## A two-phase problem with Robin conditions on the free boundary. (Un problème à frontière libre à deux phases avec conditions au bord de Robin.)(English. French summary)Zbl 1453.35200

Summary: 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.

### MSC:

 35R35 Free boundary problems for PDEs 35B65 Smoothness and regularity of solutions to PDEs 35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators 35J25 Boundary value problems for second-order elliptic equations 49Q10 Optimization of shapes other than minimal surfaces

### Keywords:

Steiner symmetric minimizers
Full Text:

### References:

 [1] Bucur, Dorin; Giacomini, Alessandro, Shape optimization problems with Robin conditions on the free boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33, 6, 1539-1568 (2016) · Zbl 1352.49045 [2] Bucur, Dorin; Luckhaus, Stephan, Monotonicity formula and regularity for general free discontinuity problems, Arch. Rational Mech. Anal., 211, 2, 489-511 (2014) · Zbl 1283.49056 [3] Caffarelli, Luis A.; Kriventsov, Dennis, A free boundary problem related to thermal insulation, Comm. Partial Differential Equations, 41, 7, 1149-1182 (2016) · Zbl 1351.35268 [4] Caffarelli, Luis A.; Soria-Carro, María; Stinga, Pablo Raúl, Regularity for $$C^{1,\alpha }$$ interface transmission problems (2020) [5] Dong, H., A simple proof of regularity for $$C^{1,\alpha }$$ interface transmission problems (2020) [6] Evans, Lawrence C.; Gariepy, Ronald F., Measure theory and fine properties of functions (2015), CRC Press: CRC Press, Boca Raton, FL · Zbl 1310.28001 [7] Maggi, Francesco, Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory, 135 (2012), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1255.49074 [8] Tamanini, Italo, Regularity results for almost minimal oriented hypersurfaces in $$\mathbb{R}^n (1984)$$, Dipartimento di Matematica dell’Università di Lecce: Dipartimento di Matematica dell’Università di Lecce, Lecce · Zbl 1191.35007 [9] Velichkov, Bozhidar, Regularity of the one-phase free boundaries (2019) · Zbl 1429.35214
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