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.


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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