Geodesic
Compactness theorems for problems with unknown boundary
Dalʹnevostočnyj matematičeskij žurnal, Tome 21 (2021) no. 1, pp. 105-112.

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The compactness theorem is proved for sequences of functions that have estimates of the higher derivatives in each subdomain of the domain of definition, divided into parts by a sequence of some curves of class $W_2^1$. At the same time, in the entire domain of determining summable higher derivatives, these sequences do not have. These results allow us to make limit transitions using approximate solutions in problems with an unknown boundary that describe the processes of phase transitions. 
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A. G. Podgaev; T. D. Kulesh. Compactness theorems for problems with unknown boundary. Dalʹnevostočnyj matematičeskij žurnal, Tome 21 (2021) no. 1, pp. 105-112. http://geodesic.mathdoc.fr/item/DVMG_2021_21_1_a8/

[1] A. G Podgaev, “On Relative Compactness Set of Abstract Function from Scale of the Banach Spaces”, Functional Analysis, Approximation Theory and Numerical Analysis, Word Scientific Publishing Co, 1994, 219–236 | MR | Zbl

[2] J. Aubin, “Un theoreme de compacte”, Compt. Rend. Acad. Sci. (Paris), 256 (1963), 5042–5044 | MR | Zbl

[3] J. Simon, “Compact Sets in the $L_p(0,T;B)$”, Annalidi Matematica pura ed applicata (IV), CXLVI (1987), 65–96 | MR | Zbl

[4] Yu. A. Dubinskii, “Slabaya skhodimost v nelineinykh ellipticheskikh i parabolicheskikh uravneniyakh”, Matem. sb., 67 (109):4 (1965)

[5] A. G. Podgaev, “Razreshimost osesimmetrichnoi zadachi dlya nelineinogo parabolicheskogo uravneniya v oblastyakh s netsilindricheskoi ili neizvestnoi granitsei. I”, Chelyab. fiz.-matem. zhurnal., 5:1 (2020), 44–55 | MR

[6] S. V. Popov, A. I. Shadrina, “Kontaktnye parabolicheskie kraevye zadachi dlya uravnenii vtorogo poryadka”, Matematicheskie zametki YaGU, 16:2 (2009), 66–77 | Zbl

[7] A. M. Meirmanov, Zadacha Stefana, Nauka, Novosibirsk, 1986 | MR