Rigidity theorems for best Sobolev inequalities

2022/06/24 by Maggi, Francesco, Neumayer, Robin, Tomasetti, Ignacio
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.12386

Abstract

For n≥ 2, p∈(1,n), the "best p-Sobolev inequality" on an open set Ω⊂ℝⁿ is identified with a family Φ_Ω of variational problems with critical volume and trace constraints. When Ω 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].

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