Harnack type estimates for nonlinear elliptic systems and applications
Annales de l'I.H.P. Analyse non linéaire, Tome 21 (2004) no. 5, pp. 543-590.
@article{AIHPC_2004__21_5_543_0,
     author = {Busca, J\'er\^ome and Sirakov, Boyan},
     title = {Harnack type estimates for nonlinear elliptic systems and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {543--590},
     publisher = {Elsevier},
     volume = {21},
     number = {5},
     year = {2004},
     doi = {10.1016/j.anihpc.2003.06.001},
     mrnumber = {2086750},
     zbl = {02116180},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2003.06.001/}
}
TY  - JOUR
AU  - Busca, Jérôme
AU  - Sirakov, Boyan
TI  - Harnack type estimates for nonlinear elliptic systems and applications
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2004
SP  - 543
EP  - 590
VL  - 21
IS  - 5
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2003.06.001/
DO  - 10.1016/j.anihpc.2003.06.001
LA  - en
ID  - AIHPC_2004__21_5_543_0
ER  - 
%0 Journal Article
%A Busca, Jérôme
%A Sirakov, Boyan
%T Harnack type estimates for nonlinear elliptic systems and applications
%J Annales de l'I.H.P. Analyse non linéaire
%D 2004
%P 543-590
%V 21
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2003.06.001/
%R 10.1016/j.anihpc.2003.06.001
%G en
%F AIHPC_2004__21_5_543_0
Busca, Jérôme; Sirakov, Boyan. Harnack type estimates for nonlinear elliptic systems and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 21 (2004) no. 5, pp. 543-590. doi : 10.1016/j.anihpc.2003.06.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2003.06.001/

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[4] Bladt M., A Markov modulated financial model, J. Comm. Statist.: Stoch. Models 14 (1998) 225-240. | MR | Zbl

[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. | MR | Zbl

[7] Caffarelli L.A., Interior estimates for fully nonlinear elliptic equations, Ann. of Math. 130 (1989) 189-213. | MR | Zbl

[8] Caffarelli L.A., Cabre X., Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Collog. Publ., vol. 43, American Mathematical Society, Providence, RI, 1995. | MR | Zbl

[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. | MR | Zbl

[10] Chen Z.Q., Zhao Z., Harnack principle for weakly coupled elliptic systems, J. Differential Equations 39 (1997) 261-282. | MR | Zbl

[11] Chen Z.Q., Zhao Z., Potential theory for elliptic systems, Ann. Probab. 24 (1) (1996) 293-319. | MR | Zbl

[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. | MR | Zbl

[13] Crandall M.G., Kocan M., Świech A., Lp theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations 25 (11&12) (2000) 1997-2053. | MR | Zbl

[14] De Figueiredo D.G., Monotonicity and symmetry of solutions of elliptic systems in general domains, Nonlinear Differential Equations Appl. 1 (1994) 119-123. | MR | Zbl

[15] De Figueiredo D.G., Mitidieri E., Maximum principles for linear elliptic systems, Rend. Inst. Mat. Univ. Trieste (1992) 36-66. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[18] Duffin R.J., Nehari Z., Note on polyharmonic functions, Proc. Amer. Math. Soc. 12 (1961) 110-115. | MR | Zbl

[19] Friedman A., On n-metaharmonic functions and harmonic functions of infinite order, Proc. Amer. Math. Soc. 8 (1957) 223-229. | MR | Zbl

[20] Ghermanescu M., Sur les valeurs moyennes des fonctions, Math. Ann. 119 (1944) 288-320. | MR | Zbl

[21] Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. Math. Stud., vol. 105, Princeton University Press, 1983. | MR | Zbl

[22] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1998. | Zbl

[23] Grunau H., Sweers G., Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997) 589. | MR | Zbl

[24] Grunau H., Sweers G., Classical solutions for some higher order semilinear elliptic equations under weak growth conditions, Nonlinear Anal. 28 (1997) 799-807. | MR | Zbl

[25] Hess P., On the eigenvalue problem for weakly coupled elliptic systems, Arch. Rational Mech. Anal. 81 (2) (1983) 151-159. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[30] Krylov , Nonlinear Elliptic and Parabolic Equations of Second Order, Coll. Math. Appl., 1987. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[33] Mandras , Diseguanza di Harnack per sistemi elliptici debolmente accopiati, Boll. Un. Mat. Ital. A 14 (5) (1977) 313-332. | MR | Zbl

[34] Murray J.D., Mathematical Biology, Springer-Verlag, 1993. | MR | Zbl

[35] Muscalu C., On the Harnack principle for strongly elliptic systems with nonsmooth coefficients, Comm. Pure Appl. Math. 52 (1999) 1213-1230. | MR | Zbl

[36] Nicolescu M., Les fonctions polyharmoniques, Hermann & Cie, Paris, 1936. | Zbl

[37] Ovčarenko I.E., On multiply superharmonic functions, Uspekhi Math. Nauk 16 (3) (1961) 197-200, (in Russian). | MR | Zbl

[38] Serrin J., Local behaviour of solutions of quasilinear equations, Acta Math. 111 (1964) 247-302. | MR | Zbl

[39] Smyrnelis E.P., Une propriété de moyenne des fonctions biharmoniques, Bull. Sci. Math. (2) 109 (1985) 103-111. | MR | Zbl

[40] Sweers G., Strong positivity in C(Ω ¯) for elliptic systems, Math. Z. 209 (1992) 251-271. | Zbl

[41] Wang L., On the regularity theory of fully nonlinear parabolic equations: I, Comm. Pure Appl. Math. 45 (1) (1992) 27-76. | MR | Zbl

Cité par Sources :