A two-phase problem with Robin conditions on the free boundary
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1-25.

We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an existence and a regularity result for minimizers of the associated variational problem. Finally, in the appendix, we give an example of a class of Steiner symmetric minimizers.

Nous étudions pour la première fois un problème à frontière libre à deux phases pour lequel la solution satisfait à une condition de Robin au bord. Nous considérons le cas où la solution est continue au bord et nous montrons un résultat d’existence et de régularité pour les minimiseurs du problème variationnel associé. Enfin, nous donnons dans l’appendice un exemple d’une classe de minimiseurs avec une symétrie de Steiner.

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Accepted:
Published online:
DOI: 10.5802/jep.139
Classification: 35R35,  49Q10
Keywords: Free boundary problems, two-phase, Robin boundary conditions, regularity
Serena Guarino Lo Bianco 1; Domenico Angelo La Manna 2; Bozhidar Velichkov 3

1 Università degli studi di Napoli “Federico II”, Dipartimento di Agraria Via Università 100, 80055 Portici (NA), Italia
2 University of Jyvaskylä, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, Finland
3 Dipartimento di Matematica, Università di Pisa Largo Bruno Pontecorvo, 5, 56127 Pisa, Italia
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Serena Guarino Lo Bianco; Domenico Angelo La Manna; Bozhidar Velichkov. A two-phase problem with Robin conditions on the free boundary. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1-25. doi : 10.5802/jep.139. https://jep.centre-mersenne.org/articles/10.5802/jep.139/

[1] D. Bucur & A. Giacomini - “Shape optimization problems with Robin conditions on the free boundary”, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) no. 6, p. 1539-1568 | Article | MR: 3569242 | Zbl: 1352.49045

[2] D. Bucur & S. Luckhaus - “Monotonicity formula and regularity for general free discontinuity problems”, Arch. Rational Mech. Anal. 211 (2014) no. 2, p. 489-511 | Article | MR: 3149064 | Zbl: 1283.49056

[3] L. A. Caffarelli & D. Kriventsov - “A free boundary problem related to thermal insulation”, Comm. Partial Differential Equations 41 (2016) no. 7, p. 1149-1182 | Article | MR: 3528530 | Zbl: 1351.35268

[4] L. A. Caffarelli, M. Soria-Carro & P. R. Stinga - “Regularity for C 1,α interface transmission problems”, 2020 | 2004.07322

[5] H. Dong - “A simple proof of regularity for C 1,α interface transmission problems”, 2020 | 2004.09365

[6] L. C. Evans & R. F. Gariepy - Measure theory and fine properties of functions, Textbooks in Math., CRC Press, Boca Raton, FL, 2015 | Zbl: 1310.28001

[7] F. Maggi - Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory, Cambridge Studies in Advanced Math., vol. 135, Cambridge University Press, Cambridge, 2012 | Article | MR: 2976521 | Zbl: 1255.49074

[8] I. Tamanini - Regularity results for almost minimal oriented hypersurfaces in n , Dipartimento di Matematica dell’Università di Lecce, Lecce, 1984 | Zbl: 1191.35007

[9] B. Velichkov - “Regularity of the one-phase free boundaries” (2019), Lecture notes, available at http://cvgmt.sns.it/paper/4367/

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