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1 June 2020 Uniqueness of the blowup at isolated singularities for the Alt–Caffarelli functional
Max Engelstein, Luca Spolaor, Bozhidar Velichkov
Duke Math. J. 169(8): 1541-1601 (1 June 2020). DOI: 10.1215/00127094-2019-0077

Abstract

We prove the uniqueness of blowups and C1,log-regularity for the free-boundary of minimizers of the Alt–Caffarelli functional at points where one blowup has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a C1,α-regularity result. We also show that these additional assumptions are satisfied by the De Silva–Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the Łojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.

Citation

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Max Engelstein. Luca Spolaor. Bozhidar Velichkov. "Uniqueness of the blowup at isolated singularities for the Alt–Caffarelli functional." Duke Math. J. 169 (8) 1541 - 1601, 1 June 2020. https://doi.org/10.1215/00127094-2019-0077

Information

Received: 21 March 2018; Revised: 11 September 2019; Published: 1 June 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07226646
MathSciNet: MR4101738
Digital Object Identifier: 10.1215/00127094-2019-0077

Subjects:
Primary: 35R35
Secondary: 35J60

Keywords: Bernoulli problem , epiperimetric inequality , free boundary , monotonicity formula , Singular points

Rights: Copyright © 2020 Duke University Press

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Vol.169 • No. 8 • 1 June 2020
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