Regularity of interfaces for a Pucci type segregation problem

Caffarelli, L.; Patrizi, S.; Quitalo, V.; Torres, M.

doi:10.1016/j.anihpc.2018.11.002

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

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 939-975.

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Résumé

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 $\partial {u > 0} \cap Ω$ and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition $u_{ν_{+}}^{+} = u_{ν_{-}}^{-}$ , 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 $\partial {u > 0} \cap Ω$ form an open set of class $C^{1, α}$ . 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].

MR Zbl

DOI : 10.1016/j.anihpc.2018.11.002

Classification : 35J60, 35R35, 35B65, 35Q92
Mots-clés : Fully nonlinear elliptic systems, Pucci operators, Regularity for viscosity solutions, Segregation of populations, Regularity of the free boundary

Affiliations des auteurs :

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

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     title = {Regularity of interfaces for a {Pucci} type segregation problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {939--975},
     publisher = {Elsevier},
     volume = {36},
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     year = {2019},
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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. http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.002/

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