Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 6 (2014), 1421-1463.

Parabolic boundary Harnack principles in domains with thin Lipschitz complement

Arshak Petrosyan and Wenhui Shi

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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 } ⊂ ℝ n − 1 × ℝ

for parabolically Lipschitz functions f on ℝn−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 C1,α-regularity of the free boundary in the parabolic Signorini problem.

Article information

Source
Anal. PDE, Volume 7, Number 6 (2014), 1421-1463.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731588

Digital Object Identifier
doi:10.2140/apde.2014.7.1421

Mathematical Reviews number (MathSciNet)
MR3270169

Zentralblatt MATH identifier
1304.35374

Subjects
Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35R35: Free boundary problems 35K85: Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40]

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

Citation

Petrosyan, Arshak; Shi, Wenhui. Parabolic boundary Harnack principles in domains with thin Lipschitz complement. Anal. PDE 7 (2014), no. 6, 1421--1463. doi:10.2140/apde.2014.7.1421. https://projecteuclid.org/euclid.apde/1513731588


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