\(L_{p}\)-solvability of the Dirichlet problem for second order parabolic equations
type: Item
instance of: scholarly article
title: \(L_{p}\)-solvability of the Dirichlet problem for second order parabolic equations
zbMATH Open document ID: 1221.35182
author: A. N. Gordeev
author: Yu. A. Alkhutov
published in: Journal of Mathematical Sciences (New York)
publication date: 2011-07-18T00:00:00Z
review text: This paper concerns weak solutions to divergence form equations of the type \[ \partial_t u+ \sum^n_{j,k=1}\partial_j a_{jk}(x, t)\partial_k u= \sum^n_{j=1} \partial_j f_j(x,t)\quad\text{in }\Omega_T:= \Omega\times [0,T] \] with homogeneous initial and Dirichlet boundary conditions. Here \(\Omega\subset\mathbb{R}^n\) is a bounded domain, the coefficient functions \(a_{jk}:\overline\Omega\times [0,T]\to \mathbb{R}\) are continuous and satisfy a uniform ellipticity condition, and \(f\in L^p(Q_T; \mathbb{R}^n)\). There are formulated regularity conditions on \(\partial\Omega\) (in terms of a so-called harmonic measure on \(\partial\Omega)\) which are necessary and sufficient for unique solvability and for the a priori estimates \[ \| u\|_{L^p(Q_T)}+ \|\nabla_x u\|_{L^p(Q_T;\mathbb{R}^n)}\leq \text{const}\| f\|_{L^p(Q_T; \mathbb{R}^n)} \] for all \(p\geq 2\).
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5924724
zbMATH Keywords: divergence form equations
zbMATH Keywords: regularity conditions
zbMATH Keywords: harmonic measure
MaRDI profile type: Publication
full work available at URL: s10958-011-0202-4
OpenAlex ID: W1970106967
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DOI: 10.1007/S10958-011-0202-4
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description: scientific article; zbMATH DE number 5924724