Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

TL;DR

This paper develops geometric Sobolev inequalities involving curvature and gradient, then applies them to derive regularity estimates for semi-stable solutions of p-Laplace equations, including bounds for extremal solutions in convex domains.

Contribution

It introduces new Sobolev and Morrey inequalities involving curvature, leading to novel a priori estimates for semi-stable p-Laplace solutions, especially extremal solutions in convex domains.

Findings

u* is bounded in L-infinity if n ≤ p+2

u* belongs to L^{np/(n-p-2)} and W^{1,p}_0 if n > p+2

New geometric inequalities relate curvature and gradients in Sobolev spaces

Abstract

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \textit{a priori} estimates for semi-stable solutions of $-\Delta_p u= g(u)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^\infty(\Omega)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(\Omega)\cap W^{1,p}_0(\Omega)$ if $n>p+2$.