# American Institute of Mathematical Sciences

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 & Applied Analysis, 2012, 11 (3) : 861-883. doi: 10.3934/cpaa.2012.11.861
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##### References:
 [1] Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - B, 2004, 4 (2) : 391-406. doi: 10.3934/dcdsb.2004.4.391 [10] Wenmin Sun, Jiguang Bao. New maximum principles for fully nonlinear ODEs of second order. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 813-823. doi: 10.3934/dcds.2007.19.813 [11] Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006 [12] Fabio Punzo. Phragmèn-Lindelöf principles for fully nonlinear elliptic equations with unbounded coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1439-1461. doi: 10.3934/cpaa.2010.9.1439 [13] Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004 [14] Bei Hu, Lishang Jiang, Jin Liang, Wei Wei. A fully non-linear PDE problem from pricing CDS with counterparty risk. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2001-2016. doi: 10.3934/dcdsb.2012.17.2001 [15] David Gómez-Ullate, Niky Kamran, Robert Milson. Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 85-106. doi: 10.3934/dcds.2007.18.85 [16] Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1025-1040. doi: 10.3934/cpaa.2010.9.1025 [17] Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335 [18] Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841 [19] Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691 [20] Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017

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