Abstract: I will describe recent results dealing with a nonlocal version of the De Giorgi conjecture on flatness of level sets for solutions of semi linear equ The first results obtained in this direction were for the fractional laplacian,$ the Caffarelli-Silvestre extension. Here we will consider general kernels for w$ an extension is not available.

Alberto Tesei (Sapienza Unviersità di Roma)
Equazioni paraboliche quasilineari con energia non convessa

Soluzioni a valori misure di Young e di Radon intervengono in modo naturale nello studio di equazioni paraboliche quasilineari con energia non convessa (o equazioni forward-backward). Saranno presentati risultati recenti, ottenuti in collaborazione con M. Bertsch e F. Smarrazzo, concernenti esistenza e proprietà qualitative delle soluzioni per una classe di tali equazioni con regolarizzazione pseudoparabolica.

Jan Kristensen (University of Oxford)
Morse-Sard type theorems for Sobolev mappings

The Morse-Sard theorem, and the generalizations by Dubovitskiui and Federer, have numerous applications and belong to the core results of multivariate calculus for smooth mappings. In this talk we discuss extensions of these results to suitable classes of Sobolev mappings. The potential lack of differentiability of these mappings warrants new interpretations, and requires one to prove that the Sobolev mappings enjoy Luzin N type properties with respect to lower dimensional Hausdorff contents. The talk is based on joint work with Jean Bourgain (Princeton) and Mikhail Korobkov (Novosibirsk).

Giovanni Bellettini (Università di Roma Tor Vergata)
Alcuni risultati sul funzionale dell'area del grafico di mappe discontinue dal piano in se

Si illustreranno alcuni recenti risultati sul funzionale dell'area del grafico di mappe discontinue del piano in se$ i legami di questo problema con il problema di Plateau ambientato tra le mappe semicartesiane, e alcuni fenomeni di nonlocalita'

Manuel Del Pino (Universidad de Chile)
Bubbling blow-up in critical parabolic problems

We construct solutions with finite and infinite type-II blow-up (and analyze their stability) in two related parabo$ the standard semilinear heat equation with a power nonlinearity at the critical exponent in a bounded domain in $\m$ domain into the sphere $\mathbb{S}^2$. Both problems have stationary states with energy scaling-invariance in en$

Marco Degiovanni (Università Cattolica del Sacro Cuore)
Grado topologico ed equazioni quasilineari con secondo membro misura

Si considera un'equazione quasilineare con secondo membro misura il cui primo membro non definisce un operatore coercivo. Si dimostra un risultato di esistenza del tipo alternativa di Fredholm. Per questo scopo viene introdotta una variante di grado topologico adatta a tale contesto.

Roberto Feola (Università di Roma La Sapienza)
Quasi-periodic solutions for fully nonlinear NLS

We consider a class of fully nonlinear, autonomous and reversible Schrödinger equations on the circle and we prove the existence and the stability of Cantor families of (small amplitude) quasi-periodic solutions. The proof is based on a combination of different ideas: (i) we perform a 'weak' Birkhoff normal form step in order to find an approximately invariant manifold on which the dynamics is approximately integrable; (ii) we introduce a suitable generalization of a KAM nonlinear iteration for 'tame' and 'unbounded' vector fields based on the invertibility of the linearized equation in a neighborhood of the origin; (iii) we exploit the 'pseudo-differential structure' of the vector field and we prove the invertibility of the linearized operator, using a regularization procedure which conjugates the operator to a differential operator with constant coefficients plus a bounded remainder. This latter step is obtained through transformations generated by torus diffeomorphisms and pseudo-differential operators. Then we use a KAM-like reducibility scheme that reduces to constant coefficients the linearized operator at the solution. This gives the linear stability.

Thierry Paul (CMLS, Ecole Polytechnique)
Flows below Cauchy-Lipschitz as asymptotic limits of PDE's ones

P.N. Srikanth (TIFR Centre for Applicable Mathematics - Bangalore)
Multi orbit concentration

In this lecture we consider a singularly perturbed semilinear elliptic problem with power non-linearity in Annular Domains in $\mathbb R^{2n}$ and show the existence of two orthogonal $\mathbb S^{n-1}$ concentrating solutions.We will discuss some issues involved in the proof in the context of $\mathbb S^1$ concentrating solutions of similar nature.

