Reverse Faber-Krahn inequality for a truncated Laplacian operator

Abstract

In this talk we will consider a reverse Faber-Krahn inequality for the principal eigenvalue $\mu_1(\Omega)$ of the fully nonlinear operator

\[ \mathcal{P}_N^+ u := \lambda_N(D^2 u), \]

where $\Omega \subset \mathbb{R}^N$ is a bounded, open convex set, and $\lambda_N(D^2 u)$ is the largest eigenvalue of the Hessian matrix of $u$. The result will be a consequence of the isoperimetric inequality

\[ \mu_1(\Omega) \leq \frac{\pi^2}{\text{diam}(\Omega)^2}. \]

Moreover, we will discuss the minimization of $\mu_1$ under various kinds of constraints. The results have been obtained in collaboration with Julio D. Rossi and Ariel Salort (Buenos Aires).

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