Optimal regularity for phase transition problems with convection
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, p. 715-740
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In this paper we consider a steady state phase transition problem with given convection v. We prove, among other things, that the weak solution is locally Lipschitz continuous provided that 𝐯=Dξ and ξ is a harmonic function. Moreover, for continuous casting problem, i.e. when v is constant vector, we show that Lipschitz free boundaries are C 1 regular surfaces.

DOI : https://doi.org/10.1016/j.anihpc.2014.03.003
Classification:  35R35,  35J60,  35R37,  80A22
Keywords: Free boundary, Stefan problem, Phase transition, Convection, Lipschitz regularity, Viscosity solution
@article{AIHPC_2015__32_4_715_0,
     author = {Karakhanyan, Aram L.},
     title = {Optimal regularity for phase transition problems with convection},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {4},
     year = {2015},
     pages = {715-740},
     doi = {10.1016/j.anihpc.2014.03.003},
     zbl = {1329.35361},
     mrnumber = {3390081},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_4_715_0}
}
Karakhanyan, Aram L. Optimal regularity for phase transition problems with convection. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 715-740. doi : 10.1016/j.anihpc.2014.03.003. http://www.numdam.org/item/AIHPC_2015__32_4_715_0/

[1] V. Alexiades, A.D. Solomon, Mathematical Modeling of Melting and Freezing Processes, Taylor & Francis (1993)

[2] H.W. Alt, L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Math. 325 (1981), 105 -144 | Zbl 0449.35105

[3] H.W. Alt, L.A. Caffarelli, A. Friedman, A free boundary problem for quasi-linear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 no. 4 (1984), 1 -44 | Numdam | MR 752578 | Zbl 0554.35129

[4] H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, Trans. Am. Math. Soc. 282 (1984), 431 -461 | Zbl 0844.35137

[5] I. Athanasopoulos, L. Caffarelli, S. Salsa, Regularity of the free boundary in parabolic phase-transition problems, Acta Math. 176 (1996), 245 -282 | MR 1397563 | Zbl 0891.35164

[6] J. Bear, Dynamics of Fluids in Porous Media, Courier Dover Publications (1988) | Zbl 1191.76002

[7] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I, Lipschitz free boundaries are C 1,α , Rev. Mat. Iberoam. 3 (1987), 139 -162 | MR 990856 | Zbl 0676.35085

[8] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part II, Flat free boundaries are Lipschitz, Commun. Pure Appl. Math. 42 (1989), 55 -78 | MR 973745 | Zbl 0676.35086

[9] L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 no. 4–5 (1998), 384 -402 | MR 1658612 | Zbl 0928.49030

[10] L. Caffarelli, D. Jerison, C. Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math. (2) 155 no. 2 (2002), 369 -404 | Zbl 1142.35382

[11] L. Caffarelli, L. Karp, H. Shahgholian, Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math. (2) 151 no. 1 (2000), 269 -292 | MR 1745013 | Zbl 0960.35112

[12] L. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems, Grad. Stud. Math. vol. 68 , AMS (2005) | MR 2145284 | Zbl 1083.35001

[13] X. Chen, F. Yi, Regularity of the free boundary of a continuous casting problem, Nonlinear Anal. 21 no. 6 (1993), 425 -438 | MR 1240895 | Zbl 0796.35179

[14] H. Federer, Geometric Measure Theory, Springer (1996) | MR 257325 | Zbl 0176.00801

[15] M. Feldman, Regularity of Lipschitz free boundaries in two-phase problems for fully nonlinear elliptic equations, Indiana Univ. Math. J. 50 no. 3 (2001), 1171 -1200 | MR 1871352 | Zbl 1037.35104

[16] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley & Sons (1982) | MR 679313 | Zbl 0564.49002

[17] G. Hile, A. Stanoyevitch, Gradient bounds for harmonic functions Lipschitz on the boundary, Appl. Anal. 73 no. 1–2 (1999), 101 -113 | MR 1735016 | Zbl 1016.31002

[18] N. Landkof, Foundations of Modern Potential Theory, Springer (1973) | MR 350027

[19] R. Nochetto, A class of non-degenerate two-phase Stefan problems in several space variables, Commun. Partial Differ. Equ. 12 no. 1 (1987), 21 -45 | MR 869101 | Zbl 0624.35085

[20] Ch.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin (2008) | MR 2492985

[21] J.-F. Rodrigues, Variational Methods in the Stefan problem, Lect. Notes Math. vol. 1584 (1994), 147 -212 | MR 1321833 | Zbl 0819.35154

[22] J.-F. Rodrigues, F. Yi, On a two-phase continuous casting Stefan problem with nonlinear flux, Eur. J. Appl. Math. 1 no. 3 (1990), 259 -278 | MR 1117351 | Zbl 0723.35086

[23] L. Simon, Lectures on Geometric Measure Theory, Centre for Mathematical Analysis, Australian National University (1984) | MR 756417

[24] M. Grüter, K.-O. Widman, The Green function for uniformly elliptic equations, Manuscr. Math. 37 (1968), 303 -342 | MR 657523

[25] B.G. Thomas, Modeling of the continuous casting of steel—past, present, and future, Metall. Mater. Trans. B 33B (2002), 395 -812