The role of the mean curvature in a Hardy-Sobolev trace inequality

TL;DR

This paper investigates how the boundary mean curvature affects the Hardy-Sobolev trace inequality on bounded domains, establishing the existence of minimizers when the curvature is negative at the singular point.

Contribution

It introduces the influence of boundary mean curvature into the Hardy-Sobolev trace inequality and proves minimizer existence under specific curvature conditions.

Findings

Existence of minimizers when boundary mean curvature is negative at the singular point.

Extension of Hardy-Sobolev trace inequality to bounded domains considering curvature effects.

Connection between boundary geometry and functional inequality optimality.

Abstract

The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.