The second eigenvalue of the fractional $p-$Laplacian

TL;DR

This paper investigates the second eigenvalue of the fractional p-Laplacian in bounded domains, providing variational characterizations, regularity discussions, and showing the non-existence of optimal shapes for a volume-constrained minimization problem.

Contribution

It extends the mountain pass characterization to the nonlocal nonlinear fractional p-Laplacian and analyzes the shape optimization problem, revealing the non-existence of optimal shapes.

Findings

Second eigenvalue is well-defined and characterized variationally.

The mountain pass characterization extends to nonlocal nonlinear setting.

No optimal shape exists for the volume-constrained minimization, with minimizing sequences involving disjoint balls.

Abstract

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ 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 \[ \inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}. \] 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.