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2007 The Dirichlet problem in convex bounded domains for operators in non-divergence form with $L^\infty$-coefficients
Matthias Hieber, Ian Wood
Differential Integral Equations 20(7): 721-734 (2007).

Abstract

Consider the Dirichlet problem for elliptic and parabolic equations in non-divergence form with variable coefficients in convex bounded domains of $\mathbb R^n$. We prove solvability of the elliptic problem and maximal $L^q$-$L^p$-estimates for the solution of the parabolic problem provided the coefficients $a_{ij} \in L^\infty$ satisfy a Cordes condition and $p \in (1,2]$ is close to $2$. This implies that in two dimensions, i.e., $n=2$, the elliptic Dirichlet problem is always solvable if the associated operator is uniformly strongly elliptic, and $p \in (1,2]$ is close to $2$, for maximal $L^q$-$L^p$-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.

Citation

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Matthias Hieber. Ian Wood. "The Dirichlet problem in convex bounded domains for operators in non-divergence form with $L^\infty$-coefficients." Differential Integral Equations 20 (7) 721 - 734, 2007.

Information

Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1212.42032
MathSciNet: MR2333653

Subjects:
Primary: 35J25
Secondary: 35K20, 47D06

Rights: Copyright © 2007 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.20 • No. 7 • 2007
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