Nassif Ghoussoub (University of British Columbia)
On the structure of optimal martingale transport plans in general dimensions

I will describe the profile of optimal solutions of the martingale counterpart of the Monge mass transport problem. These are one-step martingales that maximize or minimize the expected value of the modulus of their increment among all martingales having two prescribed convex ordered probability measures as marginals. While there is a great deal of results -mostly established by the mathematical financial community- when the marginals are probabilities on the real line, much less is known in the richer and more delicate higher dimensional setting.

Alexandre Boritchev (Université Claude Bernard Lyon 1)
Hyperbolicity of the minimisers for the stochastic Burgers equation

We consider the stochastic Burgers equation from a Lagrangian viewpoint. In other words, we study the dynamical behaviour of the energy minimisers which give the variational behaviour of the solution. Under non-degeneracy assumptions on the random forcing, we prove hyperbolicity of these minimisers, considerably simplifying the proof in [E, Khanin, Mazel, Sinai, Annals, 2000]. Finally, we will speak about the relationship between this problem and the convergence to the stationary measure for the solutions of the equations. This is a joint work with K. Khanin (Toronto).

Camillo De Lellis (Universität Zürich)
The regularity of 2-dimensional area-minimizing integral currents

Building upon the Almgren's big regularity paper, Chang proved in the eighties that the singularities of area-minimizing integral 2-dimensional currents are isolated. His proof relies on a suitable improvement of Almgren's center manifold and its construction is only sketched. In recent joint works with Emanuele Spadaro and Luca Spolaor we give a complete proof of the existence of the center manifold needed by Chang and extend his theorem to two classes of currents which are "almost area minimizing" in a suitable sense.

Adimurthi (TIFR Centre for Applicable Mathematics - Bangalore)
Uniqueness of large solutions in a ball for n-Laplace equation with critical non linearity

Brezis posed the problem of uniqueness for solutions in a ball for Brezis- Nirenberg problem. it was solved by many people, The main ingredient is the clever way using the Pohozaev Identity. But in dimension two the non linearity is of exponential type and Pohozaev identity is in effective when the exponent is critical. Here I would like to discuss this case in generality and prove the uniqueness and non degeneracy of positive solutions for large solutions.

Catherine Bandle (Universität Basel)
On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds

We investigate existence and nonexistence of stationary stable non constant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results. This is a joint research with Paolo Mastrolia, Dario Monticelli and Fabio Punzo.

Matteo Focardi (Università di Firenze)
Endpoint regularity of 2-d Mumford-Shah minimizers

We discuss an epsilon-regularity result at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers obtained in a joint work with C. De Lellis (U. Zuerich). As an outcome of our analysis, if in a ball $B_r(x)$ the jump set of a given Mumford-Shah minimizer is sufficiently close in the Hausdorff distance to a radius of $B_r(x)$, then in a smaller ball the jump set is a connected arc terminating at some interior point and $C^{1,\alpha}$ up to the tip.

Pierpaolo Esposito (Università di Roma 3)
Non-topological condensates for the self-dual Chern-Simons-Higgs model

We discuss vortex configurations in the abelian self-dual Chern-Simons-Higgs model, where topological invariants can just describe a part of the picture. We construct non-topological condensates (=doubly periodic vortex-configurations) towards a deeper understanding of the periodic setting. Joint work with M. del Pino, M. Musso and P. Figueroa.

Felix Otto (Max Planck Institute for Mathematics in the Sciences, Leipzig)
A quantitative theory of stochastic homogenization

The topic of stochastic homogenization of elliptic partial differential equations in divergence form is classical. It is about the homogeneous large-scale behavior of heterogeneous media, like conductive media or elastic media, that are characterized in stochastic terms. Our interest grew out of quantifying the error scaling in the engineer's concept of a ``representative volume element'', which allows to approximately extract the homogeneous coefficients. Meanwhile, the connections with classical regularity theory (attached to the names De Giorgi, Nash, Campanato, Meyers ... ) and with concepts of concentration of measure (as for instance captured by the Logarithmic Sobolev Inequalities) have emerged in a clearer way. Not only is the regularity theory for uniformly elliptic coefficient fields $a$ a key ingredient, but stochastic homogenization sheds a new light on a generic large-scale behavior of $a$-harmonic functions - which is more regular than suggested by the classical counter-examples. We also advocate to exploring more the synergies between the treatment of quenched noise (like the random coefficients in stochastic homogenization) and thermal noise (like in statistical mechanics or stochastic partial differential equations).

