Harnack type estimates for nonlinear elliptic systems and applications

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 21, Issue 5, 2004, Pages 543-590

  Busca, Jérôme ^a   Sirakov, Boyan ^b  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; LINEAR EQUATIONS; MATHEMATICAL OPERATORS; PROBLEM SOLVING; THEOREM PROVING; VISCOSITY;

HARNACK TYPE ESTIMATES; HIGHER ORDER EQUATIONS; INEQUALITIES; NONLINEAR ELLIPTIC SYSTEMS;

NONLINEAR SYSTEMS;

EID: 4344560340     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2003.06.001     Document Type: Article

References (41)

1. Arapostathis A., Ghosh M., Marcus St. Harnack's inequality for cooperative weakly coupled elliptic systems. Comm. Partial Differential Equations. 24:(9-10):1999;1555-1571.
2. Berestycki H., Nirenberg L., Varadhan S.R.S. The principal eigenvalue and maximum principle for second order elliptic operators in general domains. Comm. Pure Appl. Math. 47:(1):1994;47-92.
3. Birindelli I., Mitidieri E., Sweers G. Existence of the principal eigenfunction for cooperative elliptic systems in a general domain. Differential Equations (Differentsial'nye Uravneniya). 35:(3):1999;. (in Russian).
4. Bladt M. A Markov modulated financial model. J. Comm. Statist.: Stoch. Models. 14:1998;225-240.
5. M. Bladt, P. Padilla, Nonlinear financial models: finite Markov modulation and limits, Preprint.
6. Busca J. Existence results for Bellman equations and maximum principles in unbounded domains. Comm. Partial Differential Equations. 24:(11-12): 1999;2023-2042.
7. Caffarelli L.A. Interior estimates for fully nonlinear elliptic equations. Ann. of Math. 130:1989;189-213.
8. Caffarelli L.A., Cabre X. Fully Nonlinear Elliptic Equations. Amer. Math. Soc. Collog. Publ. vol. 43:1995;American Mathematical Society, Providence, RI.
9. Caffarelli L.A., Crandall M.G., Kocan M., Świech A. On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49:1996;365-397.
10. Chen Z.Q., Zhao Z. Harnack principle for weakly coupled elliptic systems. J. Differential Equations. 39:1997;261-282.
11. Chen Z.Q., Zhao Z. Potential theory for elliptic systems. Ann. Probab. 24:(1):1996;293-319.
12. Crandall M.G., Ishii H., Lions P.-L. User's guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc. 27:(1):1992;1-67.
13. p theory for fully nonlinear uniformly parabolic equations. Comm. Partial Differential Equations. 25:(11&12):2000;1997-2053.
14. de Figueiredo D.G. Monotonicity and symmetry of solutions of elliptic systems in general domains. Nonlinear Differential Equations Appl. 1:1994;119-123.
15. de Figueiredo D.G., Mitidieri E. Maximum principles for linear elliptic systems. Rend. Inst. Mat. Univ. Trieste. 1992;36-66.
16. de Figueiredo D.G., Mitidieri E. Maximum principles for cooperative elliptic systems. C. R. Acad. Sci. Paris, Ser. I. 310:(2):1990;49-52.
17. de Giorgi E. Un esempio di estremali discontinue per un problema variazionale di tipo elliptico. Boll. Un. Mat. Ital. 1:(4):1968;135-137.
18. Duffin R.J., Nehari Z. Note on polyharmonic functions. Proc. Amer. Math. Soc. 12:1961;110-115.
19. Friedman A. On n -metaharmonic functions and harmonic functions of infinite order. Proc. Amer. Math. Soc. 8:1957;223-229.
20. Ghermanescu M. Sur les valeurs moyennes des fonctions. Math. Ann. 119:1944;288-320.
21. Giaquinta M. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Ann. Math. Stud. vol. 105:1983;Princeton University Press.
22. Gilbarg D., Trudinger N.S. Elliptic Partial Differential Equations of Second Order. 2nd edition, Revised Third Printing:1998;Springer-Verlag.
23. Grunau H., Sweers G. Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. Math. Ann. 307:1997;589.
24. Grunau H., Sweers G. Classical solutions for some higher order semilinear elliptic equations under weak growth conditions. Nonlinear Anal. 28:1997;799-807.
25. Hess P. On the eigenvalue problem for weakly coupled elliptic systems. Arch. Rational Mech. Anal. 81:(2):1983;151-159.
26. Hildebrandt S., Widman K.-O. On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 4:(1):1977;145-178.
27. Ishii H., Koike S. Viscosity solutions for monotone systems of second-order elliptic PDE's. Comm. Partial Differential Equations. 16:(6 & 7):1991;1095-1128.
28. Jensen R. The maximum principle for viscosity solutions of fully nonlinear seoncd order partial differential equations. Arch. Rational Mech. Anal. 101:1988;1-27.
29. Jensen R., Lions P.L., Souganidis P.E. A uniqueness result for viscosity solutions of fully nonlinear second order partial differential equations. Proc. Amer. Math. Soc. 4:1988;975-978.
30. Krylov. Nonlinear Elliptic and Parabolic Equations of Second Order. Coll. Math. Appl. 1987.
31. Lasry J.-M., Lions P.-L. Large deviations for diffusion process coupled by a jump process. C. R. Acad. Sci. Paris, Sér. I. 321:1995;849-854.
32. Lenhart S.M., Belbas S.A. A system of nonlinear PDE's arising in the optimal control of stochastic systems. SIAM J. Appl. Math. 43:1983;465-475.
33. Mandras. Diseguanza di Harnack per sistemi elliptici debolmente accopiati. Boll. Un. Mat. Ital. A. 14:(5):1977;313-332.
34. Murray J.D. Mathematical Biology. 1993;Springer-Verlag.
35. Muscalu C. On the Harnack principle for strongly elliptic systems with nonsmooth coefficients. Comm. Pure Appl. Math. 52:1999;1213-1230.
36. Nicolescu M. Les fonctions polyharmoniques. 1936;Hermann & Cie, Paris.
37. Ovčarenko I.E. On multiply superharmonic functions. Uspekhi Math. Nauk. 16:(3):1961;197-200. (in Russian).
38. Serrin J. Local behaviour of solutions of quasilinear equations. Acta Math. 111:1964;247-302.
39. Smyrnelis E.P. Une propriété de moyenne des fonctions biharmoniques. Bull. Sci. Math. (2). 109:1985;103-111.
40. Sweers G. Strong positivity in C( Ω ) for elliptic systems. Math. Z. 209:1992;251-271.
41. Wang L. On the regularity theory of fully nonlinear parabolic equations: I. Comm. Pure Appl. Math. 45:(1):1992;27-76.