Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems

TL;DR

This paper proves power- and log-concavity properties of positive viscosity solutions to certain degenerate elliptic PDEs in convex domains, revealing new structural insights into their solutions.

Contribution

It establishes that specific transformations of solutions are concave, introducing new methods like a weak comparison principle and a Hopf-type Lemma for these equations.

Findings

Positive solutions' transformations are concave (power and log)

Methods include weak comparison principle and Hopf-type Lemma

Results apply to degenerate elliptic PDEs in convex domains

Abstract

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of $-|\nabla u|^\alpha \Delta_p^N u = 1$ has the property that $u^\frac{\alpha + 1}{\alpha + 2}$ is a concave function. Secondly we consider positive solutions of the eigenvalue problem $-|\nabla u|^\alpha \Delta_p^N u = \lambda |u|^{\alpha} u$, in which case $\log u$ turns out to be concave. The methods provided include a weak comparison principle and a Hopf-type Lemma.