Reverse Faber-Krahn inequality for a truncated laplacian operator

In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $mu_1(Omega)$ of the fully nonlinear eigenvalue problem [ label{eq} left{begin{array}{r c l l} -lambda_N(D^2 u) & = & mu u & text{in }Omega, u & = & 0 & text{on }partial Omega. end{array}right. ] Here $ lambda_N(D^2 u)$ stands for the largest eigenvalue of the Hessian matrix of $u$. More precisely, we prove that, for an open, bounded, convex domain $Omega subset mathbb{R}^N$, the inequality [ mu_1(Omega) leq frac{pi^2}{[text{diam}(Omega)]^2} = mu_1(B_{text{diam}(Omega)/2}),] where $text{diam}(Omega)$ is the diameter of $Omega$, holds true. The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint. Furthermore, we discuss the minimization of $mu_1(Omega)$ under different kinds of constraints.

Compactness and dichotomy in nonlocal shape optimization

We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions concentration-compactness principle, we prove that either an optimal shape exists, or there exists a minimizing sequence consisting of two pieces whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.