Lorenzo Brasco

The second eigenvalue of the fractional

Abstract We consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set Ω ⊂ ℝ n

p-

{\Omega\subset\mathbb{R}^{n}}

Laplacian

, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue λ 2 ⁢ ( Ω )

The fractional Cheeger problem

{\lambda_{2}(\Omega)}

Faber–Krahn inequalities in sharp quantitative form

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 inf ⁡ { λ 2 ⁢ ( Ω ) : | Ω | = c } .

On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds

\inf\{\lambda_{2}(\Omega):|\Omega|=c\}.

A Benamou–Brenier Approach to Branched Transport

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

A SHARP ESTIMATE A LA CALDERON-ZYGMUND FOR THE p−LAPLACIAN

{c/2}

An inequality à la Szegő–Weinberger for the p-Laplacian on convex sets

whose mutual distance tends to infinity.

Lipschitz regularity for local minimizers of some widely degenerate problems

Given an open and bounded set