Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities

TL;DR

This paper establishes a Brunn-Minkowski type inequality for the first Dirichlet eigenvalue of a weighted p-operator in Gauss space and shows that positive eigenfunctions are log-concave in convex domains.

Contribution

It introduces a new inequality for the eigenvalues of a weighted p-operator in Gauss space and proves log-concavity of eigenfunctions in convex domains.

Findings

Proved a Brunn-Minkowski type inequality for the first eigenvalue.

Established log-concavity of positive eigenfunctions in convex domains.

Extended geometric properties of solutions to elliptic PDEs in Gauss space.

Abstract

We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -\Delta_{p,\gamma}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] where $p>1$, in the class of bounded Lipschitz domains in $\mathbb{R}^n$. We also prove that any corresponding positive eigenfunction is log-concave if the domain is convex.