show the existence of second derivatives for the solutions of the corresponding relaxed problems. This applies to prove that the problem of mixing two materials in order to minimize the first eigenvalue of the corresponding operator has not an unrelaxed solution in general.

Elena Beretta (Politecnico di Milano)
On the determination of finitely many parameters in some elliptic equations and systems from boundary measurements

In the talk I will describe some nonlinear severly ill-posed inverse boundary value problems involving elliptic equations and elliptic systems with applications to medical imaging, non destructive testing of materials and seismology. More precisely, one wants to determine some coecient appearing in an elliptic equation or system in a bounded domain $\Omega \subset \mathcal R^3$ from observations of solutions and its derivatives on $\partial \Omega$. In particular, I will focus my attention on the conductivity problem, the Gelfand-Calderon problem and the elasticity inverse problem reviewing some of the main results concerning uniqueness and continuous dependence. In the second part of the talk I will concentrate on the issue of continuous dependence, crucial for eective reconstruction, describing some recent results where Lipschitz continuous dependence estimates have been derived in the case of coecients that are nite linear combinations of functions $\psi_j$ ; j = 1; ;N, dened on a Lipschitz partition $D_N = \cup_{j-1}^N D_j$ of $\Omega$, for example when $\psi_j = \chi_{D_j}$, $j = 1; ;N$. This is quite natural having in mind a nite element scheme used in the reconstruction procedure. A crucial role is played by the Lipschitz constant appearing in the estimates and its dependence on the a priori parameters in particular on the mesh size $r = r(N)$ of the partition $D_N$. I will present some recent results concerning the Gelfand-Calderon problem; in this case an explicit optimal bound of the Lipschitz constant with respect to the mesh size is derived. The results are obtained in collaboration with E. Francini, M. de Hoop, L. Qiu, O. Scherzer and S. Vessella.

Laurent Vèron (Université Fran¨ois Rabelais)
Local and global behaviour of solutions of some quasilinear Hamilton Jacobi equations

We study the boundary behaviour of the solutions of (E) $\Delta_p u+|\nabla u|^q=0$ in a domain $\Omega \subset \mathbb R^N$, when $N\geq p> q>p-1$. We first recall the results obtained in the case $p=2$: boundary trace, boundary isolated or removable singularities. In the case $p\neq 2$, we show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$ there exist positive solutions of (E) with an isolated singularity on $\partial \Omega$ and that these solutions belong to two types of singular solutions. If $q_*\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular solutions are classified according the two types of singular solutions that we have constructed. An extension to a geometric framework is also presented as a consequence of a general estimate. The case p=2 is a joint paper with T. Nguyen Phuoc. The case $p\neq 2$ corresponds to a work in progress with M.F. Bidaut-Vèron and M.

Marco Barchiesi (Università di Napoli)
Stability of the isoperimetric inequality and Polya-Szego inequality under Steiner rearrangement

I shall start with a quick review of the basic properties of Steiner symmetrization of sets and functions. Through some recently developed analytical techniques, I will give a characterization of the cases of equality. Then, I will prove a sharp quantitative version of the inequalities, in the case of convex sets and log-concave functions.

Alessio Porretta (Roma Tor Vergata)
Soluzioni deboli per Fokker-Planck e Mean Field Games

La teoria dei mean-field games proposta da Lasry e Lions, e in parallelo da Huang, Caines e Malhame', a partire dal 2006, e' un modello di campo medio per dinamiche con grandi popolazioni di piccoli agenti identici le cui singole strategie dipendono dalla legge di distribuzione della massa. Il modello macroscopico risulta in un sistema di PDE in cui un'equazione di Fokker-Planck e' accoppiata a un'equazione (backward) di Hamilton-Jacobi. Nel seminario illustrero' diversi aspetti legati in particolare all'esistenza e unicita' di soluzioni deboli, alla buona positura dell'equazione di Fokker-Planck con drift L^2, nonche' applicazioni al problema della pianificazione, in cui si cerca di realizzare un trasporto ottimo per la densita' di massa attraverso strategie ottimali per il costo corrente degli agenti.

Scott N. Armstrong (CEREMADE Université Paris-Dauphine)
Optimal estimates in stochastic homogenization for nondivergence form equations

We present some recent results concerning the homogenization of uniformly elliptic equations in nondivergence form. The equations are assumed to have coefficients which are independent at unit distance. We give optimal results on the order of the error in homogenization in every dimension, measured in $L^\infty$ and Holder spaces up to $C^{1,\alpha}$, $\alpha\leq 1$. As a corollary, we obtain the existence of stationary correctors exist in dimensions five and higher (and their nonexistence, in general, in dimensions four and smaller). Finally, we give regularity results which state that a generic equation has essentially the same regularity as Laplace's equation, up to $C^{1,1}$.

