Geodesic
Regularity of interfaces for a Pucci type segregation problem
Caffarelli, L.1 ; Patrizi, S.1 ; Quitalo, V.2 ; Torres, M.3
1The University of Texas at Austin, Department of Mathematics – RLM 8.100, 2515 Speedway – Stop C1200, Austin, TX 78712-1202, United States of America
2CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
3Purdue University, Department of Mathematics, 150 N. University Street, West Lafayette, IN 47907-2067, United States of America
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 939-975
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We show the existence of a Lipschitz viscosity solution u in Ω to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface ∂{u>0}∩Ω and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition uν++=uν−−, and hence u is a solution to a two-phase free boundary problem. We show that we can apply the classical method of sup-convolutions developed by the first author in [5, 6], and generalized by Wang [20, 21] and Feldman [11] to fully nonlinear operators, to conclude that the regular points in ∂{u>0}∩Ω form an open set of class C1,α. A novelty in our problem is that we have different operators, F+ and F−, on each side of the free boundary. In the particular case when these operators are the Pucci's extremal operators M+ and M−, our results provide an alternative approach to obtain the stationary limit of a segregation model of populations with nonlinear diffusion in [19].

DOI : 10.1016/j.anihpc.2018.11.002
Classification : 35J60, 35R35, 35B65, 35Q92
Keywords: Fully nonlinear elliptic systems, Pucci operators, Regularity for viscosity solutions, Segregation of populations, Regularity of the free boundary

Caffarelli, L.  1   ; Patrizi, S.  1   ; Quitalo, V.  2   ; Torres, M.  3

1 The University of Texas at Austin, Department of Mathematics – RLM 8.100, 2515 Speedway – Stop C1200, Austin, TX 78712-1202, United States of America
2 CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
3 Purdue University, Department of Mathematics, 150 N. University Street, West Lafayette, IN 47907-2067, United States of America 
@article{AIHPC_2019__36_4_939_0,
     author = {Caffarelli, L. and Patrizi, S. and Quitalo, V. and Torres, M.},
     title = {Regularity of interfaces for a {Pucci} type segregation problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {939--975},
     year = {2019},
     publisher = {Elsevier},
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Caffarelli, L.; Patrizi, S.; Quitalo, V.; Torres, M. Regularity of interfaces for a Pucci type segregation problem. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 939-975. doi: 10.1016/j.anihpc.2018.11.002

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