Higher order parabolic boundary Harnack inequality in C1 and Ck, α domains

Teo Kukuljan

doi:10.3934/dcds.2021207

Article Contents

2022, Volume 42, Issue 6: 2667-2698. Doi: 10.3934/dcds.2021207

This issue Previous Article Global well-posedness for fractional Sobolev-Galpern type equations Next Article On involution kernels and large deviations principles on

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Higher order parabolic boundary Harnack inequality in C¹ and C^{k, α} domains

Teo Kukuljan

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

Received: August 31, 2021

Published: June 2022

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

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Abstract

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 C^{k, 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.

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Mathematics Subject Classification: Primary: 35K10; Secondary: 35R35.

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