The known techniques to extend Lipschitz mappings expanded considerably in the last decade, but still very little is known about the possibility to choose these extensions continuously. In other words is the restriction operator open?
We'll discuss the motivation of the problem and present first examples which indicate that the openess can quite drastically fall.

We deal with the PDE \(\lambda_j(D^2 u) = 0\), in \(\Omega\), with u=g, on \(\partial \Omega\). Here \(\lambda_1(D^2 u) \leq ... \leq \lambda_N (D^2 u)\) are the ordered eigenvalues of the Hessian \(D^2 u\). The equation \(\lambda_1 (D^2u)=0\) is just the PDE verified by the convex envelope inside \(\Omega\) of the boundary datum g. Our main result is to show a necessary and sufficient condition on the domain so that the problem has a continuous solution for every continuous datum g. We also introduce a related two-player zero-sum game whose values approximate solutions to this PDE problem.

In this talk we will present the classical local Monge-Ampére equation and some of its applications to optimal transport and differential geometry. We will discuss the degeneracy of the equation and the challenges it poses for regularity of solutions. Finally, we will consider a nonlocal analogue of the Monge-Ampére operator, recently introduced in a joint work with Luis Caffarelli.

The Kolmogorov-Arnold-Moser theory covers different classical results in dynamical systems about the stability of quasiperiodic motions. On the other hand, the correspondence principle claims that the high energy behavior of a quantum system should be governed by its classical counterpart. In this talk we state and prove a KAM theorem on small perturbations of vector fields on the torus from the point of view of the spectral properties of the associated quantum system. We also characterize the set of quantum limits, namely the weak limit measures of sequences of L^2 mass densities of eigenfunctions for the system as the energy grows to inifinity.

We study the behavior of nonnegative solutions to the porous medium equation on Cartan-Hadamard manifolds, i.e. complete, simply connected Riemannian manifolds with everywhere nonpositive sectional curvatures. A well-known result ensures that Cartan-Hadamard manifolds are diffeomorphic to R^n. The subclass of manifolds we consider here can have unbounded negative curvature at spatial infinity: accordingly, we divide the corresponding geometrical settings into quasi-Euclidean, quasi-hyperbolic, super-hyperbolic and borderline critical cases. We focus on compactly-supported initial data, which give rise to a growing free boundary. We establish upper and lower pointwise estimates on the solutions, that only depend on the assumed power-type upper bound on the sectional curvatures and lower bound on the Ricci curvature, respectively. If the two bounds match our results are sharp. In particular, we can (approximately) locate the free boundary and estimate the growth rate of the support. In the super-hyperbolic range we also obtain a convergence theorem towards a separable solution involving the solution of a suitable sublinear elliptic equation, which is investigated separately. The talk is based on joint works with G. Grillo and J.L. Vázquez.

It is well-known that minimizers of variational integrals, even under favourable conditions, might not be fully regular nor unique in the multi-dimensional vectorial case. However under suitable smallness conditions it is possible to regain both. In this talk we discuss such results under natural conditions on the integrand.