Parabolic boundary harnack principles in domains with thin lipschitz complement

Arshak Petrosyan, 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) : xn−1 ≤ f (x 00 , t), xn = 0} ⊂ 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,α-regularity of the free boundary in the parabolic Signorini problem.

Original languageEnglish
Pages (from-to)1421-1463
Number of pages43
JournalAnalysis and PDE
Volume7
Issue number6
DOIs
Publication statusPublished - 1 Jan 2014
Externally publishedYes

Keywords

  • Backward boundary Harnack principle
  • Heat equation
  • Kernel functions
  • Parabolic boundary Harnack principle
  • Parabolic signorini problem
  • Regularity of the free boundary
  • Thin free boundaries

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