Sobolev inequalities with jointly concave weights on convex cones

2020/03/26 by Balogh, Zoltán M., Gutiérrez, Cristian E., Kristály, Alexandru · 1 citation
#35A23 #46E35 #47J20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.12157

Abstract

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form (∫_E |u(x)|^q ω(x) dx)^1/q≤ K₀ (∫_E |∇ u(x)|^p σ(x) dx)^1/p, u∈ C₀^∞(\mathbb Rⁿ), \rm (WSI) where p≥ 1 and q>0 is the corresponding Sobolev critical exponent. Here E⊆ \mathbb Rⁿ is an open convex cone, and ω,σ:E→ (0,∞) are two homogeneous weights verifying a general concavity-type structural condition. The constant K₀= K₀(n, p, q, ω, σ) >0 is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that K₀ is optimal in (WSI) if and only if ω and σ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided.

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• The weighted isoperimetric inequality and Sobolev inequality outside convex sets

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