Thin Sets and Boundary Behavior of Solutions of the Helmholtz Equation |
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Authors: | Gowrisankaran Kohur Singman David |
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Affiliation: | (1) Department of Mathematics, McGill University, c[Montreal, Quebec, Canada, H3A 2K6;(2) Department of Mathematics, George Mason Universitat, Fairfax, USA, 22030 |
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Abstract: | The Martin boundary for positive solutions of the Helmholtz equation in n-dimensional Euclidean space may be identified with the unit sphere. Let v denote the solution that is represented by Lebesgue surface measure on the sphere. We define a notion of thin set at the boundary and prove that for each positive solution of the Helmholtz equation, u, there is a thin set such that u/v has a limit at Lebesgue almost every point of the sphere if boundary points are approached with respect to the Martin topology outside this thin set. We deduce a limit result for u/v in the spirit of Nagel–Stein (1984). |
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Keywords: | Helmholtz thin boundary behavior admissible approach region. |
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