| Résume | For $n\ge 2$ and $p\in(1,n)$ the "best $p$-Sobolev inequality" on an open set $\Om\subset\R^n$ is identified with a family $\Phi_\Om$ of variational problems with critical volume and trace constraints. In joint work with Neumayer and Tomasetti we prove that, if $\Om$ is bounded, then (i) for every $n$ and $p$, there exist generalized minimizers of $\Phi_\Om$, having at most one boundary concentration point, and: (ii) if $n> 2\,p$, then there exist (classical) minimizers. We then establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities", thus giving the first affirmative answers to a question raised in joint work with Villani (JGA 2005). A key ingredient in our analysis is the complete characterization of $\Phi_{Half Space}$ obtained in joint work with Neumayer (JFA 2017). Connections with the Yamabe problem are also discussed.
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