Parabolic boundary harnack principles in domains with thin lipschitz complement

Arshak Petrosyan, Wenhui Shi

Research output: Contribution to journalArticleResearchpeer-review

6 Citations (Scopus)


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
Issue number6
Publication statusPublished - 1 Jan 2014
Externally publishedYes


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

Cite this