Adriano Pisante (Università di Roma)
Disuguaglianze di Sobolev migliorate e decomposizione in profili in spazi di Sobolev frazionari

Presenteremo nuovi miglioramenti della disuguaglianza di Sobolev in spazi frazionari tipo $H^s$ in termini di norme di Morrey. Da queste si prova immediatamente esistenza di funzioni ottimali per la disuguaglianza di Sobolev frazionaria usuale come pure se ne descrive il comportamento delle successioni ottimizzanti. Piu' in generale tale disuguaglianza permette di ottenere una prova alternativa della decomposizione asintotica in profili per successioni in $H^s$ (dovuta originariamente a Gerard con una dimostrazione differente) usando l'approccio astratto in termini di spazi di dislocazioni introdotto da Tintarev. La mancanza di compattezza per le successioni viene analizzata anche direttamente descrivendo la concentrazione in termini di misure di difetto alla P.L.Lions. Come applicazione modello discuteremo il comportamento asintotico di una famiglia di problemi sottocritici, provando concentrazione per la corrispondente famiglia di estremali, comportamento ben noto quando s e' intero. Il lavoro e' in collaborazione con Giampiero Palatucci.

Gianluca Crippa (Università di Basilea)

Ordinary Differential Equations and Singular Integrals

Given a Lipschitz vector field, the classical Cauchy-Lipschitz theory gives existence, uniqueness and regularity of the associated ODE flow. In recent years, much attention has been devoted to extensions of such theory to cases in which the vector field is less regular than Lipschitz, but still belongs to some "weak differentiability classes". In this talk, I will review the main points of an approach involving quantitative estimates for flows of Sobolev vector fields (joint work with Camillo De Lellis) and describe further extensions to a case involving singular integrals of L1 functions (joint work with Francois Bouchut) and to a case endowed with a "split structure" (joint work with Anna Bohun and Francois Bouchut).

Luca Rossi (Università di Padova)
Generalized principal eigenvalues for elliptic operators in non-compact scenarios

What is the analogue of the principal eigenvalue for elliptic operators with non-compact resolvents? Focusing on the case where the lack of compactness is due to the unboundedness of the domain, we show that the answer depends on the property one is looking for: existence of a positive eigenfunction, simplicity, lower bound of the spectrum, characterization of the maximum principle. Indeed, there is not a unique notion fulfilling all such properties in general. In the last part of the talk we present some recent results concerning degenerate elliptic operators.

Federico Cacciafesta (Sapienza Università di Roma)
Sul metodo del moltiplicatore per equazioni dispersive a coefficienti variabili

A partire dalla sua introduzione alla fine degli anni '60 ad opera di Morawetz, il metodo del moltiplicatore è stato largamente e floridamente usato per ottenere vari importanti risultati nell'ambito delle PDE dispersive. In particolare, esso permette di dimostrare con tecniche dirette e elementari stime di "local smoothing" per varie equazioni (onde, Schroedinger, Dirac), anche in presenza di potenziali esterni. In questo seminario daremo una panoramica del metodo e presenteremo alcune recenti applicazioni, in collaborazione con P. D'Ancona e R. Lucà, alle equazioni a coefficienti variabili.

Haïm Brezis (Université Pierre-et-Marie-Curie)
Henri Poincaré, a founding father of the modern theory of PDEs

I will discuss some contributions of Henri Poincaré which have had a major impact on the development of PDEs in the 20th century; in particular, the solution of the Laplace equation, the spectral theory for the Laplacian, and much more. Poincaré also had prophetic insights about the role of PDEs within Mathematics.

Claudio Muñoz (Université Paris-Sud)
On the stability of breathers

The purpose of this talk is to discuss several stability properties of a class of very particular structures associated to well-known integrable models, usually referred as "breathers". I will explain why breathers are stable objects, at the same level of regularity as solitons. This is joint work with Miguel Angel Alejo (IMPA).

Mariapia Palombaro (Università dell'Aquila)
Analisi multi-scala di dislocazioni in etero-strutture sottili

Le dislocazioni sono uno dei difetti piu' comuni dei solidi cristallini e la loro presenza influenza il comportamento dei materiali in svariati modi. Per esempio, nell'elettronica dei semiconduttori le dislocazioni giocano un ruolo cruciale nello sviluppo di etero-strutture sottili, ottenute dalla combinazione di due o piu' materiali cristallini. Difatti, un mismatch troppo grande fra le strutture cristalline utilizzate puo' dar luogo alla formazione di dislocazioni all'interfaccia fra le diverse componenti. Nel seminario presenteremo una giustificazione matematica rigorosa della formazione di dislocationi nelle "nanowires heterostructures", ovvero eterostrutture sottili sviluppate longitudinalmente. Vedremo come, per un dato mismatch, l'energia del sistema favorisca la formazione di dislocazioni alle deformazioni elastiche quando il raggio della sezione del materiale e' sufficientemente grande. L'analisi e' presentata sia nel setting continuo che discreto.

