Attainability of the best constant of Hardy-Sobolev inequality with full boundary singularities

TL;DR

This paper investigates when the best constant in a Hardy-Sobolev inequality with boundary singularities is attained, linking geometric domain properties to attainability across different dimensions.

Contribution

It characterizes the attainability of the best constant based on domain geometry, including curvature and convexity, extending understanding of boundary singularity inequalities.

Findings

Best constant achieved iff domain is non-convex in 2D.

Achievability linked to negative mean curvature in higher dimensions.

Best constant not achieved for domains close to a ball in $C^2$ sense.

Abstract

We consider a type of Hardy-Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a conformally invariant one using the conformal radius of the domain. The best constant of such inequality on a smooth bounded domain is achieved if and only if the domain is non-convex. In higher dimensions, the best constant is achieved if the domain has negative mean curvature somewhere. If the mean curvature vanishes but is non-umbilic somewhere, we also establish the attainability for some special cases. In the other direction, we also show the best constant is not achieved if the domain is sufficiently close to a ball in $C^2$ sense.