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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

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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

http://arxiv.org/pdf/math/0701447

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A brief introduction can be found in M. Taylor's PDEs, vol. 3., Chapter 15.6

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