# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2018152

## The two membranes problem for fully nonliear operators

 1 Mathematics Department, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712, USA 2 Departamento de Matemática, Universidad Nacional de Mar del Plata/Conicet, Deán Funes 3600, Mar del Plata, Buenos Aires 7600, Argentina

* Corresponding author

Received  August 2017 Revised  February 2018 Published  April 2018

We study the two membranes problem for two different fully nonlinear operators. We give a viscosity formulation for the problem and prove existence of solutions. Then we prove a general regularity result and the optimal $C^{1, 1}$ regularity when the operators are the Pucci extremal operators. We also give an example that shows that no regularity for the free boundary is to be expected to hold in general.

Citation: Luis Caffarelli, Luis Duque, Hernán Vivas. The two membranes problem for fully nonliear operators. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2018152
##### References:
 [1] A. Azevedo, J.-F. Rodrigues and L. Santos, The N-membranes problem for quasilinear degenerate systems, Interfaces Free Bound., 7 (2005), 319-337. [2] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, 1995. doi: 10.1090/coll/043. [3] L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. [4] L. Caffarelli, D. De Silva and O. Savin, The two membranes problem for different operators, Annales de l'Institut Henri Poincare, 34 (2017), 899-932. doi: 10.1016/j.anihpc.2016.05.006. [5] S. Carillo, M. Chipot and G. Vergara-Caffarelli, The N-membrane problem with nonlocal constraints, J. Math. Anal. Appl., 308 (2005), 129-139. doi: 10.1016/j.jmaa.2004.11.024. [6] M. Chipot and G. Vergara-Caffarelli, The N-membranes problem, Appl. Math. Optim., 13 (1985), 231-249. doi: 10.1007/BF01442209. [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977. [8] E. Indrei and A. Minne, Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33 (2016), 1259-1277. doi: 10.1016/j.anihpc.2015.03.009. [9] D. Kinderlehrer and G. Stampacchia, An Introuction to Variational Inequalities and Their Applications, Reprint of the 1980 original, Classics in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451. [10] L. Silvestre, The two membranes problem, Comm. Partial Differential Equations, 30 (2005), 245-257. doi: 10.1081/PDE-200044490. [11] G. Vergara-Caffarelli, Regolarita di un problema di disequazioni variazionali relativo a due membrane, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 50 (1971), 659-662.

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##### References:
 [1] A. Azevedo, J.-F. Rodrigues and L. Santos, The N-membranes problem for quasilinear degenerate systems, Interfaces Free Bound., 7 (2005), 319-337. [2] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, 1995. doi: 10.1090/coll/043. [3] L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. [4] L. Caffarelli, D. De Silva and O. Savin, The two membranes problem for different operators, Annales de l'Institut Henri Poincare, 34 (2017), 899-932. doi: 10.1016/j.anihpc.2016.05.006. [5] S. Carillo, M. Chipot and G. Vergara-Caffarelli, The N-membrane problem with nonlocal constraints, J. Math. Anal. Appl., 308 (2005), 129-139. doi: 10.1016/j.jmaa.2004.11.024. [6] M. Chipot and G. Vergara-Caffarelli, The N-membranes problem, Appl. Math. Optim., 13 (1985), 231-249. doi: 10.1007/BF01442209. [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977. [8] E. Indrei and A. Minne, Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33 (2016), 1259-1277. doi: 10.1016/j.anihpc.2015.03.009. [9] D. Kinderlehrer and G. Stampacchia, An Introuction to Variational Inequalities and Their Applications, Reprint of the 1980 original, Classics in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451. [10] L. Silvestre, The two membranes problem, Comm. Partial Differential Equations, 30 (2005), 245-257. doi: 10.1081/PDE-200044490. [11] G. Vergara-Caffarelli, Regolarita di un problema di disequazioni variazionali relativo a due membrane, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 50 (1971), 659-662.
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