Max Fathi (Université Pierre et Marie Curie)
A gradient flow approach to large deviations for diffusion processes

In the 80s, De Giorgi introduced the notion of abstract gradient flows, which allowed to define a notion of solutions to ordinary differential equations of the form $x' = -\hbox{grad} F(x)$ on metric spaces (rather than Riemannian manifolds for the usual definition). In 2005, Ambrosio, Gigli and Savare showed that when we consider the space of probability measures on R d endowed with the Wasserstein metric, this notion allows to give an alternate formulation for Fokker-Planck equations. These equations are the PDEs whose solutions are the flow of marginals of solutions of stochastic differential equations of the form $dX = -\hbox{grad} H(X)\,dt + dB$. In this talk, I will explain how we can use this notion to study large deviations for sequences of SDEs. The main result is that proving a large deviation principle is equivalent to studying the limit of a sequence of functionals that appear in the abstract gradient flow formulation for Fokker-Planck equations. As an application, I will show how to obtain large deviations from the hydrodynamic scaling limit for a system of interacting continuous spins in a random environment.

Luigi Chierchia (Università di Roma 3)
Periodic and quasi-periodic solutions in nearly-integrable Hamiltonian systems

In 1892 H. Poincaré conjectured that periodic orbits are dense in the phase space of a general (analytic) nearly-integrable Hamiltonian systems. I shall discuss a weaker "asymptotic" version of Poincaré's conjecture and its connection with the measure of maximal quasi-periodic solutions.

John King (University of Nottingham)
Asymptotic behaviour of a semilinear elliptic equation

The well-studied power-law-nonlinearity elliptic PDE is known to exhibit a variety of interesting effects, including non-existence, non-uniqueness and concentration phenomena. The application of formal-asymptotic and symmetry methods in characterising such behaviour in terms of the domain geometry will be illustrated, focussing on the infinite strip. Implications for 'fast' nonlinear diffusion will be noted.

Stefania Patrizi (WIASS, Berlin)
On a long range segregation model

Segregation phenomena occurs in many areas of mathematics and science: from equipartition problems in geometry, to social and biological processes (cells, bacteria, ants, mammals) to finance (sellers and buyers). There is a large body of literature studying segregation models where the interaction between species is punctual. There are many processes though, where the growth of a population at a point is inhibited by the populations in a full area surrounding that point. The work we present is a first attempt to study the properties of such a segregation process. This is a joint paper with Luis Caffarelli and V?ronica Quitalo.

Sergio Conti (University of Bonn)
On the theory of relaxation for variational problems with constraints on the determinant

We consider vectorial variational problems of the form $E[u]=\int W(Du)dx$, typical for example of nonlinear elasticity and plasticity, which include constraints on the determinant. Specifically, the energy density $W$ is assumed to diverge outside of the set of matrices with positive determinant, or, alternatively, outside of the set of matrices with determinant equal 1. If $W$ is not quasiconvex then $E$ is not lower semicontinuous and does not, in general, have minimizers. Low-energy states can be studied via the relaxation of $E$. We discuss how, in some situations of physical interest, the relaxation of $E$ can be explicitly characterized in terms of the quasiconvex envelope of $W$. This talk is based on joint work with Georg Dolzmann (Regensburg).

Graziano Crasta (Sapienza Università di Roma)
Geometric problems related to the inhomogeneous infinity Laplace equation

We discuss some recent results related to the homogeneous Dirichlet problem for the infinity Laplace equation with constant source in a bounded domain. We characterize the geometry of domains for which an overdetermined problem admits a viscosity solution. An essential tool is a regularity result for viscosity solutions in convex domains, obtained by the convex envelope method introduced by Alvarez, Lasry and Lions. Based on some recent joint works with Ilaria Fragala' (Politecnico di Milano)

Daniele Andreucci (Sapienza Università di Roma)
Large time behavior for diffusion equations in unbounded domains

The asymptotic decay rate and other qualitative features of solutions to parabolic equations set in noncompact domains of RN, or Riemannian manifolds, depend in general on a suitable notion of the geometry of the domain at infinity, as well as on the structure of the equation. We report on some recent and on-going work on the interplay of these two factors (joint project with prof. A.Tedeev).

