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Transactions of the American Mathematical Society

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

 
 

 

Interior and boundary continuity of weak solutions of degenerate parabolic equations


Author: William P. Ziemer
Journal: Trans. Amer. Math. Soc. 271 (1982), 733-748
MSC: Primary 35K60; Secondary 35K65
DOI: https://doi.org/10.1090/S0002-9947-1982-0654859-7
MathSciNet review: 654859
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Abstract: In this paper we consider degenerate parabolic equations of the form $ ({\ast})$

$\displaystyle \beta {(u)_t} - \operatorname{div} A(x,\,t,\,u,\,{u_x}) + B(x,\,t,\,u,\,{u_x}) \ni 0$

where $ A$ and $ B$ are, respectively, vector and scalar valued Baire functions defined on $ U \times {R^1} \times {R^n}$, where $ U$ is an open subset of $ {R^{n + 1}}(x,\,t)$. The functions $ A$ and $ B$ are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation $ \beta \subset {R^1} \times {R^1}$ so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of $ ({\ast})$ are continuous throughout $ U$. In the event that $ U = \Omega \times (0,\,T)$ where $ \Omega $ is an open set of $ {R^n}$, it is also shown that a weak solution is continuous at $ ({x_0},{t_0}) \in \partial \Omega \times (0,\,T)$ provided $ {x_0}$ is a regular point for the Laplacian on $ \Omega $.

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DOI: https://doi.org/10.1090/S0002-9947-1982-0654859-7
Article copyright: © Copyright 1982 American Mathematical Society

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