Differential and Integral Equations

The Dirichlet problem in convex bounded domains for operators in non-divergence form with $L^\infty$-coefficients

Matthias Hieber and Ian Wood

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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.

Article information

Differential Integral Equations Volume 20, Number 7 (2007), 721-734.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]


Hieber, Matthias; Wood, Ian. The Dirichlet problem in convex bounded domains for operators in non-divergence form with $L^\infty$-coefficients. Differential Integral Equations 20 (2007), no. 7, 721--734. https://projecteuclid.org/euclid.die/1356039406.

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