Extremal functions for Morrey's inequality in convex domains

2016/09/26 by Hynd, Ryan, Lindgren, Erik · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.08186

Abstract

For a bounded domain Ω⊂ ℝⁿ and p>n, Morrey's inequality implies that there is c>0 such that c‖u‖^p_∞≤ ∫_Ω|Du|^pdx for each u belonging to the Sobolev space W^1,p₀(Ω). We show that the ratio of any two extremal functions is constant provided that Ω is convex. We also explain why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this property. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the p-Laplacian.

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