An isoperimetric inequality for extremal Sobolev functions

Tom Carroll; Jesse Ratzkin

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arXiv · 1208.1352

An isoperimetric inequality for extremal Sobolev functions

Tom Carroll·Jesse Ratzkin

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Abstract

Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev inequality, which finds a lower bound for their (p-1)-norm in terms of their p-norm. This inequality is sharp, and it is an equality if and only if the domain is a round ball. Our result generalizes a theorem of Payne and Rayner and our proof relies on integral rearrangements and an analysis of the ODE corresponding to the radial case.

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Tom Carroll, Jesse Ratzkin. 2012-08-07. An isoperimetric inequality for extremal Sobolev functions. https://arxiv.org/abs/1208.1352

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