Revista Matemática Iberoamericana

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Volume 26, Issue 3, 2010, pp. 1013–1034
DOI: 10.4171/RMI/625

Published online: 2010-12-31

Elliptic equations in the plane satisfying a Carleson measure condition

Martin Dindoš[1] and David J. Rule[2]

(1) Edinburgh University, UK
(2) Heriot-Watt University, Edinburgh, UK

In this paper we settle (in dimension $n=2$) the open question whether for a divergence form equation $\div (A\nabla u) = 0$ with coefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the $L^p$ Neumann and Dirichlet regularity problems are solvable for some values of $p\in (1,\infty)$. The related question for the $L^p$ Dirichlet problem was settled (in any dimension) in 2001 by Kenig and Pipher [Kenig, C.E. and Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45 (2001), no. 1, 199-217].

Keywords: Elliptic equations, Carleson measure condition, Neumann problem, regularity problem, distributional inequalities, inhomogeneous equation

Dindoš Martin, Rule David: Elliptic equations in the plane satisfying a Carleson measure condition. Rev. Mat. Iberoam. 26 (2010), 1013-1034. doi: 10.4171/RMI/625