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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A strong type of regularity for the $\textrm {PWB}$ solution of the Dirichlet problem
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by D. H. Armitage PDF
Proc. Amer. Math. Soc. 61 (1976), 285-289 Request permission

Abstract:

Let ${H_f}$ be the Perron-Wiener-Brelot solution of the Dirichlet problem for a resolutive function $f$ on the boundary $\partial \Omega$ of a bounded domain $\Omega$ in ${E^n}$. A point $y$ of $\partial \Omega$ will be called strongly regular if ${H_f}(x) \to f(y)(x \to y)$ whenever $f$ is resolutive and continuous at $y$. Necessary and sufficient conditions for strong regularity are given.
References
  • Marcel Brelot, Sur la mesure harmonique et le problème de Dirichlet, Bull. Sci. Math. (2) 69 (1945), 153–156 (French). MR 16187
  • —, Éléments de la théorie classique du potentiel, Centre de Documentation Universitaire, Paris, 1969.
  • Nicolaas du Plessis, An introduction to potential theory, University Mathematical Monographs, No. 7, Hafner Publishing Co., Darien, Conn.; Oliver and Boyd, Edinburgh, 1970. MR 0435422
  • L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0261018
  • John T. Kemper, A boundary Harnack principle for Lipschitz domains and the principle of positive singularities, Comm. Pure Appl. Math. 25 (1972), 247–255. MR 293114, DOI 10.1002/cpa.3160250303
  • Ü. Kuran, Harmonic majorizations in half-balls and half-spaces, Proc. London Math. Soc. (3) 21 (1970), 614–636. MR 315148, DOI 10.1112/plms/s3-21.4.614
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 61 (1976), 285-289
  • MSC: Primary 31B20
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0427658-1
  • MathSciNet review: 0427658