American Institute of Mathematical Sciences

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

The two membranes problem for fully nonlinear operators

Luis Caffarelli 1, , Luis Duque 1, and  Hernán Vivas 2,,

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.

Keywords: Free boundary problems, fully nonlinear.
Mathematics Subject Classification: 35R35, 35J60.
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. Azevedo, J.-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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. Azevedo, J.-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. 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.  Google Scholar

[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.  Google Scholar

[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.  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

[1]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020
[2]

Roland Schnaubelt. Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1193-1230. doi: 10.3934/dcds.2015.35.1193
[3]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013
[4]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920
[5]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621
[6]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
[7]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025
[8]

Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386
[9]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[10]

Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations & Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017
[11]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673
[12]

Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365
[13]

Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799
[14]

Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179
[15]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239
[16]

Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233
[17]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[18]

Yuri Latushkin, Jan Prüss, Ronald Schnaubelt. Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 595-633. doi: 10.3934/dcdsb.2008.9.595
[19]

Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030
[20]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (211)
  • HTML views (395)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences