Extremal functions for Morrey’s inequality in convex domains
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208_2018_Article_1775.pdf
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Author(s)
Lindgren, Erik
Hynd, Ryan C
Date Issued
November 2018
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Citation
Hynd, Ryan, and Erik Lindgren. “Extremal Functions for Morrey’s Inequality in Convex Domains.” Mathematische Annalen, Nov. 2018. © 2018 The Authors
Version
Final published version
Abstract
For a bounded domain Ω ⊂ R[superscript n] and p>n , Morrey’s inequality implies that there is c>0 such that c∥u∥p[subscript ∞]≤∫[subscript Ω]|Du|p[subscript dx] for each u belonging to the Sobolev space W[superscript 1,p][subscript 0](Ω) . We show that the ratio of any two extremal functions is constant provided that Ω is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green’s function for the p-Laplacian.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00208-018-1775-8
