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Necessary and sufficient conditions for absolute continuity of elliptic- harmonic measure. (English) Zbl 0551.35024

Let A(x) be a continuous, symmetric, real, elliptic matrix-valued function of \(x\in R^{n+1}\). Consider the Dirichlet problem of the equation \((1)\quad div(A(x) \nabla u(x))=0\) in a bounded domain \(D\subset R^{n+1}\) of class \(C^ 1\). The authors give a necessary and sufficient condition on the modulus of continuity of the coefficients of (1) in order that the corresponding Poisson kernel for the Dirichlet problem exists. The condition on the coefficients is global continuity together with the property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function \(\eta\) (t) satisfying Dini-type condition \(\int_{o}\eta^ 2(t)t^{-1}dt<\infty\). The treatment is based on small perturbations of a constant coefficient equation and the main tool in the proof of sufficiency is a multilinear Littlewood-Paley estimate. The authors also consider the properties of the Poisson kernel implied by the necessary and sufficient condition.
Reviewer: A.Lahtinen

MSC:

35J25 Boundary value problems for second-order elliptic equations
35C15 Integral representations of solutions to PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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