Savaré, Visintin: Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase

Savaré, Giuseppe and Visintin, Augusto:
Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase (Convergenza variazionale di equazioni di diffusione non lineari: applicazioni ai problemi di cambiamento di fase in capacità concentrata)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 8 (1997), fasc. n.1, p. 49-89, (English)
pdf (3.29 MB), djvu (776 Kb). | MR1484545 | Zbl 0888.35139

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Si studia la formulazione variazionale del problema di Stefan in due corpi adiacenti, in uno dei quali la conducibilità termica tende all'infinito. Utilizzando e sviluppando alcuni concetti e metodi della teoria della \( \Gamma \)-convergenza e delle equazioni di evoluzione astratte negli spazi di Hilbert, si riesce a giustificare il modello limite, che rientra nella classe dei problemi in «capacità concentrata».

Referenze Bibliografiche

[1] E. ACERBI - G. BUTTAZZO, Reinforcement problems in the calculus of variations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 3, 1986, 273-284. | fulltext EuDML | fulltext mini-dml | MR 853383 | Zbl 0607.73018

[2] D. ANDREUCCI, Existence and uniqueness of solutions to a concentrated capacity problem with change of phase. Europ. J. Appl. Math., 1, 1990, 330-351. | fulltext (doi) | MR 1117356 | Zbl 0734.35161

[3] H. ATTOUCH, Variational Convergence for Functions and Operators. Pitman, London 1984. | MR 773850 | Zbl 0561.49012

[4] T. AUBIN, Non linear Analysis on Manifolds. Monge-Ampère Equations. Springer, New York 1982. | Zbl 0512.53044

[5] H. BREZIS, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. Proc. Symp. by the Mathematics Research Center, Madison, Wisconsin, Academic Press, New York 1971, 101-156. | MR 394323 | Zbl 0278.47033

[6] H. BREZIS, Opérateurs maximaux monotones et sémi-groupes de contractions dans les espaces de Hilbert. North Holland, Amsterdam 1973. | Zbl 0252.47055

[7] H. BREZIS - L. A. CAFFARELLI - A. FRIEDMAN, Reinforcement problems for elliptic equations and variational inequalities. Ann. Mat. pura e appl., 123 (IV), 1980, 219-246. | fulltext (doi) | MR 581931 | Zbl 0434.35079

[8] J. R. CANNON - G. H. MEYER, On diffusion in a fractured medium. SIAM J. Appl. Math., 20(3), 1971, 434-448. | Zbl 0266.35002

[9] P. G. CIARLET, Plates and junctions in Elastic Multi-Structures. Masson-Springer-Verlag, Paris 1990. | MR 1071376 | Zbl 0706.73046

[10] F. H. CLARKE, Optimization and Nonsmooth Analysis. Wiley, New York 1983. | MR 709590 | Zbl 0582.49001

[11] P. COLLI - J. F. RODRIGUES, Diffusion through thin layers with high specific heat. Asymptotic Anal., 3, 1990, 249-263. | MR 1076450 | Zbl 0724.35010

[12] G. DAL MASO, An Introduction to \( \Gamma \)-Convergence. Birkhäuser, Boston 1993. | fulltext (doi) | MR 1201152

[13] A. DAMLAMIAN, Some results on the multi-phase Stefan problem. Comm. P.D.E., 2, 1977, 1017-1044. | MR 487015 | Zbl 0399.35054

[14] A. DAMLAMIAN, How to homogenize a nonlinear diffusion equation: Stefan's problem. SIAM J. Math. Anal., 12, 1981, 306-313. | fulltext (doi) | MR 613313 | Zbl 0468.35052

[15] M. C. DELFOUR - J. P. ZOLÉSIO, Shape analysis via oriented distance functions. J. Funct. Anal., 86, 1989, 129-201. | fulltext (doi) | MR 1279299 | Zbl 0814.49032

[16] M. C. DELFOUR - J. P. ZOLÉSIO, A boundary differential equation for thin shells. J. Differential Equations, to appear. | Zbl 0827.73038

[17] E. DI BENEDETTO - R. E. SHOWALTER, Implicit degenerate evolution equations and applications. SIAM J. Math. Anal., 12, 1981, 731-751. | fulltext (doi) | MR 625829 | Zbl 0477.47037

[18] I. EKELAND - R. TEMAM, Analyse Convexe et Problèmes Variationnels. Dunod, Gauthier-Villars, Paris 1974. | MR 463993 | Zbl 0281.49001

