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2020 Parabolic $L^p$ Dirichlet boundary value problem and VMO-type time-varying domains
Martin Dindoš, Luke Dyer, Sukjung Hwang
Anal. PDE 13(4): 1221-1268 (2020). DOI: 10.2140/apde.2020.13.1221


We prove the solvability of the parabolic Lp Dirichlet boundary value problem for 1<p for a PDE of the form ut= div(Au)+Bu 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.


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Martin Dindoš. Luke Dyer. Sukjung Hwang. "Parabolic $L^p$ Dirichlet boundary value problem and VMO-type time-varying domains." Anal. PDE 13 (4) 1221 - 1268, 2020.


Received: 28 October 2018; Revised: 12 March 2019; Accepted: 18 April 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07221202
MathSciNet: MR4109905
Digital Object Identifier: 10.2140/apde.2020.13.1221

Primary: 35K10, 35K20
Secondary: 35R05

Rights: Copyright © 2020 Mathematical Sciences Publishers


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Vol.13 • No. 4 • 2020
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