Vol. 13, No. 4, 2020

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Parabolic $L^p$ Dirichlet boundary value problem and VMO-type time-varying domains

Martin Dindoš, Luke Dyer and Sukjung Hwang

Vol. 13 (2020), No. 4, 1221–1268
Abstract

We prove the solvability of the parabolic Lp Dirichlet boundary value problem for 1 < p for a PDE of the form ut = div(Au) + B u on time-varying domains where the coefficients A = [aij(X,t)] and B = [bi] satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard.

Furthermore, we establish that if the coefficients A, B of the operator satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic Lp Dirichlet boundary value problem is solvable for all 1 < p . This result is related to results in papers by Maz’ya, Mitrea and Shaposhnikova, and Hofmann, Mitrea and Taylor, where the fact that the boundary of the domain has a normal in VMO or near VMO implies invertibility of certain boundary operators in Lp for all 1 < p , which then (using the method of layer potentials) implies solvability of the Lp boundary value problem in the same range for certain elliptic PDEs.

Our result does not use the method of layer potentials since the coefficients we consider are too rough to use this technique, but remarkably we recover Lp solvability in the full range of p’s as in the two papers mentioned above.

Keywords
parabolic boundary value problems, $L^p$ solvability, VMO-type domains
Mathematical Subject Classification 2010
Primary: 35K10, 35K20
Secondary: 35R05
Milestones
Received: 28 October 2018
Revised: 12 March 2019
Accepted: 18 April 2019
Published: 13 June 2020
Authors
Martin Dindoš
School of Mathematics
The University of Edinburgh and Maxwell Institute of Mathematical Sciences
Edinburgh
United Kingdom
Luke Dyer
School of Mathematics
The University of Edinburgh and Maxwell Institute of Mathematical Sciences
Edinburgh
United Kingdom
Sukjung Hwang
Department of Mathematics
Yonsei University
Seoul
South Korea