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, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71 (2009), 6054-6067. 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), 835-858. 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 Poincaré. Analyse Non Linéaire, 21 (2004), 543-590.  Google Scholar

[4]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Volume 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 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), 425-461. 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), 597-638. 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'Académie des Sciences. Série I. Mathématique, 310 (1990), 49-52.  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), 231-260.  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), 523-543. 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), 2751-2770. 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), 365-389. 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), 182-192.  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), 1095-1128. doi: 10.1080/03605309108820791.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 299-312.  Google Scholar

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.  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), 835-858. 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 Poincaré. Analyse Non Linéaire, 21 (2004), 543-590.  Google Scholar

[4]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Volume 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 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), 425-461. 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), 597-638. 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'Académie des Sciences. Série I. Mathématique, 310 (1990), 49-52.  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), 231-260.  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), 523-543. 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), 2751-2770. 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), 365-389. 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), 182-192.  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), 1095-1128. doi: 10.1080/03605309108820791.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 299-312.  Google Scholar

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