Vol. 7, No. 6, 2014

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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
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

$E=\left\{\left(x,t\right):{x}_{n-1}\le f\left({x}^{\prime \prime },t\right),{x}_{n}=0\right\}\subset {ℝ}^{n-1}×ℝ$

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

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 ${C}^{1,\alpha }$-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