Parabolic Lp Dirichlet boundary value problem and VMO-type time-varying domains

Dindoš, Martin; Dyer, Luke; Hwang, Sukjung

doi:10.2140/apde.2020.13.1221

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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

DOI: 10.2140/apde.2020.13.1221

Abstract

We prove the solvability of the parabolic $L^{p}$ Dirichlet boundary value problem for $1 < p \leq \infty$ for a PDE of the form $u_{t} = div (A \nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients $A = [a_{i j} (X, t)]$ and $B = [b_{i}]$ 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 $L^{p}$ Dirichlet boundary value problem is solvable for all $1 < p \leq \infty$ . 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 $L^{p}$ for all $1 < p \leq \infty$ , which then (using the method of layer potentials) implies solvability of the $L^{p}$ 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 $L^{p}$ 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

Vol. 13, No. 4, 2020