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**Parabolic boundary Harnack principles in domains with thin Lipschitz complement.**
*(English)*
Zbl 1304.35374

In this interesting paper the authors study forward and backward boundary Harnack principles for nonnegative solutions of the heat equation in certain domains with thin Lipschitz complement. Such free boundaries are also known as thin free boundaries and are motivated by the parabolic Signorini problem.

The boundary Harnack principles give the possibility of proving that the thin Lipschitz free boundaries have Hölder-continuous spatial normals, following the original idea introduced by We have to point out that the elliptic counterparts of the results in this paper are very well known; see Athanasopoulos and Caffarelli. The authors prove two types of boundary Harnack principles for parabolic equations: the forward one (also known as the Carleson estimate) and the backward one.

The boundary Harnack principles give the possibility of proving that the thin Lipschitz free boundaries have Hölder-continuous spatial normals, following the original idea introduced by We have to point out that the elliptic counterparts of the results in this paper are very well known; see Athanasopoulos and Caffarelli. The authors prove two types of boundary Harnack principles for parabolic equations: the forward one (also known as the Carleson estimate) and the backward one.

Reviewer: Vincenzo Vespri (Firenze)

### MSC:

35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35R35 | Free boundary problems for PDEs |

35B45 | A priori estimates in context of PDEs |

### Keywords:

backward boundary Harnack principle; parabolic Signorini problem; thin free boundaries; regularity of free boundary; Carleson estimate
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\textit{A. Petrosyan} and \textit{W. Shi}, Anal. PDE 7, No. 6, 1421--1463 (2014; Zbl 1304.35374)

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