Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space
Abstract
In this paper we study Hardy-Sobolev inequalities on hypersurfaces of , 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.
Keywords
Cite
@article{arxiv.1912.09282,
title = {Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space},
author = {Xavier Cabre and Pietro Miraglio},
journal= {arXiv preprint arXiv:1912.09282},
year = {2020}
}
Comments
In version 2, an equality within the proof of the Michael-Simon inequality has been corrected and it is now an inequality. This is explained in a new footnote. The correction does not alter at all the rest of the proof