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2014 Parabolic boundary Harnack principles in domains with thin Lipschitz complement
Arshak Petrosyan, Wenhui Shi
Anal. PDE 7(6): 1421-1463 (2014). 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 f ( x , t ) , x n = 0 } n 1 ×

for parabolically Lipschitz functions f on n2×.

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 C1,α-regularity of the free boundary in the parabolic Signorini problem.

Citation

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Arshak Petrosyan. Wenhui Shi. "Parabolic boundary Harnack principles in domains with thin Lipschitz complement." Anal. PDE 7 (6) 1421 - 1463, 2014. https://doi.org/10.2140/apde.2014.7.1421

Information

Received: 8 December 2013; Accepted: 27 August 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1304.35374
MathSciNet: MR3270169
Digital Object Identifier: 10.2140/apde.2014.7.1421

Subjects:
Primary: 35K20
Secondary: 35K85 , 35R35

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

Rights: Copyright © 2014 Mathematical Sciences Publishers

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