In this paper we settle (in dimension ) the open question whether for a divergence form equation with coefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the Neumann and Dirichlet regularity problems are solvable for some values of . The related question for the 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].
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Martin Dindoš, David J. Rule, Elliptic equations in the plane satisfying a Carleson measure condition. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 1013–1034DOI 10.4171/RMI/625