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On the domain dependence of solutions to the two-phase Stefan problem. (English) Zbl 1067.35155

In mathematical physics and numerical analysis of partial differential equations (PDEs), one is often to study the behaviour of solutions to PDEs defined on domains approximated by simpler ones in some sense. Then, it is of interest whether or not the problems defined on the different domains have solutions that are close to each other. The authors prove in the paper that weak solutions to the multidimensional two-phase Stefan problem defined on a sequence of spatial domains \(\Omega_n\) converge to a solution of the same problem on a domain \(\Omega\) that is the limit of \(\Omega_n\) in the sense of Mosco. Further it is shown that the corresponding free (or moving) boundaries converge in the sense of Lebesgue measure. This is a remarkable paper as most of the existing theory of PDEs in this respect deals with elliptic problems and much less seems to be known for evolution equations. The results of the paper are of significance for the numerical analysis of moving boundary problems as well.

MSC:

35R35 Free boundary problems for PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K65 Degenerate parabolic equations
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References:

[1] D. Blanchard, H. Redwane: Solutions renormalisées d’equations paraboliques à deux non linéarités. C. R. Acad. Sci. Paris, Sér. I, 319 (1994), 831-835. · Zbl 0810.35038
[2] L. Boccardo, F. Murat: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Analysis 19 (6) (1992), 581-597. · Zbl 0783.35020
[3] D. Bucur, J. P. Zolesio: \(N\)-dimensional shape optimization under capacity constraints. J. Differential Equations 123 (1995), 544-522.
[4] M. G. Crandall, H. Ishii and P.-L. Lions: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1) (1992), 1-67. · Zbl 0755.35015
[5] G. Dal Masso and U. Mosco: Wiener’s criterion and \(\Gamma \) convergence. Appl. Math. Optim. 15, 15-63. · Zbl 0644.35033
[6] E. N. Dancer: The effect of domain shape on the number of positive solutions of certain nonlinear equations, II. J. Differential Equations 87 (1990), 316-339. · Zbl 0729.35050
[7] D. Daners: Domain perturbation for linear and nonlinear parabolic equations. J. Differential Equations 129 (2) (1996), 358-402. · Zbl 0868.35059
[8] E. DiBenedetto: Interior and boundary regularity for a class of free boundary problems. Free boundary problems: theory and applications, II., A. Fasano, M. Primicerio (eds.), Research Notes in Math., vol. 78, Pitman, Boston, 1983, pp. 383-396. · Zbl 0516.35081
[9] A. Friedman: Variational Principles and Free-Boundary Problems. John Wiley, New York, 1982. · Zbl 0564.49002
[10] J. K. Hale and J. Vegas: A nonlinear parabolic equation with varying domain. Arch. Rat. Mech. Anal. 86 (1984), 99-123. · Zbl 0569.35048
[11] A. Henrot: Continuity with respect to the domain for the Laplacian: a survey. Control Cybernet. 23 (3) (1994), 427-443. · Zbl 0822.35029
[12] A. M. Meirmanov: The Stefan Problem. De Gruyter, Berlin, 1992. · Zbl 0751.35052
[13] O. Pironneau: Optimal Shape Design for Elliptic Systems. Springer-Verlag, Berlin, 1984. · Zbl 0534.49001
[14] J. Rauch, M. Taylor: Potential and scattering theory on wildly perturbed domains. J. Functional Anal. 18 (1975), 27-59. · Zbl 0293.35056
[15] V. Šverák: On optimal shape design. J. Math. Pures Appl. 72 (1993), 537-551. · Zbl 0849.49021
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