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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Fatou theorem for the solution of the heat equation at the corner points of a cylinder


Author: Kin Ming Hui
Journal: Trans. Amer. Math. Soc. 333 (1992), 607-642
MSC: Primary 35K05; Secondary 35A05, 35C99
MathSciNet review: 1091707
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Abstract: In this paper the author proves existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain $ D \times (0,\infty )$ where $ D$ is a bounded smooth domain in $ {R^n}$ with zero lateral values. A unique representation of the strong solution is given in terms of measures $ \mu $ on $ D$ and $ \lambda $ on $ \partial D$. We also show that the strong solution $ u(x,t)$ of the heat equation in a cylinder converges a.e. $ {x_0} \in \partial D \times \{ 0\} $ as $ (x,t)$ converges to points on $ \partial D \times \{ 0\} $ along certain nontangential paths.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1091707-4
PII: S 0002-9947(1992)1091707-4
Keywords: Heat equation, initial-Dirichlet problem, Fatou theorem at corner points
Article copyright: © Copyright 1992 American Mathematical Society