Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T17:29:17.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the quasi-steady one phase Stefan problem

Published online by Cambridge University Press:  14 November 2011

Bento Louro  and
José-Francisco Rodrigues
Bento Louro
Affiliation:
Centro de Matemática e Aplicaçõoes Fundamentais, Av. Prof. Gama Pinto, 2, 1699 Lisboa Codex, Portugal
José-Francisco Rodrigues
Affiliation:
Centro de Matemática e Aplicaçõoes Fundamentais, Av. Prof. Gama Pinto, 2, 1699 Lisboa Codex, Portugal
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper presents some regularity results on the solution and on the free boundary for the one phase Stefan problem with zero specific heat in the framework of the variational inequalities formulation. In particular we show the Hölder continuity of the free boundary. Estimates on the rate of convergence when the specific heat vanishes are given for the variational solutions and for the free boundaries.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 263 - 275
Copyright
Copyright © Royal Society of Edinburgh 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Borsuk, M. V.. A priori estimates and solvability of second order quasilinear elliptic equations in a composite domain with nonlinear boundary conditions and a conjunction condition. Proc. Steklov Inst. Math. 103 (1968), 13–51.Google Scholar
2Brézis, H.. Problèmes unilateraux, J. Math. Pures Appl. 51 (1972), 1–168.Google Scholar
3Brézis, H.. Opérateurs maximaux monotones. Notas de Matemática 50 (Amsterdam: North Holland, 1973).Google Scholar
4Brézis, H.. Solutions with compact support of variational inequalities. Russian Math. Surveys 29, 2 (1974), 103–108.CrossRefGoogle Scholar
5Crank, J.. Free and moving boundary problems (Oxford: Clarendon Press, 1984).Google Scholar
6Duvaut, G.. Résolution d'un probleme de Stefan. C.R. Acad. Sci. Paris Sér. I. Math. 276 (1973), 1461–1463.Google Scholar
7Elliot, C. M.. On a variational inequality formulation of an electrochemical machining moving boundary problem and its approximation by the finite element method. J. Inst. Math. Appl. 25 (1980), 121–131.CrossRefGoogle Scholar
8Elliot, C. M. and Janoský, V.. A variational inequality approach to the Hele–Shaw flow with a moving boundary. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 93–107.CrossRefGoogle Scholar
9Friedman, A.. Variational principles and free–boundary problems (New York: J. Wiley, 1982).Google Scholar
10Friedman, A. and Kinderlehrer, D.. A one phase Stefan problem. Indiana Univ. Math. J. 24 (1975), 1005–1035.CrossRefGoogle Scholar
11Gustafsson, B.. Applications of variational inequalities to a moving boundary problem for Hele–Shaw flows. SIAM. J. Math. Anal. 16 (1985), 279–300.CrossRefGoogle Scholar
12Maderna, C. and Salsa, S.. Some special properties of solutions to obstacle problems. Rend. Sem. Mat. Univ. Padova 71 (1984), 121–129.Google Scholar
13Nirenberg, L.. An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa 20 (1966), 733–737.Google Scholar
14Rodrigues, J. F.. Stabilité de la frontière libre dans le problème de la digue. C.R. Acad. Sci. Paris Sér. I. Math. 294 (1982), 565–568.Google Scholar
15Rodrigues, J. F.. Further remarks on the stability of the obstacle problem (to appear).Google Scholar