Vol. 13, No. 4, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
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

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.

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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