Geometric-type Sobolev inequalities and applications to the regularity\n of minimizers

2011/11/11 by Xavier Cabré, Cabre, Xavier, Manel Sanchón +1
Computer Science · Mathematics · #35B65 #35K57 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1111.2801

openalex publication_date 2011/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is twofold. We first prove a weighted Sobolev\ninequality and part of a weighted Morrey's inequality, where the weights are a\npower of the mean curvature of the level sets of the function appearing in the\ninequalities. Then, as main application of our inequalities, we establish new\nL^q and W^1,q estimates for semi-stable solutions of -\Δ u=g(u) in\na bounded domain \Ω of \ℝⁿ. These estimates lead to an\nL^2n/(n-4)(\Ω) bound for the extremal solution of -\Δ u=\λ\nf(u) when n\≥ 5 and the domain is convex. We recall that extremal\nsolutions are known to be bounded in convex domains if n\≤ 4, and that\ntheir boundedness is expected ---but still unkwown--- for n\≤ 9.\n

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