[19] L. C. EVANS - R. GARIEPY, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, CRC Press, 1992. | MR 1158660 | Zbl 0804.28001

[20] A. FASANO - M. PRIMICERIO - L. RUBINSTEIN, A model problem for heat conduction with a free boundary in a concentrated capacity. J. Inst. Maths. Applics., 26, 1980, 327-347. | MR 605396 | Zbl 0456.35093

[21] D. GILBARG - N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin 1983. | MR 737190 | Zbl 0562.35001

[22] H. LE DRET, Problèmes variationnels dans les multi-domaines. Masson, Paris 1991. | MR 1130395

[23] J. L. LIONS, Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod, Gauthier-Villars, Paris 1969. | MR 259693 | Zbl 0189.40603

[24] J. L. LIONS - E. MAGENES, Non Homogeneous Boundary Value Problems and Applications I, II. Springer Verlag, Berlin 1972. | Zbl 0223.35039

[25] E. MAGENES, On a Stefan problem on a boundary of a domain. In: M. MIRANDA (ed.), Partial Differential Equations and Related Subjects. Longman Scient. Techn., 1992, 209-226. | MR 1190942 | Zbl 0803.35170

[26] E. MAGENES, Some new results on a Stefan problem in a concentrated capacity. Rend. Mat. Acc. Lincei, s. 9, v. 3, 1992, 23-34. | fulltext bdim | fulltext mini-dml | MR 1159996 | Zbl 0767.35110

[27] E. MAGENES, The Stefan problem in a concentrated capacity. In: P. E. RICCI (ed.), Atti Simp. Int. «Problemi attuali dell'Analisi e della Fisica Matematica». Dip. di Matematica, Univ. «La Sapienza», Roma 1993, 155-182. | MR 1249096 | Zbl 0803.35171

[28] E. MAGENES, Regularity and approximation properties for the solution of a Stefan problem in a concentrated capacity. Proc. Int. Workshop Variational Methods, Nonlinear Analysis and Differential Equations. E.C.I.G., Genova, 1994, 88-106.

[29] E. MAGENES, On a Stefan problem in a concentrated capacity. In: P. MARCELLINI - G. TALENTI - E. VESENTINI (eds.), P.D.E. and Applications. Marcel Dekker, Inc., 1996, 237-253. | MR 1371595 | Zbl 0864.35127

[30] E. MAGENES, Stefan problems in a concentrated capacity. Adv. Math. Comp. and Appl., Proc. AMCA 95, N.C.C. Pubbl., Novosibirsk 1996, 82-90. | MR 1701426 | Zbl 0803.35171

[31] U. MOSCO, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math., 3, 1969, 510-585. | MR 298508 | Zbl 0192.49101

[32] U. MOSCO, On the continuity of the Young-Fenchel transformation. J. Math. Anal. Appl., 35, 1971, 518-535. | MR 283586 | Zbl 0253.46086

[33] H. PHAM HUY - E. SANCHEZ-PALENCIA, Phénomènes de transmission à travers des couches minces de conductivité élevée. J. Math. Anal, and Appl., 47, 1974, 284-309. | MR 400916 | Zbl 0286.35007

[34] L. RUBINSTEIN, The Stefan problem: Comments on its present state. J. Inst. Maths. Applics., 24, 1979, 259-277. | MR 550476 | Zbl 0434.35086

[35] E. SANCHEZ-PALENCIA, Problèmes de perturbations liés aux phénomènes de conduction à travers des couches minces de grande résistivité. J. Math. pures et appl., 53, 1974, 251-270. | MR 364917 | Zbl 0273.35007

[36] E. SANCHEZ-PALENCIA, Non Homogeneous Media and Vibration Theory. Lect. Notes in Phys. 127, Springer, Berlin-Heidelberg-New York 1980. | MR 578345 | Zbl 0432.70002

[37] M. SHILLOR, Existence and continuity of a weak solution to the problem of a free boundary in a concentrated capacity. Proc. Roy. Soc. Edinburgh, Sect. A. 100, 1985, 271-280. | fulltext (doi) | MR 807706 | Zbl 0591.35086

[38] A. VISINTIN, Partial differential equations in domains with self contact. Rend. Sem. Mat. Univ. Padova, 81, 1989, 37-48. | fulltext EuDML | fulltext mini-dml | MR 1020184 | Zbl 0696.35031

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Referenze Bibliografiche