Michele Correggi (Sapienza Università di Roma)
On the Ginzburg-Landau Functional in the Surface Superconductivity Regime

We shall discuss the behavior of type II superconductors in the framework of Ginzburg-Landau theory for an applied magnetic field varying between the second and third critical fields. In this regime superconductivity is restricted to a thin layer along the boundary of the sample and we will prove that the Ginzburg-Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. In the special case of disc samples, we will derive pointwise estimates on the Ginzburg-Landau order parameter, thereby proving a conjecture due to Xing-Bin Pan.

Alfonso Sorrentino (Università di Roma 3)
Invariant Lagrangian graphs, Hamilton-Jacobi equation and action-minizing properties of Tonelli Hamiltonians

In the study of Hamiltonian systems a special role is played by invariant Lagrangian submanifolds. These objects arise quite naturally in many physical and geometric problems and besides sharing a deep relation with the dynamics of the system, they are also closely related to classical and `weak' solutions of the corresponding Hamilton-Jacobi equation(s). When does a Hamiltonian system possess an invariant smooth Lagrangian graph or a family thereof? In this talk I shall discuss how this very interesting (and difficult) question can be approached from different perspectives and describe several results related to the so-called Principle of least Lagrangian action.

Lucia De Luca (Sapienza Università di Roma)
Statics and dynamics of dislocations: A variational approach

Dislocations are line defects in crystals and they are considered the main mechanism of plastic deformations in metals. We will consider straight dislocations, so that their positions are completely identified by the intersections of the dislocation line with an orthogonal plane. In this talk we will present a purely variational approach, based on Gamma-convergence, to the study of the asymptotic behavior of the static energy induced by a finite system of dislocations as the atomic scale goes to zero. For a special class of dislocations, the so-called screw dislocations, we derive also an interaction between the defects, which drives their dynamics. The essential tool in all the results we will present is given by the analogy between dislocations and vortices in superconductors, studied within the Ginzburg-Landau framework. The results are obtained in collaboration with R. Alicandro, A. Garroni and M. Ponsiglione.


Giovanni Scilla (Sapienza Università di Roma)
Variational motion of discrete interfaces

We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional periodic environment, by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuous analysis. In the case of a homogeneous environment, recently treated by Braides, Gelli and Novaga, the effective continuous motion is a flat motion related to the crystalline perimeter obtained by $\Gamma$-convergence from the ferromagnetic energies, with an additional discontinuous dependence on the curvature, giving in particular a pinning threshold. In a joint work with A. Braides, we show that, in general, the motion does not depend only on the $\Gamma$-limit, but also on geometrical features that are not detected in the static description. In particular, we show how the pinning threshold is influenced by the microstructure and that the effective motion is described by a new homogenized velocity. In the last part of the talk, I would like to present also the results of an ongoing joint work with A. Braides: we use a discrete approximation of the motion by crystalline curvature to define an evolution of sets from a single point (nucleation) following a criterion of ``maximization'' of the perimeter, formally giving a backward version of the motion by crystalline curvature.

Luisa Moschini (Sapienza Università di Roma)
Sharp Trace Hardy-Soboev-Maz'ya Inequalities and the Fractional Laplacian

We establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space our results cover the full range of the exponent $s\in (0,1)$ of the fractional Laplacians. Thus we answer in particular an open problem raised by Frank and Seiringer, which has been also answered but only in the range of $s\in (1/2,1)$ and through different approaches by Dyda in 2010, Sloane in 2010 and by Dyda and Frank in 2011. The results presented in this talk are contained in two different articles in collaboration with S. Filippas and A. Tertikas, the second one only recently submitted.

Denis Bonheure (Université Libre de Bruxelles)
Some questions from the nonlinear theory of electromagnetism of Born-Infeld

In this talk, I will discuss some questions related to the nonlinear theory of electromagnetism formulated by Born and Infeld in 1934. I will discuss the link between this theory and the curvature operators in the Euclidean and in the Lorentz-Minkowski space. I will address the solvability of the electrostatic Born-Infeld equation with sources (and a model driven by the nonlinear Klein-Gordon equation) emphasizing the open questions and the partial recent progress we made. Finally, I will present some more academic results related to the curvature operator in the Lorentz-Minkowski space and the solvability of some scalar field type equations.