Geometric-type Sobolev inequalities and applications to the regularity of minimizers

TL;DR

This paper develops weighted Sobolev inequalities involving mean curvature and applies them to derive regularity estimates for semi-stable solutions of nonlinear elliptic equations, including bounds for extremal solutions in convex domains.

Contribution

It introduces new geometric Sobolev inequalities based on mean curvature and applies them to improve regularity results for solutions of elliptic PDEs.

Findings

Established weighted Sobolev and Morrey's inequalities with curvature weights.

Derived new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions.

Obtained bounds for extremal solutions in convex domains for dimensions $n \\geq 5$.

Abstract

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-\Delta u=g(u)$ in a bounded domain $\Omega$ of $\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(\Omega)$ bound for the extremal solution of $-\Delta u=\lambda f(u)$ when $n\geq 5$ and the domain is convex. We recall that extremal solutions are known to be bounded in convex domains if $n\leq 4$, and that their boundedness is expected ---but still unkwown--- for $n\leq 9$.