Partial regularity of solutions to a class of degenerate systems
HTML articles powered by AMS MathViewer
- by Xiangsheng Xu PDF
- Trans. Amer. Math. Soc. 349 (1997), 1973-1992 Request permission
Abstract:
We consider the system $\displaystyle \frac {\partial u }{\partial t}-\Delta u=\sigma \left ( u\right ) \left | \nabla \varphi \right | ^2$, $\mathrm {div} \left ( \sigma \left ( u\right ) \nabla \varphi \right ) =0$ in $Q_T\equiv \Omega \times \left ( 0,T\right ]$ coupled with suitable initial-boundary conditions, where $\Omega$ is a bounded domain in $\mathbf {R}^N$ with smooth boundary and $\sigma \left ( u\right )$ is a continuous and positive function of $u$. Our main result is that under some conditions on $\sigma$ there exists a relatively open subset $Q_0$ of $Q_T$ such that $u$ is locally Hölder continuous on $Q_0$, the interior of $Q_T\backslash Q_0$ is empty, and $u$ is essentially bounded on $Q_T\backslash Q_0$.References
- S. N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness uniqueness, blowup, SIAM J. Math. Anal. 25 (1994), no. 4, 1128–1156. MR 1278895, DOI 10.1137/S0036141092233482
- Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
- Xinfu Chen, Existence and regularity of solutions of a nonlinear nonuniformly elliptic system arising from a thermistor problem, J. Partial Differential Equations 7 (1994), no. 1, 19–34. MR 1269721
- Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
- E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993), no. 5, 1107–1134. MR 1246185, DOI 10.2307/2375066
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- L. C. Young, On an inequality of Marcel Riesz, Ann. of Math. (2) 40 (1939), 567–574. MR 39, DOI 10.2307/1968941
- S. S. Chern (ed.), Seminar on nonlinear partial differential equations, Mathematical Sciences Research Institute Publications, vol. 2, Springer-Verlag, New York, 1984. Papers from the seminar held at the Mathematical Sciences Research Institute, Berkeley, Calif., May 9, 1983. MR 765224, DOI 10.1007/978-1-4612-1110-5
- José-Francisco Rodrigues, Obstacle problems in mathematical physics, North-Holland Mathematics Studies, vol. 134, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 114. MR 880369
- Peter Shi, Meir Shillor, and Xiangsheng Xu, Existence of a solution to the Stefan problem with Joule’s heating, J. Differential Equations 105 (1993), no. 2, 239–263. MR 1240396, DOI 10.1006/jdeq.1993.1089
- Peter Shi and Steve Wright, Higher integrability of the gradient in linear elasticity, Math. Ann. 299 (1994), no. 3, 435–448. MR 1282226, DOI 10.1007/BF01459793
- Hong Xie and W. Allegretto, $C^\alpha (\overline \Omega )$ solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem, SIAM J. Math. Anal. 22 (1991), no. 6, 1491–1499. MR 1129396, DOI 10.1137/0522096
- Xiangsheng Xu, A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type, Comm. Partial Differential Equations 18 (1993), no. 1-2, 199–213. MR 1211731, DOI 10.1080/03605309308820927
- Xiangsheng Xu, A $p$-Laplacian problem in $L^1$ with nonlinear boundary conditions, Comm. Partial Differential Equations 19 (1994), no. 1-2, 143–176. MR 1257001, DOI 10.1080/03605309408821012
- G. Yuan, Regularity of solutions of the nonstationary thermistor problem, Applicable Anal. 53(1994), 149–169.
- Guang Wei Yuan and Zu Han Liu, Existence and uniqueness of the $C^\alpha$ solution for the thermistor problem with mixed boundary value, SIAM J. Math. Anal. 25 (1994), no. 4, 1157–1166. MR 1278896, DOI 10.1137/S0036141092237893
Additional Information
- Xiangsheng Xu
- Affiliation: Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762
- Email: xxu@math.msstate.edu
- Received by editor(s): September 26, 1994
- Received by editor(s) in revised form: November 27, 1995
- Additional Notes: This work was supported in part by an NSF grant (DMS942448).
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 1973-1992
- MSC (1991): Primary 35B65, 35K65
- DOI: https://doi.org/10.1090/S0002-9947-97-01734-0
- MathSciNet review: 1373648