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)

 
 

 

Free boundary convergence in the homogenization of the one-phase Stefan problem


Author: José-Francisco Rodrigues
Journal: Trans. Amer. Math. Soc. 274 (1982), 297-305
MSC: Primary 35R35; Secondary 35K05
DOI: https://doi.org/10.1090/S0002-9947-1982-0670933-3
MathSciNet review: 670933
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the one phase Stefan problem in a "granular" medium, i.e., with nonconstant thermal diffusity, and we study the asymptotic behaviour of the free boundary with respect to homogenization. We prove the convergence of the coincidence set in measure and in the Hausdorff metric. We apply this result to the free boundary and we obtain the convergence in mean for the star-shaped case and the uniform convergence for the one-dimensional case, respectively. This gives an answer to a problem posed by J. L. Lions in [L].


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

  • [B] G. A. Beer, Hausdorff metric and convergence in measure, Michigan Math. J. 21 (1974), 63-64. MR 0367161 (51:3403)
  • [BLP] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978. MR 503330 (82h:35001)
  • [C] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1978), 155-184. MR 0454350 (56:12601)
  • [CF] L. A. Caffarelli and A. Friedman, Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J. 28 (1979), 53-70. MR 523623 (80i:35104)
  • [CT] P. Charrier and G. M. Troianiello, Un résultat d'existence et de régularité pour les solutions fortes d'un problème unilatéral d'évolution avec obstacle dépendent du temps, C. R. Acad. Sci. Paris Ser. A 281 (1975), 621-623. MR 0382826 (52:3708)
  • [Ch] F. Chiarenza, Principio di massimo forte per sottosoluzionni di equazioni paraboliche di tipo variazionale, Matematiche (Catania) 32 (1978), 32-43.
  • [CR] M. Codegone and J. F. Rodrigues, Convergence of the coincidence set in homogenization of the obstacle problem, Ann. Fac. Sci. Toulouses 3 (1981), 275-285. MR 658736 (83h:49014)
  • [De] C. Dellacherie, Ensembles analytiques. Capacités. Mesures de Hausdorff, Lecture Notes in Math., vol. 295, Springer-Verlag, Berlin, 1972. MR 0492152 (58:11301)
  • [D] G. Duvaut, Résolution d'un problème de Stefan, C. R. Acad. Sci. Paris 276 (1973), 1461-1463. MR 0328346 (48:6688)
  • [FK] A. Friedman and D. Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J. 24 (1975), 1005-1035. MR 0385326 (52:6190)
  • [KN] D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Comm. Pure Appl. Math. 31 (1978), 257-282. MR 480348 (82b:35152)
  • [KS] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980. MR 567696 (81g:49013)
  • [LSU] G. A. Ladyženskaya, V. A. Solonnikov, and N. N. Uralceva, Linear and quasi-linear equations of parabolic type, Transl. Math. Monos., vol. 23, Amer. Math. Soc., Providence, R. I., 1968.
  • [L] J. L. Lions, Asymptotic behaviour of solutions of variational inequalities with highly oscillating coefficients, Lecture Notes in Math., vol. 503, Springer-Verlag, Berlin, 1976, pp. 30-55. MR 0600341 (58:29088)
  • [M1] F. Murat, Sur l'homogéneisation d'inéquations elliptiques du $ 2$ème ordre relatives au convexe $ k({\Psi _1},{\Psi _2})$, Thèse d'Etat, Université Paris VI, 1976.
  • [M2] -, Oral communication, Paris, June 1980.
  • [R] J. F. Rodrigues, Sur le comportement asymptotique de la solution et de la frontière libre d'une inéquation variationelle parabolique, Ann. Fac. Sci. Toulouse 4 (1982) (to appear). MR 701732 (84e:35089)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35R35, 35K05

Retrieve articles in all journals with MSC: 35R35, 35K05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0670933-3
Keywords: Free boundary problems, Parabolic variational inequalities, homogenization, Stefan problem
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society