Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the steady-state continuous casting Stefan problem with nonlinear cooling


Authors: Michel Chipot and José-Francisco Rodrigues
Journal: Quart. Appl. Math. 40 (1983), 476-491
MSC: Primary 35K05; Secondary 35R35, 80A20
DOI: https://doi.org/10.1090/qam/693880
MathSciNet review: 693880
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Abstract: A steady-state one-phase Stefan problem corresponding to the solidification process of an ingot of pure metal by continuous casting with nonlinear lateral cooling is considered via the weak formulation introduced in [5] for the dam problem. Two existence results are obtained, for a general nonlinear flux and for a maximal monotone flux. Comparison results and the regularity of the free boundary are discussed. An uniqueness theorem is given for the monotone case.


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  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] Hans Wilhelm Alt, The fluid flow through porous media. Regularity of the free surface, Manuscripta Math. 21 (1977), no. 3, 255–272. MR 0449170, https://doi.org/10.1007/BF01167879
  • [3] C. Baiocchi and A. Capelo, Disequazioni variazionali e quasi variazional. Applicazioni a problemi di frontiera libera, I, II, Quarderni dell'U.M.I., Pitágora Ed. Bologna (1978)
  • [4] H. Brézis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, Math. Studies, 5, North Holland (1973)
  • [5] H. Brézis, D. Kinderlehrer and G. Stampacchia, Sur une nouvelle formulation du problème de l'écoulement à travers une digne, C.R. Acad. Sc. Paris 287-A, 711-714 (1978)
  • [6] T. Brière, Application des méthodes variationnelles à la cristallisation d'un métal par passage dans une gaîne de reproidissement, Ann. Fac. Sc. Toulouse 2, 219-247 (1980)
  • [7] Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), no. 3-4, 155–184. MR 0454350, https://doi.org/10.1007/BF02392236
  • [8] José Carrillo Menendez and Michel Chipot, On the dam problem, J. Differential Equations 45 (1982), no. 2, 234–271. MR 665999, https://doi.org/10.1016/0022-0396(82)90068-7
  • [9] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches Mathématiques, No. 21. MR 0464857
  • [10] D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 373–391. MR 0440187
  • [11] David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
  • [12] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris (1969)
  • [13] M. K. V. Murthy and G. Stampacchia, A variational inequality with mixed boundary conditions, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 188–224 (1973). MR 0333430, https://doi.org/10.1007/BF02760237
  • [14] Guido Stampacchia, Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hölderiane, Ann. Mat. Pura Appl. (4) 51 (1960), 1–37 (Italian). MR 0126601, https://doi.org/10.1007/BF02410941
  • [15] José-Francisco Rodrigues, Sur un problème à frontière libre stationnaire traduisant la cristallisation d’un métal, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 17, A823–A825 (French, with English summary). MR 580575
  • [16] L. I. Rubenšteĭn, The Stefan problem, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by A. D. Solomon; Translations of Mathematical Monographs, Vol. 27. MR 0351348

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DOI: https://doi.org/10.1090/qam/693880
Article copyright: © Copyright 1983 American Mathematical Society


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