May  2012, 11(3): 861-883. doi: 10.3934/cpaa.2012.11.861

Bernstein estimates: weakly coupled systems and integral equations

1. 

Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

2. 

Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received  October 2010 Revised  November 2011 Published  December 2011

In this paper we extend the classical Bernstein estimates for systems of weakly coupled fully non-linear elliptic equations as well as scalar elliptic equations with non-local integral terms and singular kernels.
Citation: Diogo A. Gomes, Gabriele Terrone. Bernstein estimates: weakly coupled systems and integral equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 861-883. doi: 10.3934/cpaa.2012.11.861
References:
[1]

T. Arnarson, B. Djehiche, M. Poghosyan and H. Shahgholian, A PDE approach to regularity of solutions to finite horizon optimal switching problems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71 (2009), 6054-6067. doi: 10.1016/j.na.2009.05.063.

[2]

G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Mathematical Journal, 61 (1990), 835-858. doi: 10.1215/S0012-7094-90-06132-0.

[3]

J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 21 (2004), 543-590.

[4]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Volume 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1995.

[5]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Inventiones Mathematicae, 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[6]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62 (2009), 597-638. doi: 10.1002/cpa.20274.

[7]

D. G. de Figueiredo and E. Mitidieri, Maximum principles for cooperative elliptic systems, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 310 (1990), 49-52.

[8]

D. G. de Figueiredo and B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Mathematische Annalen, 333 (2005), 231-260.

[9]

B. Djehiche and S. Hamadène, On a finite horizon starting and stopping problem with risk of abandonment, International J. of Theoretical & Applied Finance, 12 (2009), 523-543. doi: 10.1142/S0219024909005312.

[10]

B. Djehiche, S. Hamadène and A. Popier, A finite horizon optimal multiple switching problem, SIAM Journal on Control and Optimization, 48 (2009), 2751-2770. doi: 10.1137/070697641.

[11]

L. C. Evans and A. Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Transactions of the American Mathematical Society, 253 (1979), 365-389. doi: 10.2307/1998203.

[12]

S. Hamadène and M. Jeanblanc, On the starting and stopping problem: application in reversible investments, Mathematics of Operations Research, 32 (2007), 182-192.

[13]

H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Communications in Partial Differential Equations, 16 (1991), 1095-1128. doi: 10.1080/03605309108820791.

[14]

A. Quaas and B. Sirakov, Solvability of monotone systems of fully nonlinear elliptic PDE's, Comptes Rendus Mathématique. Académie des Sciences. Paris, 346 (2008), 641-644. doi: 10.1016/j.crma.2008.04.008.

[15]

B. Sirakov, Some estimates and maximum principles for weakly coupled systems of elliptic PDE, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009), 3039-3046. doi: 10.1016/j.na.2008.12.026.

[16]

Wei-an Liu and Hua Chen, Viscosity solutions of nonlinear systems of degenerated elliptic equations of second order, Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 19 (2000), 927-951.

[17]

Weian Liu, Lu Gang, Hua Chen and Yang Yin, Viscosity solutions of fully nonlinear degenerated elliptic systems, Communications in Applied Analysis. An International Journal for Theory and Applications, 7 (2003), 299-312.

show all references

References:
[1]

T. Arnarson, B. Djehiche, M. Poghosyan and H. Shahgholian, A PDE approach to regularity of solutions to finite horizon optimal switching problems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71 (2009), 6054-6067. doi: 10.1016/j.na.2009.05.063.

[2]

G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Mathematical Journal, 61 (1990), 835-858. doi: 10.1215/S0012-7094-90-06132-0.

[3]

J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 21 (2004), 543-590.

[4]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Volume 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1995.

[5]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Inventiones Mathematicae, 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[6]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Communications on Pure and Applied Mathematics, 62 (2009), 597-638. doi: 10.1002/cpa.20274.

[7]

D. G. de Figueiredo and E. Mitidieri, Maximum principles for cooperative elliptic systems, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 310 (1990), 49-52.

[8]

D. G. de Figueiredo and B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Mathematische Annalen, 333 (2005), 231-260.

[9]

B. Djehiche and S. Hamadène, On a finite horizon starting and stopping problem with risk of abandonment, International J. of Theoretical & Applied Finance, 12 (2009), 523-543. doi: 10.1142/S0219024909005312.

[10]

B. Djehiche, S. Hamadène and A. Popier, A finite horizon optimal multiple switching problem, SIAM Journal on Control and Optimization, 48 (2009), 2751-2770. doi: 10.1137/070697641.

[11]

L. C. Evans and A. Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Transactions of the American Mathematical Society, 253 (1979), 365-389. doi: 10.2307/1998203.

[12]

S. Hamadène and M. Jeanblanc, On the starting and stopping problem: application in reversible investments, Mathematics of Operations Research, 32 (2007), 182-192.

[13]

H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Communications in Partial Differential Equations, 16 (1991), 1095-1128. doi: 10.1080/03605309108820791.

[14]

A. Quaas and B. Sirakov, Solvability of monotone systems of fully nonlinear elliptic PDE's, Comptes Rendus Mathématique. Académie des Sciences. Paris, 346 (2008), 641-644. doi: 10.1016/j.crma.2008.04.008.

[15]

B. Sirakov, Some estimates and maximum principles for weakly coupled systems of elliptic PDE, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009), 3039-3046. doi: 10.1016/j.na.2008.12.026.

[16]

Wei-an Liu and Hua Chen, Viscosity solutions of nonlinear systems of degenerated elliptic equations of second order, Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 19 (2000), 927-951.

[17]

Weian Liu, Lu Gang, Hua Chen and Yang Yin, Viscosity solutions of fully nonlinear degenerated elliptic systems, Communications in Applied Analysis. An International Journal for Theory and Applications, 7 (2003), 299-312.

[1]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure and Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

[2]

Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032

[3]

Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141

[4]

A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373

[5]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[6]

Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517

[7]

Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343

[8]

Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198

[9]

T. Gilbert, J. R. Dorfman. On the parametric dependences of a class of non-linear singular maps. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 391-406. doi: 10.3934/dcdsb.2004.4.391

[10]

Xiaoming He, Xin Zhao, Wenming Zou. Maximum principles for a fully nonlinear nonlocal equation on unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4387-4399. doi: 10.3934/cpaa.2020200

[11]

Wenmin Sun, Jiguang Bao. New maximum principles for fully nonlinear ODEs of second order. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 813-823. doi: 10.3934/dcds.2007.19.813

[12]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure and Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006

[13]

Fabio Punzo. Phragmèn-Lindelöf principles for fully nonlinear elliptic equations with unbounded coefficients. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1439-1461. doi: 10.3934/cpaa.2010.9.1439

[14]

Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems and Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004

[15]

Bei Hu, Lishang Jiang, Jin Liang, Wei Wei. A fully non-linear PDE problem from pricing CDS with counterparty risk. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2001-2016. doi: 10.3934/dcdsb.2012.17.2001

[16]

David Gómez-Ullate, Niky Kamran, Robert Milson. Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 85-106. doi: 10.3934/dcds.2007.18.85

[17]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841

[18]

Tarek Saanouni. Non-linear bi-harmonic Choquard equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5033-5057. doi: 10.3934/cpaa.2020221

[19]

Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691

[20]

Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1025-1040. doi: 10.3934/cpaa.2010.9.1025

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]