Donatella Donatelli (Università degli Studi dell'Aquila)
Plasma Oscillations: analysis of dispersive behavior and acoustic waves

We perform a rigorous analysis of the quasineutral limit for a hydrodynamical models of viscous plasmas represented by the Navier Stokes Poisson system in 3-D. A common feature of this kind of limits in the ill prepared data framework is the high plasma oscillations, namely the presence of high frequency time oscillations of the acoustic waves. Moreover and other issue which makes the limiting behaviour analysis very hard is the presence of very stiff terms due to the electric field. We shall provide a detailed mathematical description of the different behaviors of the various vector fields acting in our system, what and which are the relationship between high frequency interacting waves, dispersive behavior and the different roles of time and space oscillations.

Giovanni Franzina (Sapienza Università di Roma)
The isoperimetric problem with densities

The question whether or not there exist isoperimetric regions with densities can be very trivial or extremely difficult to consider depending on the assumptions on the density functions. We survey some known facts about this topic and we present an existence result obtained in collaboration with Guido De Philippis (ENS Lyon) and Aldo Pratelli (FAU Erlangen).

Filippo Gazzola (Politecnico di Milano)
Equazioni nonlineari della piastra che modellano la statica dei ponti sospesi

Partendo dalla teoria classica dell'elasticit? si ricavano diversi possibili modelli che descrivono la statica dei ponti sospesi. Le equazioni differenziali risultanti sono ellittiche nonlineari del quart'ordine con condizioni al contorno di tipo misto. Dopo avere analizzato lo spettro dell'equazione lineare, verranno discusse equazioni semilineari, quasilineari e non locali.

Nicola Gigli (SISSA, Trieste)
Una versione astratta della disuguaglianza di Lewy-Stampacchia ed alcune applicazioni

Nel talk discutero' come formulare in maniera astratta, nel setting di spazi vettoriali topologici muniti di una struttura di reticolo, la disuguaglianza classica di Lewy-Stampacchia. Il vantaggio di questa formulazione, oltre alla generalita', e' nella semplificazione della dimostrazione. Presentero' poi alcuni applicazioni di questo risultato. Da un lavoro con Mosconi.

Bernard Dacorogna (École Polytechnique Fédérale de Lausanne)
A Dirichlet problem involving the divergence operator

Given a vector field $a$ and a function $f$, we want to find a vector field $u$ such that \begin{eqnarray*} \begin{cases} \hbox{div}\,u+\left\langle a;u\right\rangle =f & \hbox{in } \Omega\\ u=0 & \text{on }\partial\Omega. \end{cases} \end{eqnarray*} This is a joint work with Gyula Csato.

David Sarrocco (Sapienza Università di Roma)
Evolution of microstructures for a damage model

Starting from the damage model for elastic material introduced by Francfort&Marigo, we will present results that combines efficiently the notion of quasi-static evolution with processes of homogenization. We will consider 3 different kind of energy derived from such model: a 1D oscillating energy describing the evolution of a two two-phase material, an elastic energy penalized by the perimeter of the damage region, and finally we study the problem in a dynamic framework considering also a kinetic term. Existence and convergence results will be shown. Moreover we wil present a different way to study such evolutions through a threshold criterion, showing some results in this direction.

Anne-Sophie de Suzzoni (Université Paris 13)
Invariant measure for the non linear Schrödinger equation on $\mathbb R$ displaying a local non linearity

We consider the equation on $\mathbb R$ $$ i\partial_t u -\Delta u + \chi |u|^2u = 0 $$ where $\chi$ is a smooth function decaying at infinity. The aim of this talk is to build an invariant measure for this equation supported below $L^2$. We will in particular explain in which way it differs from the periodic setting and the difficulties due to the infinite speed of propagation. This is a joint work with F. Cacciafesta.

