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Proceedings of the American Mathematical Society

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



Schwarz reflection principles for solutions of parabolic equations

Author: David Colton
Journal: Proc. Amer. Math. Soc. 82 (1981), 87-94
MSC: Primary 35B60; Secondary 35K10
MathSciNet review: 603607
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Abstract: A reflection principle is obtained for solutions of the heat equation defined in a cylindrical domain of the form $ \Omega \times (0,T)$ where $ \Omega $ is a ball in $ {{\mathbf{R}}^n}$ and the solution vanishes on $ \partial \Omega \times (0,T)$. It is shown that the domain of dependence of the solution at a point outside the cylinder $ \Omega \times (0,T)$ is a line segment contained inside $ \Omega \times (0,T)$. In the case $ n = 2$ this result is generalized to the case of analytic solutions of parabolic equations with analytic coefficients defined in an arbitrary bounded simply connected cylinder $ D \times (0,T)$ where the solution vanishes on a portion of $ \partial D \times (0,T)$.

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Article copyright: © Copyright 1981 American Mathematical Society