Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $.

References [Enhancements On Off] (What's this?)

  • [AS] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81-123. MR 0244638 (39:5952)
  • [CD] J. R. Cannon and E. DiBenedetto, On the existence of weak solutions to an $ n$-dimensional Stefan problem with non-linear boundary conditions, SIAM J. Math. Anal. 11 (1980). MR 579555 (81j:35058)
  • [CE] L. A. Caffarelli and L. C. Evans, Continuity of the temperature in the two-phase Stefan problem (to appear). MR 683353 (84g:35070)
  • [CF] L. A. Cafarelli and A. Friedman, Continuity of the density of a gas flow in a porous medium (to appear).
  • [D1] E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations (to appear). MR 663969 (83k:35045)
  • [D2] -, Continuity of weak solutions to a general porous media equation (to appear).
  • [DZ] D. J. Deignan and W. P. Ziemer, Strong differentiability properties of Bessel potentials, Trans. Amer. Math. Soc. 225 (1977), 113-122. MR 0422645 (54:10631)
  • [GZ1] R. Gariepy and W. P. Ziemer, Behavior at the boundary of solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal. 56 (1974), 372-384. MR 0355332 (50:7807)
  • [GZ2] -, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal. 67 (1977), 25-39. MR 0492836 (58:11898)
  • [LSU] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and quasilinear equations of parabolic type, Transl. Math. Mono., vol. 23, Amer. Math. Soc, Providence, R. I., 1968.
  • [LU] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968. MR 0244627 (39:5941)
  • [S] P. Sacks, Existence and regularity of solutions of the inhomogeneous porous medium equations, Math. Research Center, Technical report, December, 1980.
  • [T] N. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205-266. MR 0226168 (37:1758)
  • [Z] W. P. Ziemer, Behavior at the boundary of solutions of quasilinear parabolic equations, J. Differential Equations 35 (1980), 291-305. MR 563383 (81c:35073)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K60, 35K65

Retrieve articles in all journals with MSC: 35K60, 35K65


Additional Information

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

American Mathematical Society