Mohammad Al Haj (Sapienza Università di Roma)
Existence of traveling waves for Lipschitz discrete dynamics: monostable case as a limit of bistable cases

We study discrete monostable dynamics with general Lipschitz non-linearities. This includes also degenerate non-linearities. In the positive monostable case, we show the existence of a branch of traveling waves solutions for velocities $c\geq c^{+},$ with non existence of solutions for $c< c^+$. We also give certain sufficient conditions to insure that $c^{+}\geq0$ and we give an example when $c^{+}<0.$ We as well prove a lower bound of $c^{+},$ precisely we show that $c^{+}\geq c^{*},$ where $c^{*}$ is associated to a linearized problem at infinity. On the other hand, under a KPP condition we show that $c^{+}\leq c^{*}.$ We also give an example where $c^{+}>c^{*}.$ This model of discrete dynamics can be seen as a generalized Frenkel-Kontorova model for which we can also add a driving force parameter $\sigma.$ We show that $\sigma$ can vary in an interval $[\sigma^{-},\sigma^{+}].$ For $\sigma\in(\sigma^{-},\sigma^{+})$ this corresponds to a bistable case, while for $\sigma=\sigma^{+}$ this is a positive monostable case, and for $\sigma=\sigma^{-}$ this is a negative monostable case. We study the velocity function $c=c(\sigma)$ as $\sigma$ varies in $[\sigma^{-},\sigma^{+}].$ In particular for $\sigma=\sigma^{+}$ (resp. $\sigma=\sigma^{-}$), we find vertical branches of traveling waves solutions with $c\geq c^{+}$ (resp. $c\leq c^{-}$). The results have been obtained in collaboration with R. Monneau

Filippo Cagnetti (Sussex University)
The rigidity problem for symmetrization inequalities

Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting, it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.

Fabio Cavalletti (Scuola Normale Superiore di Pisa)
A variational time discretization for Euler equations

I will present a variational time discretization for the multi-dimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each timestep requires the minimization of a functional measuring the acceleration of fluid elements, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations. This is a joint work with Marc Sedjro and Michael Westdickenberg.

Biagio Cassano (Sapienza Università di Roma)
Scattering in the energy space for nonlinear Schrödinger equations

We study the theory of Scattering in the energy space for various nonlinear Schr?dinger equations. In dimension 3 or bigger we consider a variable coefficients equation, for a gauge invariant, defocusing nonlinearity of power type on an exterior domain with Dirichlet boundary conditions. In order to prove scattering, we prove a bilinear smoothing (interaction Morawetz) estimate for the solution and, under the conditional assumption that Strichartz estimates are valid for the linear flow, we prove global well posedness in the energy space for energy subcritical powers, and scattering provided the power is mass supercritical. When the domain is the whole space, by extending the Strichartz estimates due to Tataru, we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space. In low dimension spaces of dimension 1,2 or 3, we simplify the scattering theory in $\mathbb{R}^n$ for the Schrödinger equation, generalizing it to the system framework. Joint work with Piero D'Ancona and Mirko Tarulli.


Thuong Nguyen (Sapienza Università di Roma)
Asymptotic Behavior of Singularly Perturbed Control System: non-periodic setting

In this talk we are interested in asymptotic behavior of singularly perturbedcontrol system in the non-periodic setting. More precisely, we consider the value function of finite horizon optimal control problem (Bolza form) associated with singularly perturbed control system, and aim at characterizing its weak semilimits as viscosity sub- and supersolutions of a limiting Hamilton-Jacobi-Bellman equation (also called effective HJB equation). This PDE approach is extensively studied in a series of papers by Alvarez and Bardi in the periodic setting ([AB03], [AB10]). Our contribution is to extend the results of Alvarez and Bardi to the nonperiodic case. The key idea is to replace the periodicity on the datum by coercivity on the running cost, and we only need the local version of boundedtime controllability used in [AB10]. The remarkable novelty of our work is to approximate the Bellman Hamiltonian (convex, but non-coercive in the momentum) by a suitable sequence of convex, coercive Hamiltonians and then use some basic tools of Aubry-Mather theory developed by Fathi and Siconolfi (see [FS05]) for these convex, coercive Hamiltonians. We finally obtain some similar results as those of Alvarez and Bardi. Joint work with Antonio Siconolfi.

Ederson Moreira dos Santos (Universidade de São Paulo)
Hénon type equations and concentration on spheres

Abstract: In this talk, I will present the concentration profile of various kind of symmetric solutions of some semilinear elliptic problems arising in astrophysics and in diffusion phenomena. Motivated by these elliptic equations and exploiting their symmetry, I will discuss about solutions that concentrate and blow up at points and around spheres as the concentration parameter tends to infinity.

