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A Wiener test for nondivergence structure, second-order elliptic equations. (English) Zbl 0583.35034

The purpose of this paper is to prove a Wiener test which is necessary and sufficient for a boundary point \(y\in \partial \Omega\) to be regular for the equation \[ Lu=\sum^{n}_{i,j=1}a_{ij}(x)D^ 2_{x_ ix_ j}u+\sum^{n}_{i=1}b_ i(x)D_{x_ i}u=0 \] in a bounded domain \(\Omega\) in \(R^ n\). This result supplements a Wiener-type series test due to Landis, which is sufficient for regularity. As an application the author reproves a sufficient condition for regularity of \(y\in \partial \Omega\), also due to Landis: Theorem 4.1. Suppose \(y\in \partial \Omega\) and \(meas (\Omega^ c\cap B_ r(y))\geq c\cdot meas B_ r(y)\) for all \(r\in (0,\rho)\) and some \(c>0\). Then y is a regular point for \(Lu=0\) in \(\Omega\).
Reviewer: J.Wloka

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B65 Smoothness and regularity of solutions to PDEs
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