Bernstein estimates: weakly coupled systems and integral equations

Diogo A. Gomes; Gabriele Terrone

doi:10.3934/cpaa.2012.11.861

Article Contents

2012, Volume 11, Issue 3: 861-883. Doi: 10.3934/cpaa.2012.11.861

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Bernstein estimates: weakly coupled systems and integral equations

Diogo A. Gomes^1, and
Gabriele Terrone^2,

1.
Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa
2.
Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa

Received: September 30, 2010

Revised: October 31, 2011

Published: April 2012

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Abstract

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.

Keywords:

Bernstein estimate,
systems of fully non-linear elliptic equations,
weak coupling,
singular integral operators,
fractional Laplacian,
maximum principles.

Mathematics Subject Classification: Primary: 35B45, 35R09; Secondary: 35G50, 35B50.

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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, 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.