Carlo Mantegazza (Scuola Normale Superiore di Pisa)
Some variations on Ricci Flow

Abstract: I will present and discuss some results and problems about flows of metrics on Riemannian manifolds correlated to Ricci flow: - The "renormalization group" flow, truncated at the second order term. The Ricci flow is its trucation at the first order (joint work with L. Cremaschi). - The "Ricci-Bourguignon" flow, which is a perturbation of the Ricci flow equation by an extra term proportional to the product of the scalar curvature with the metric tensor (joint work with G. Catino, L. Cremaschi, Z. Djadli, L. Mazzieri). - A "noname" flow that I and Nicola Gigli introduced using the theory of optimal transport of mass, which is "tangent" to the Ricci flow at the initial time and which can be defined also for nonsmooth metric spaces.

Alberto Tesei (Sapienza, Università di Roma)
Soluzioni a valori misure di equazioni paraboliche forward-backward

Equazioni paraboliche quasilineari con diffusione di segno variabile appaiono in importanti contesti applicativi (transizioni di fase, trattamento di immagini, dinamica di popolazioni, oceanografia). Molto lavoro e' stato dedicato allo studio di opportune regolarizzazioni dei relativi problemi di valori iniziali. Nello studio di tali problemi, un aspetto importante e' che l'equazione puo' sviluppare singolarita' per tempi positivi anche se il dato iniziale e' regolare. Cio' richiede l'introduzione e lo studio di soluzioni a valori misure di Radon opportunamente definite. Nel seminario presenteremo alcuni risultati in tal senso, ottenuti recentemente in collaborazione con M. Bertsch, M. M. Porzio e F. Smarrazzo.

Denis Bonheure (Université libre de Bruxelles)
Higher Order Functional Inequalities and the 1-Biharmonic Operator

Two models extensively studied by the community in elliptic PDE in the past 25 years are the Allen-Cahn equation ($\alpha =1$) and the stationary Schrödinger equation ($\alpha =-1$) $$ -\Delta u = \alpha (u-u^3). $$ The aim of my talk will be to address some perspectives on fourth order extensions of these models, namely on mixed diffusion equations $$ \gamma \Delta^2 u -\Delta u = \alpha (u-u^3). $$ The parameter $\gamma$ is positive $\alpha = 1$ corresponds to the extended Fisher-Kolmogorov equation (EFK) and $\alpha = -1$ to the stationary fourth order nonlinear Schrödinger equation (4NLS) with Kerr nonlinearity. I will first explain the phenomenological interests of these mixed diffusion models and then I will review a (certainly non exhaustive) list of classical results for the second order models and their counterparts for the mixed diffusion models, emphasizing the new difficulties and some central open questions. The talk is based on various works in progress with F. Hamel, E. Moreira dos Santos, H. Tavares, J. Foldes, A. Saldaña, R. Alves do Nascimento and M. Ghimenti.

Bernhard Ruf (Università di Milano)
Higher Order Functional Inequalities and the 1-Biharmonic Operator

We study optimal embeddings for the space of functions whose Laplacian belongs to $L^1(\Omega)$, where $\Omega \subset \mathcal R^N$ is a bounded domain. This function space turns out to be strictly larger than the Sobolev space $W^{2,1}(\Omega)$ in which the whole set of second order derivatives is considered. In particular, in the limiting Sobolev case, when $N = 2$, we establish a sharp embedding inequality into the Zygmund space $L_{exp}(\Omega)$. This result enables us to improve the Brezis-Merle regularity estimate for the Dirichlet problem $\Delta u = f(x) \in L^1(\Omega)$, $u = 0$ on $\partial \Omega$. We then study the operator associated to this problem, the 1-biharmonic operator.

Paolo Marcellini (Università di Firenze)
A variational approach to parabolic systems

We consider a purely variational approach to time dependent problems, yielding the existence of global parabolic minimizers. These evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals. In the simplest situation, the method provides the existence of global weak solutions to Cauchy-Dirichlet problems of parabolic systems. This is a joint collaboration with V. B?gelein and F. Duzaar.