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Transactions of the American Mathematical Society

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

 

 

Normal derivative for bounded domains with general boundary


Authors: Guang Lu Gong, Min Ping Qian and Martin L. Silverstein
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
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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

DOI: https://doi.org/10.1090/S0002-9947-1988-0951628-0
Keywords: Normal derivative, boundary conditions, Brownian motion, Dirichlet spaces
Article copyright: © Copyright 1988 American Mathematical Society