Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space

TL;DR

This paper establishes universal Hardy-Sobolev inequalities on hypersurfaces in Euclidean space, involving mean curvature, with simplified proofs and new inequalities applicable to elliptic PDE regularity and improvements over existing Hardy inequalities.

Contribution

It provides simplified proofs of known Sobolev inequalities and introduces two new Hardy inequalities on hypersurfaces, including one derived from elliptic PDE regularity and an improved Hardy inequality.

Findings

Unified Hardy-Sobolev inequalities with universal constants

New Hardy inequalities relevant to elliptic PDE regularity

Improved Hardy and Hardy-Poincaré inequalities

Abstract

In this paper we study Hardy-Sobolev inequalities on hypersurfaces of $\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael-Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a "ground state" substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy-Poincar\'e inequality.