Preprint
Inserted: 24 jun 2022
Last Updated: 24 jun 2022
Year: 2022
Abstract:
For $n\ge 2$, $p\in(1,n)$, 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 https://cvgmt.sns.it/paper/555/.
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