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

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


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