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A Regularity Result for the p-Laplacian Near Uniform Ellipticity

Carlo Mercuri Orcid Logo, Giuseppe Riey, Berardino Sciunzi

SIAM Journal on Mathematical Analysis, Volume: 48, Issue: 3, Pages: 2059 - 2075

Swansea University Author: Carlo Mercuri Orcid Logo

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DOI (Published version): 10.1137/16m1058546

Abstract

We consider weak solutions to a class of Dirichlet boundary value problems involving the $p$-Laplace operator, and prove that the second weak derivatives have summability as high as it is desirable, provided p is sufficiently close to 2. And as a consequence the Holder exponent of the gradients appr...

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Published in: SIAM Journal on Mathematical Analysis
ISSN: 0036-1410 1095-7154
Published: Society for Industrial & Applied Mathematics (SIAM) 2016
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URI: https://cronfa.swan.ac.uk/Record/cronfa28298
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Abstract: We consider weak solutions to a class of Dirichlet boundary value problems involving the $p$-Laplace operator, and prove that the second weak derivatives have summability as high as it is desirable, provided p is sufficiently close to 2. And as a consequence the Holder exponent of the gradients approaches 1. We show that this phenomenon is driven by the classical Calderon-Zygmund constant. We believe that this result is particularly interesting in higher dimensions, and it is related to the optimal regularity of $p$-harmonic mappings. which is still an open question.
Keywords: quasi-linear elliptic equations, regularity theory, degenerate elliptic equations
College: College of Science
Issue: 3
Start Page: 2059
End Page: 2075