Homogenization of one-phase Stefan-type problems in periodic and random media
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- by Inwon C. Kim and Antoine Mellet PDF
- Trans. Amer. Math. Soc. 362 (2010), 4161-4190 Request permission
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.References
- M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math. 323 (1981), 53–67. MR 611442, DOI 10.1515/crll.1981.323.53
- D. G. Aronson, On the Green’s function for second order parabolic differential equations with discontinous coefficients, Bull. Amer. Math. Soc. 69 (1963), 841–847. MR 155109, DOI 10.1090/S0002-9904-1963-11059-9
- 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
- Luis A. Caffarelli and Avner Friedman, Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J. 28 (1979), no. 1, 53–70. MR 523623, DOI 10.1512/iumj.1979.28.28004
- Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
- Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
- J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
- Gianni Dal Maso and Luciano Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4) 144 (1986), 347–389 (English, with Italian summary). MR 870884, DOI 10.1007/BF01760826
- Gianni Dal Maso and Luciano Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math. 368 (1986), 28–42. MR 850613
- Georges Duvaut, Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1461–A1463 (French). MR 328346
- Avner Friedman, The Stefan problem in several space variables, Trans. Amer. Math. Soc. 133 (1968), 51–87. MR 227625, DOI 10.1090/S0002-9947-1968-0227625-7
- Avner Friedman and David Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J. 24 (1974/75), no. 11, 1005–1035. MR 385326, DOI 10.1512/iumj.1975.24.24086
- Nicola Garofalo and Ermanno 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, DOI 10.1090/S0002-9947-1988-0951629-2
- Ei-ichi Hanzawa, Classical solutions of the Stefan problem, Tohoku Math. J. (2) 33 (1981), no. 3, 297–335. MR 633045, DOI 10.2748/tmj/1178229399
- S. L. Kamenomostskaja, On Stefan’s problem, Mat. Sb. (N.S.) 53 (95) (1961), 489–514 (Russian). MR 0141895
- Inwon C. Kim, Uniqueness and existence results on the Hele-Shaw and the Stefan problems, Arch. Ration. Mech. Anal. 168 (2003), no. 4, 299–328. MR 1994745, DOI 10.1007/s00205-003-0251-z
- Inwon C. Kim, Homogenization of the free boundary velocity, Arch. Ration. Mech. Anal. 185 (2007), no. 1, 69–103. MR 2308859, DOI 10.1007/s00205-006-0035-3
- I. Kim and A. Mellet, Homogenization of a Hele-Shaw problem in periodic and random media, submitted.
- Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
- O. A. Oleĭnik, A method of solution of the general Stefan problem, Soviet Math. Dokl. 1 (1960), 1350–1354. MR 0125341
- Mario Primicerio, Stefan-like problems with space-dependent latent heat, Meccanica 5 (1970), 187–190 (English, with Italian summary). MR 0372424, DOI 10.1007/BF02133573
- George C. Papanicolaou and S. R. S. Varadhan, Diffusions with random coefficients, Statistics and probability: essays in honor of C. R. Rao, North-Holland, Amsterdam, 1982, pp. 547–552. MR 659505
- J.-F. Rodrigues, The Stefan problem revisited, Mathematical models for phase change problems (Óbidos, 1988) Internat. Ser. Numer. Math., vol. 88, Birkhäuser, Basel, 1989, pp. 129–190. MR 1038069
- José-Francisco 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, DOI 10.1090/S0002-9947-1982-0670933-3
- Tomáš Roubíček, The Stefan problem in heterogeneous media, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. 6, 481–501 (English, with French summary). MR 1035339, DOI 10.1016/S0294-1449(16)30311-0
Additional Information
- Inwon C. Kim
- Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095
- MR Author ID: 684869
- 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. - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4161-4190
- MSC (2010): Primary 35Q35, 35Q80, 74Q10, 78M40
- DOI: https://doi.org/10.1090/S0002-9947-10-04945-7
- MathSciNet review: 2608400