Convegno
“VARIATIONAL AND PDE PROBLEMS IN GEOMETRIC ANALYSIS, IV”

This talk is devoted to the validity and the failure of the strong maximum principle for equations involving the k-th fractional truncated Laplacian or the k-th fractional eigenvalue, which are fully nonlinear integral operators whose nonlocality is somehow k-dimensional. We give in particular geometric characterizations of the sets of minima for nonnegative supersolutions. Based on joint works with I. Birindelli (Sapienza University), H. Ishii (Tsuda University) and D. Schiera (University of Lisbon).

In this talk, I will show what problems arise when trying to obtain Gaffney-type inequalities in subRiemannian geometry, since one cannot simply apply the classical Riemannian tools to this particular setting. I will then present some of the tools that are currently available to tackle this problem, and how they can be applied to obtain the desired results in the case of contact manifolds.

I will discuss a class of quasilinear elliptic equations involving the p-Laplace operator and nonlinearities of Sobolev-critical growth, focusing on existence, non-existence, and compactness issues related to their variational formulation.

We study existence and qualitative properties of solutions to subelliptic problems with Hardy potential and critical nonlinearities on stratified groups. We investigate both the semilinear and the quasilinear case. First, we determine the existence, Lorentz regularity and asymptotic behavior of entire solutions. By convenient transformations, we are naturally lead to study the equation satisfied by the extremal functions to some weighted Sobolev-type inequalities on groups, whose analytic expression is not known. As a byproduct, we derive existence results for the associated Brezis-Nirenberg type problem, depending on the involved parameters. We also obtain non-existence Pohozaev-type results.

In this seminar we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a critical problem driven by a mixed local and nonlocal linear operator. More precisely, given a bounded open set in R^n (with n ≥ 4), we consider a perturbed critical problem and we develop an existence theory, both in the case of linear (that is, p = 1) and superlinear (that is, p > 1) perturbations. In the particular case p = 1, we also investigate the mixed Sobolev inequality associated with (P), detecting the optimal constant, which we show that is never achieved. The results discussed in this talk are obtained in collaboration with S. Dipierro, E. Valdinoci and E. Vecchi.

The general themes of the talk are Dirichlet forms, fractional operators and associated Sobolev type spaces on groups of homogeneous type. Our results lead to explicit integral formulas of the infinitesimal generators, which are the studied fractional operators, and embedding theorems between the relevant spaces. The considered groups are not assumed to be Carnot groups or to satisfy a Hörmander type conditions. Finally, we will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.

We consider the problem of existence of extremal functions for second order Adams' inequalities with Navier boundary conditions on balls in R^n in any dimension n\geq 4. We also discuss some sharp weighted versions of Adams' inequality on the same spaces. The weights that we consider determine a supercritical exponential growth, except in the origin, and the corresponding inequalities hold for spherically symmetric functions only.

Given a complete, connected Riemannian manifold with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium equation with respect to the 2-Wasserstein distance. We produce stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm and Otto-Westdickenberg.

We consider the classical geometric problem of prescribing scalar and boundary mean curvature via conformal deformation of the metric on a n-dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if n=3 all the blow-up points are isolated and simple. In this work we prove that this is not true anymore in low dimensions (that is n=4, 5, 6, 7). In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and minimizes the norm of the trace-free second fundamental form of the boundary.