Regularity of almost-minimizers of Hölder-coefficient surface energies

David A. Simmons

doi:10.3934/dcds.2022015

Article Contents

2022, Volume 42, Issue 7: 3233-3299. Doi: 10.3934/dcds.2022015

This issue Previous Article Introducing sub-Riemannian and sub-Finsler billiards Next Article Existence of positive solutions for a class of fractional Choquard equation in exterior domain

Regularity of almost-minimizers of Hölder-coefficient surface energies

David A. Simmons

University of Washington, Department of Mathematics, Box 354350, C-138 Padelford, Seattle, WA 98195-4350, USA

Received: May 31, 2021

Published: July 2022

The author was partially supported by NSF FRG 1853993

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We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.

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Mathematics Subject Classification: 49Q10, 49Q20, 49J45, 53A05.

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