Elliptic equations in the plane satisfying a Carleson measure condition. (English) Zbl 1206.35101

Summary: 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 C. E. Kenig and J. Pipher [Publ. Mat., Barc. 45, No. 1, 199–217 (2001; Zbl 1113.35314)].


35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B65 Smoothness and regularity of solutions to PDEs


Zbl 1113.35314
Full Text: DOI Euclid


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