Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)?
I wish to avoid classical theory because I prefer Sobolev spaces and the like.
Thank you
There is a huge literature in free boundaries. Here I collected some papers (in no particular order) addressing different physical systems (coming mainly from fluid dynamics) and questions (so, not only well-posedness). Hopefully you find these papers interesting.
First of all a very nice, in my opinion, review: http://arxiv.org/pdf/1005.5329
1) Stefan
http://arxiv.org/pdf/1212.1422 http://arxiv.org/pdf/1112.5817
2) Water waves
http://arxiv.org/pdf/0910.2473 http://arxiv.org/pdf/1212.0632 http://arxiv.org/pdf/1305.4090 http://arxiv.org/pdf/0810.5340 http://arxiv.org/pdf/1106.2120 http://arxiv.org/pdf/1208.2726 http://arxiv.org/pdf/1201.4919 http://arxiv.org/pdf/1312.2917 http://arxiv.org/pdf/1401.1252 http://arxiv.org/pdf/1110.5155 http://arxiv.org/pdf/1005.4565 http://arxiv.org/pdf/math/0702015
3) Hele-Shaw & Muskat
http://arxiv.org/pdf/1102.1902 http://arxiv.org/pdf/0806.2258 http://arxiv.org/pdf/1311.7653 http://arxiv.org/pdf/1208.6213 http://arxiv.org/pdf/1303.1769 http://arxiv.org/pdf/1311.0430
4) Vortex sheet
http://arxiv.org/pdf/math/0502215
5) SQG
A brief introduction can be found in M. Taylor's PDEs, vol. 3., Chapter 15.6
The classical existence and uniqueness theorems (and thus the well posedness) for weak solutions (in some suitable sense) to the Stefan problem were proved by Shoshana Kamin (neé Shoshana L'vovna Kamenomostskaya) (see [1] and [2]) and (in a slightly generalised form) by her teacher Ol'ga Alexandrovna Ladyzhenskaya (see [4]) in the years 1958-1960.
Their solutions turned out to be functions belonging to the Sobolev space $W_{2}^{1,1}(Q_T)$ where $Q_T=\Omega\times[0,T]$, $\Omega$ being a domain in $\Bbb R^n$, fulfilling further requirements imposed by the Stefan data: a clear sketch of the details is offered by Ladyzhenskaya et al. ([3] chapter V, §V.9 pp. 496-503).
Remark.
Citing Rubinsteĭn ([a3], pp. 263-264), there are several example of physically meaningful Stefan problems for which the concept of weak solution, as it is currently defined, is unsatisfactory. In this respect, Rubinsteĭn's effort in determining a global classical solution for the $1$-dimensional problem (described in reference [a2]) and the final success of Meirmanov (described in reference [a1]) for the $n$-dimensional problem are not at all to be considered mere exercises of style or "only" regularity results: they are decisive steps in the description of the solution to the concrete problem, thus I felt wort the work of adding this remark and the associated references.
Refereces
[1] Shoshana L'vovna Kamenomostskaya, "On Stefan Problem", Nauchnye Doklady Vysshey Shkoly, Fiziko-Matematicheskie Nauki (in Russian), 1 (1): 60–62 (1958), Zbl 0143.13901.
[2] Shoshana L'vovna Kamenomostskaya, "On Stefan's problem", Matematicheskii Sbornik (in Russian), 53(95) (4): 489-514 (1961), MR 0141895, Zbl 0102.09301.
[3] Ol'ga Alexandrovna Ladyzhenskaya, Vsevolod Alekseevich Solonnikov, and Nina Nikolaevna Ural’tseva, Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith. (English) Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society (AMS), pp. XI+648 (1968), DOI:10.1090/mmono/023, MR241822, Zbl 0174.15403.
[4] Ol'ga Alexandrovna Oleinik, "A method of solution of the general Stefan problem", Doklady Akademii Nauk SSSR (in Russian), 135: 1050–1057 (1960), MR125341, Zbl 0131.09202.
Addendum references
[a1] Anvarbek Mukatovich Meirmanov, The Stefan problem, translated from the Russian by Marek Niezgódka and Anna Crowley. (English) De Gruyter Expositions in Mathematics. 3. Berlin, New York: Walter de Gruyter, pp. ix+245 (1992), DOI:10.1515/9783110846720.245, MR1154310, Zbl 0751.35052.
[a2] Lev Isaakovich Rubinsteĭn, The Stefan problem. (English) Translations of Mathematical Monographs. Vol. 27. Providence, R.I.: American Mathematical Society (AMS), VIII+419 (1971), [Zbl 0219.35043].
[a3] Lev Isaakovich Rubinsteĭn, "The Stefan problem: Comments on its present state", (English) Journal of the Institute of Mathematics and its Applications 24, 259-277 (1979), , Zbl 0434.35086.
