A non-smooth Brezis-Oswald uniqueness result

TL;DR

This paper extends the Brezis-Oswald uniqueness result to non-smooth functionals by classifying critical points of a convex, positively p-homogeneous functional with non-differentiable components, using a non-smooth Picone inequality.

Contribution

It introduces a framework for analyzing non-smooth, non-convex functionals and establishes uniqueness of critical points without relying on Euler-Lagrange equations.

Findings

Classifies non-negative critical points of a non-smooth functional.

Establishes uniqueness results using a non-smooth Picone inequality.

Provides regularity results for critical points without classical Euler-Lagrange equations.

Abstract

We classify the non-negative critical points in $W^{1,p}_0(\Omega)$ of \[ J(v)=\int_\Omega H(Dv)-F(x, v)\, dx \] where $H$ is convex and positively $p$-homogeneous, while $t\mapsto \partial_tF(x, t)/t^{p-1}$ is non-increasing. Since $H$ may not be differentiable and $F$ has a one-sided growth condition, $J$ is only l.s.c. on $W^{1,p}_0(\Omega)$. We employ a weak notion of critical point for non-smooth functionals, derive sufficient regularity of the latter without an Euler-Lagrange equation available and focus on the uniqueness part of the results in \cite{BO}, through a non-smooth Picone inequality.