# 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
##### 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, 71 (2009), 6054.  doi: 10.1016/j.na.2009.05.063.  Google Scholar [2] G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE,, Duke Mathematical Journal, 61 (1990), 835.  doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar [3] J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications,, Annales de l'Institut Henri Poincar\'e. Analyse Non Lin\'eaire, 21 (2004), 543.   Google Scholar [4] L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", Volume 43 of American Mathematical Society Colloquium Publications, (1995).   Google Scholar [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.  doi: 10.1007/s00222-007-0086-6.  Google Scholar [6] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Communications on Pure and Applied Mathematics, 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar [7] D. G. de Figueiredo and E. Mitidieri, Maximum principles for cooperative elliptic systems,, Comptes Rendus de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique, 310 (1990), 49.   Google Scholar [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.   Google Scholar [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.  doi: 10.1142/S0219024909005312.  Google Scholar [10] B. Djehiche, S. Hamadène and A. Popier, A finite horizon optimal multiple switching problem,, SIAM Journal on Control and Optimization, 48 (2009), 2751.  doi: 10.1137/070697641.  Google Scholar [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.  doi: 10.2307/1998203.  Google Scholar [12] S. Hamadène and M. Jeanblanc, On the starting and stopping problem: application in reversible investments,, Mathematics of Operations Research, 32 (2007), 182.   Google Scholar [13] H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs,, Communications in Partial Differential Equations, 16 (1991), 1095.  doi: 10.1080/03605309108820791.  Google Scholar [14] A. Quaas and B. Sirakov, Solvability of monotone systems of fully nonlinear elliptic PDE's,, Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris, 346 (2008), 641.  doi: 10.1016/j.crma.2008.04.008.  Google Scholar [15] B. Sirakov, Some estimates and maximum principles for weakly coupled systems of elliptic PDE,, Nonlinear Analysis. Theory, 70 (2009), 3039.  doi: 10.1016/j.na.2008.12.026.  Google Scholar [16] Wei-an Liu and Hua Chen, Viscosity solutions of nonlinear systems of degenerated elliptic equations of second order,, Zeitschrift f\, 19 (2000), 927.   Google Scholar [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.   Google Scholar

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##### 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, 71 (2009), 6054.  doi: 10.1016/j.na.2009.05.063.  Google Scholar [2] G. Barles, L. C. Evans and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE,, Duke Mathematical Journal, 61 (1990), 835.  doi: 10.1215/S0012-7094-90-06132-0.  Google Scholar [3] J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications,, Annales de l'Institut Henri Poincar\'e. Analyse Non Lin\'eaire, 21 (2004), 543.   Google Scholar [4] L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", Volume 43 of American Mathematical Society Colloquium Publications, (1995).   Google Scholar [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.  doi: 10.1007/s00222-007-0086-6.  Google Scholar [6] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Communications on Pure and Applied Mathematics, 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar [7] D. G. de Figueiredo and E. Mitidieri, Maximum principles for cooperative elliptic systems,, Comptes Rendus de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique, 310 (1990), 49.   Google Scholar [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.   Google Scholar [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.  doi: 10.1142/S0219024909005312.  Google Scholar [10] B. Djehiche, S. Hamadène and A. Popier, A finite horizon optimal multiple switching problem,, SIAM Journal on Control and Optimization, 48 (2009), 2751.  doi: 10.1137/070697641.  Google Scholar [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.  doi: 10.2307/1998203.  Google Scholar [12] S. Hamadène and M. Jeanblanc, On the starting and stopping problem: application in reversible investments,, Mathematics of Operations Research, 32 (2007), 182.   Google Scholar [13] H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs,, Communications in Partial Differential Equations, 16 (1991), 1095.  doi: 10.1080/03605309108820791.  Google Scholar [14] A. Quaas and B. Sirakov, Solvability of monotone systems of fully nonlinear elliptic PDE's,, Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris, 346 (2008), 641.  doi: 10.1016/j.crma.2008.04.008.  Google Scholar [15] B. Sirakov, Some estimates and maximum principles for weakly coupled systems of elliptic PDE,, Nonlinear Analysis. Theory, 70 (2009), 3039.  doi: 10.1016/j.na.2008.12.026.  Google Scholar [16] Wei-an Liu and Hua Chen, Viscosity solutions of nonlinear systems of degenerated elliptic equations of second order,, Zeitschrift f\, 19 (2000), 927.   Google Scholar [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.   Google Scholar
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