Reverse Faber–Krahn inequality for a truncated Laplacian operator

Abstract

In this paper we prove a reverse Faber–Krahn inequality for the principal eigenvalue ${\mu }_1(\Omega )$ of the fully nonlinear eigenvalue problem

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 $\mathrm{\Omega }\subset {ℝ}^N$, the inequality

$${\mu }_1(\mathrm{\Omega })\le \frac{{\pi }^2}{{[\text{diam}(\mathrm{\Omega })]}^2}={\mu }_1(B_{\text{diam}(\mathrm{\Omega })/2}),$$

where $\text{diam}(\mathrm{\Omega })$ is the diameter of $\mathrm{\Omega }$, 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 ${\mu }_1(\mathrm{\Omega })$ under different kinds of constraints.