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

TL;DR

The paper addresses Brunn-Minkowski type inequalities for the first Dirichlet eigenvalue of the Gaussian $p$-Laplacian and establishes log-concavity of the corresponding eigenfunction on convex domains. It develops a framework connecting viscosity and weak solutions in Gauss space, using sup-convolutions and a convolution-based construction to compare Rayleigh quotients. The main contributions are a Brunn-Minkowski inequality for the Gaussian $p$-Frequent eigenvalue $\lambda_{p,\gamma}$ and a simultaneous proof of log-concavity of the positive eigenfunction, both extending known Gaussian and $p$-Laplacian results. These results enhance the interaction between nonlinear PDEs in Gaussian space and convex geometry, with potential implications for isoperimetric-type inequalities and spectral theory in weighted spaces.

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.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Theorem 1.5
• Remark 1.6
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 3.1