Rigidity theorems for best Sobolev inequalities

TL;DR

This paper investigates the best Sobolev inequalities on open sets in Euclidean space, proving existence and rigidity results that characterize when equality occurs, especially comparing general domains to balls.

Contribution

It establishes existence of minimizers with at most one boundary concentration point and proves rigidity theorems confirming balls have the worst inequalities, answering a longstanding open question.

Findings

Existence of generalized minimizers with at most one boundary concentration point.

Existence of classical minimizers when n > 2p.

Rigidity results confirming balls have the worst Sobolev inequalities.

Abstract

For $n\geq 2$, $p\in(1,n)$, the "best $p$-Sobolev inequality" on an open set $\Omega\subset\mathbb{R}^n$ is identified with a family $\Phi_\Omega$ of variational problems with critical volume and trace constraints. When $\Omega$ is bounded we prove: (i) for every $n$ and $p$, the existence of generalized minimizers that have at most one boundary concentration point, and: (ii) for $n> 2\,p$, the existence of (classical) minimizers. We then establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities" by the first named author and Villani, thus giving the first affirmative answers to a question raised in [MV05].