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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Normal derivative for bounded domains with general boundary
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by Guang Lu Gong, Min Ping Qian and Martin L. Silverstein PDF
Trans. Amer. Math. Soc. 308 (1988), 785-809 Request permission

Abstract:

Let $D$ be a general bounded domain in the Euclidean space ${R^n}$. A Brownian motion which enters from and returns to the boundary symmetrically is used to define the normal derivative as a functional for $f$ with $f$, $\nabla f$ and $\Delta f$ all in ${L^2}$ on $D$. The corresponding Neumann condition (normal derivative $= 0$) is an honest boundary condition for the ${L^2}$ generator of reflected Brownian notion on $D$. A conditioning argument shows that for $D$ and $f$ sufficiently smooth this general definition of the normal derivative agrees with the usual one.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 308 (1988), 785-809
  • MSC: Primary 60J65; Secondary 35A99, 35R60
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0951628-0
  • MathSciNet review: 951628