Interior and boundary continuity of weak solutions of degenerate parabolic equations
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- by William P. Ziemer PDF
- Trans. Amer. Math. Soc. 271 (1982), 733-748 Request permission
Abstract:
In this paper we consider degenerate parabolic equations of the form $({\ast })$ \[ \beta {(u)_t} - \operatorname {div} A(x, t, u, {u_x}) + B(x, t, u, {u_x}) \ni 0\] 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 $\Omega$.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- 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