### 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 language | English |
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Pages (from-to) | 1421-1463 |

Number of pages | 43 |

Journal | Analysis and PDE |

Volume | 7 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jan 2014 |

Externally published | Yes |

### Keywords

- 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

Petrosyan, A., & Shi, W. (2014). Parabolic boundary harnack principles in domains with thin lipschitz complement.

*Analysis and PDE*,*7*(6), 1421-1463. https://doi.org/10.2140/apde.2014.7.1421