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Homogenization of one-phase Stefan-type problems in periodic and random media

Authors: Inwon C. Kim and Antoine Mellet
Journal: Trans. Amer. Math. Soc. 362 (2010), 4161-4190
MSC (2010): Primary 35Q35, 35Q80, 74Q10, 78M40
Published electronically: March 24, 2010
MathSciNet review: 2608400
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Abstract: We investigate the homogenization of Stefan-type problems with oscillating diffusion coefficients. Both cases of periodic and random (stationary ergodic) mediums are considered. The proof relies on the coincidence of viscosity solutions and weak solutions (which are the time derivatives of the solutions of an obstacle problem) for the Stefan problem. This coincidence result is of independent interest.

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  • [AK] M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math., 323 (1981), 53-67. MR 611442 (83k:28015)
  • [A] D. G. Aronson, On the Green's function for second order parabolic differential equations with discontinuous coefficients. Bull. Amer. Math. Soc. 69 (1963), 841-847. MR 0155109 (27:5049)
  • [BLP] A. Bensoussan, J.-L. Lions and G. Papanicolaou. Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications, North-Holland, Amsterdam-New York, 1978. MR 503330 (82h:35001)
  • [CF] L. Caffarelli and A. Friedman, Continuity of the temperature in the Stefan problem, Indiana U. Math. J. 28 (1979) no.1, 53-70. MR 523623 (80i:35104)
  • [CL] G. M. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1-42. MR 690039 (85g:35029)
  • [Da] G. Dal Maso, An introduction to $ \Gamma$-convergence. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston (1993). MR 1201152 (94a:49001)
  • [Do] J. Doob, Classical potential theory and its probabilistic counterpart, New-York, Springer-Verlag, 1984. MR 731258 (85k:31001)
  • [DM1] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. 144 (1986), 347-389. MR 870884 (88h:49025)
  • [DM2] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 368 (1986), 28-42. MR 850613 (88k:28021)
  • [Du] G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glaceà zéro degreé). C. R. Acad. Sci. Paris Ser. A-B 276 (1973), A1461-A1463. MR 0328346 (48:6688)
  • [F] A. Friedman, The Stefan problem in several space variables. Trans. Amer. Math. Soc. 133 (1968) 51-87. MR 0227625 (37:3209)
  • [FK] A. Friedman and D. Kinderlehrer. A one phase Stefan problem. Indiana Univ. Math. J. 24 (1974/75), 1005-1035. MR 0385326 (52:6190)
  • [GL] N. Garofalo and E. Lanconelli, Wiener's criterion for parabolic equations with variable coefficients and its consequences. Trans. Amer. Math. Soc. 308 (1988), no. 2, 811-836. MR 951629 (89k:35104)
  • [H] E.-I. Hanzawa, Classical solutions of the Stefan problem. Tôhoku Math. J. 33 (1981), 297-335. MR 633045 (82k:35065)
  • [Ka] S. L. Kamenomostskaja, On Stefan's problem. (Russian) Mat. Sb. (N.S.) 53 (1961) 489-514. MR 0141895 (25:5292)
  • [K1] I. Kim, Uniqueness and Existence result of Hele-Shaw and Stefan problem, Arch. Rat. Mech. Anal, 168 (2003), 299-328. MR 1994745 (2004k:35422)
  • [K2] I. Kim, Homogenization of the free boundary velocity, Arch. Rat. Mech. Anal, 185 (2007), 69-103. MR 2308859 (2008f:35019)
  • [KM] I. Kim and A. Mellet, Homogenization of a Hele-Shaw problem in periodic and random media, submitted.
  • [M] J. Moser, On Harnack's inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. MR 0159139 (28:2357)
  • [O] O. A. Oleĭnik, A method of solution of the general Stefan problem. Dokl. Akad. Nauk SSSR 135 1054-1057 (Russian); translated as Soviet Math. Dokl. 1 (1960) 1350-1354. MR 0125341 (23:A2644)
  • [P] M. Primicerio, Stefan-like problems with space-dependent latent heat, Meccanica 5 (1970), 187-190. MR 0372424 (51:8633)
  • [PV] G. Papanicolaou and S.R.S. Varadhan, Diffusions with random coefficients. Statistics and probability: Essays in honor of C. R. Rao, pp. 547-552, North-Holland, Amsterdam, 1982. MR 659505 (85e:60082)
  • [R1] J.-F. Rodrigues, The Stefan problem revisited. Mathematical models for phase change problems (ibidos, 1988), 129-190, Internat. Ser. Numer. Math., 88, Birkhäuser, Basel, 1989. MR 1038069 (91e:80008)
  • [R2] J.-F. Rodrigues, Free boundary convergence in the homogenization of the one-phase Stefan problem, Trans. Amer. Math. Soc. 274 (1982), no.1, 297-305. MR 670933 (83k:35084)
  • [Rou] T. Roubíček, The Stefan problem in heterogeneous media, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 481-501. MR 1035339 (91c:35167)

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Additional Information

Inwon C. Kim
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095

Antoine Mellet
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Received by editor(s): February 25, 2008
Published electronically: March 24, 2010
Additional Notes: The first author was partially supported by NSF-DMS 0700732
The second author was partially supported by NSF grant DMS-0456647.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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