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

Dindoš, Martin; Rule, David J.

Rev. Mat. Iberoam. 26, No. 3, 1013-1034 (2010).

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)].

MSC:

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

Keywords:

elliptic equations; Carleson measure condition; Neumann problem; regularity problem; distributional inequalities; inhomogeneous equation

Citations:

Zbl 1113.35314

References:

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