We consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set ω ⊂ Rn, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue λ2(ω) is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem. Infλ(ω)=c We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume c/2 whose mutual distance tends to infinity

The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincaré-Sobolev constant for the embeddings W 1,2 0 (Ω) → L q (Ω).2010 Mathematics Subject Classification. 47A75, 49Q20, 49R05.

We prove higher Hölder regularity for solutions of equations involving the fractional p−Laplacian of order s, when p ≥ 2 and 0 < s < 1. In particular, we provide an explicit Hölder exponent for solutions of the non-homogeneous equation

We prove that for p≥2, solutions of equations modeled by the fractional p-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in Wloc1,p and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation s reaches 1

Given an open and bounded set Ω ⊂ R N , we consider the problem of minimizing the ratio between the s−perimeter and the N −dimensional Lebesgue measure among subsets of Ω. This is the nonlocal version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.

Abstract. We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

Starting from a model of traffic congestion, we introduce a minimal-flow--like variational problem whose solution is characterized by a very degenerate elliptic PDE. We precisely investigate the relations between these two problems, which can be done by considering some weak notion of flow for a related ODE. We also prove regularity results for the degenerate elliptic PDE, which enables us in some cases to apply the DiPerna-Lions theory

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