Parabolic boundary Harnack principles in domains with thin Lipschitz complement

Petrosyan, Arshak; Shi, Wenhui

doi:10.2140/apde.2014.7.1421

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Parabolic boundary Harnack principles in domains with thin Lipschitz complement

Arshak Petrosyan and Wenhui Shi

Vol. 7 (2014), No. 6, 1421–1463

DOI: 10.2140/apde.2014.7.1421

Abstract

We prove forward and backward parabolic boundary Harnack principles for nonnegative solutions of the heat equation in the complements of thin parabolic Lipschitz sets given as subgraphs

E = {(x, t) : x_{n - 1} \leq f (x^{''}, t), x_{n} = 0} \subset ℝ^{n - 1} \times ℝ

for parabolically Lipschitz functions $f$ on $ℝ^{n - 2} \times ℝ$ .

We are motivated by applications to parabolic free boundary problems with thin (i.e., codimension-two) free boundaries. In particular, at the end of the paper we show how to prove the spatial $C^{1, α}$ -regularity of the free boundary in the parabolic Signorini problem.

Keywords

parabolic boundary Harnack principle, backward boundary Harnack principle, heat equation, kernel functions, parabolic Signorini problem, thin free boundaries, regularity of the free boundary

Mathematical Subject Classification 2010

Primary: 35K20

Secondary: 35R35, 35K85

Milestones

Received: 8 December 2013

Accepted: 27 August 2014

Published: 18 October 2014

Authors

Arshak Petrosyan
Department of Mathematics Purdue University West Lafayette, IN 47907 United States

Wenhui Shi
Mathematisches Institut Universität Bonn Endenicher Allee 64 D-53115 Bonn Germany

Vol. 7, No. 6, 2014