# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021207
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## Higher order parabolic boundary Harnack inequality in C1 and Ck, α domains

 Universitat de Barcelona, Departament de Matematiques i Informàtica, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain, Springfield, MO 65801-2604, USA

Received  September 2021 Early access January 2022

Fund Project: The author has received funding from the European Research Council (ERC) under the Grant Agreement No 801867, and from the Swiss National Science Foundation project 200021 178795

We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in C1 and Ck, α domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary.

As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.

Citation: Teo Kukuljan. Higher order parabolic boundary Harnack inequality in C1 and Ck, α domains. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021207
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