December  2018, 38(12): 6015-6027. doi: 10.3934/dcds.2018152

The two membranes problem for fully nonlinear 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 nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6015-6027. doi: 10.3934/dcds.2018152
References:
[1]

A. AzevedoJ.-F. Rodrigues and L. Santos, The N-membranes problem for quasilinear degenerate systems, Interfaces Free Bound., 7 (2005), 319-337.   Google Scholar

[2]

L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, 1995. doi: 10.1090/coll/043.  Google Scholar

[3]

L. CaffarelliM. G. CrandallM. 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.  Google Scholar

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L. CaffarelliD. 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.  Google Scholar

[5]

S. CarilloM. 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.  Google Scholar

[6]

M. Chipot and G. Vergara-Caffarelli, The N-membranes problem, Appl. Math. Optim., 13 (1985), 231-249.  doi: 10.1007/BF01442209.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Silvestre, The two membranes problem, Comm. Partial Differential Equations, 30 (2005), 245-257.  doi: 10.1081/PDE-200044490.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

A. AzevedoJ.-F. Rodrigues and L. Santos, The N-membranes problem for quasilinear degenerate systems, Interfaces Free Bound., 7 (2005), 319-337.   Google Scholar

[2]

L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, 1995. doi: 10.1090/coll/043.  Google Scholar

[3]

L. CaffarelliM. G. CrandallM. 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.  Google Scholar

[4]

L. CaffarelliD. 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.  Google Scholar

[5]

S. CarilloM. 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.  Google Scholar

[6]

M. Chipot and G. Vergara-Caffarelli, The N-membranes problem, Appl. Math. Optim., 13 (1985), 231-249.  doi: 10.1007/BF01442209.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Silvestre, The two membranes problem, Comm. Partial Differential Equations, 30 (2005), 245-257.  doi: 10.1081/PDE-200044490.  Google Scholar

[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.   Google Scholar

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