# The Brunn-Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators

## TL;DR

This paper establishes a Brunn-Minkowski inequality for the principal eigenvalue of certain fully nonlinear elliptic operators, extending classical results to broader classes of operators and domains.

## Contribution

It introduces a new Brunn-Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators under convexity assumptions.

## Key findings

- The principal eigenvalue satisfies a Brunn-Minkowski inequality.
- The result applies to the p-Laplacian and minimal Pucci operator.
- Existence and log-concavity of positive viscosity eigenfunctions are demonstrated.

## Abstract

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in $\mathbb{R}^n$ satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) $p$-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope method by Alvarez-Lasry-Lions. We also deal with the existence and log-concavity of positive viscosity eigenfunctions.