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

Abstract

We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -Δ_{p,γ}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] 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.

Final version. Minor corrections in Lemma 3.3 and author contact information