Regularity of $\infty$ for elliptic equations with measurablecoefficients and its consequences

Ugur G. Abdulla

doi:10.3934/dcds.2012.32.3379

Article Contents

2012, Volume 32, Issue 10: 3379-3397. Doi: 10.3934/dcds.2012.32.3379

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Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences

Ugur G. Abdulla^1,

1.
Department of Mathematics, Florida Institute of Technology, Melbourne, Florida 32901

Received: December 31, 2010

Revised: February 29, 2012

Published: September 2012

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Abstract

This paper introduces a notion of regularity (or irregularity) of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset {\mathbb R}^{N}$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the ${\mathcal A}$- harmonic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of ${\mathbb R}^{N}, N\ge 3$ is established in terms of the Wiener test for the regularity of $\infty$. It coincides with the Wiener test for the regularity of $\infty$ in the case of Laplace equation. From the topological point of view, the Wiener test at $\infty$ presents thinness criteria of sets near $\infty$ in fine topology. Precisely, the open set is a deleted neigborhood of $\infty$ in fine topology if and only if $\infty$ is irregular.

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Mathematics Subject Classification: Primary: 35J25, 31C05, 31C15, 31C40; Secondary: 60J45, 60J60.

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