#### 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