Harnack type estimates for nonlinear elliptic systems and applications
Annales de l'I.H.P. Analyse non linéaire, Volume 21 (2004) no. 5, p. 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},
     publisher = {Elsevier},
     volume = {21},
     number = {5},
     year = {2004},
     pages = {543-590},
     doi = {10.1016/j.anihpc.2003.06.001},
     zbl = {02116180},
     mrnumber = {2086750},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 21 (2004) no. 5, pp. 543-590. doi : 10.1016/j.anihpc.2003.06.001. http://www.numdam.org/item/AIHPC_2004__21_5_543_0/

[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 1708101 | Zbl 0934.35039

[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 1258192 | Zbl 0806.35129

[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 1726799 | Zbl 0940.35147

[4] Bladt M., A Markov modulated financial model, J. Comm. Statist.: Stoch. Models 14 (1998) 225-240. | MR 1617564 | Zbl 0934.60061

[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 1720774 | Zbl 0961.35021

[7] Caffarelli L.A., Interior estimates for fully nonlinear elliptic equations, Ann. of Math. 130 (1989) 189-213. | MR 1005611 | Zbl 0692.35017

[8] Caffarelli L.A., Cabre X., Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Collog. Publ., vol. 43, American Mathematical Society, Providence, RI, 1995. | MR 1351007 | Zbl 0834.35002

[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 1376656 | Zbl 0854.35032

[10] Chen Z.Q., Zhao Z., Harnack principle for weakly coupled elliptic systems, J. Differential Equations 39 (1997) 261-282. | MR 1472349 | Zbl 0882.35039

[11] Chen Z.Q., Zhao Z., Potential theory for elliptic systems, Ann. Probab. 24 (1) (1996) 293-319. | MR 1387637 | Zbl 0854.60062

[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 1118699 | Zbl 0755.35015

[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 1789919 | Zbl 0973.35097

[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 1273345 | Zbl 0822.35039

[15] De Figueiredo D.G., Mitidieri E., Maximum principles for linear elliptic systems, Rend. Inst. Mat. Univ. Trieste (1992) 36-66. | MR 1210477 | Zbl 0793.35011

[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 1044413 | Zbl 0712.35021

[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 227827 | Zbl 0155.17603

[18] Duffin R.J., Nehari Z., Note on polyharmonic functions, Proc. Amer. Math. Soc. 12 (1961) 110-115. | MR 141793 | Zbl 0097.08504

[19] Friedman A., On n-metaharmonic functions and harmonic functions of infinite order, Proc. Amer. Math. Soc. 8 (1957) 223-229. | MR 85430 | Zbl 0077.30304

[20] Ghermanescu M., Sur les valeurs moyennes des fonctions, Math. Ann. 119 (1944) 288-320. | MR 11489 | Zbl 0063.01603

[21] Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. Math. Stud., vol. 105, Princeton University Press, 1983. | MR 717034 | Zbl 0516.49003

[22] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1998. | Zbl 0562.35001

[23] Grunau H., Sweers G., Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997) 589. | MR 1464133 | Zbl 0892.35031

[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 1422185 | Zbl 0867.35031

[25] Hess P., On the eigenvalue problem for weakly coupled elliptic systems, Arch. Rational Mech. Anal. 81 (2) (1983) 151-159. | MR 682266 | Zbl 0509.35029

[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 457936 | Zbl 0353.35013

[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 1116855 | Zbl 0742.35022

[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 920674 | Zbl 0708.35019

[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 934877 | Zbl 0662.35048

[30] Krylov , Nonlinear Elliptic and Parabolic Equations of Second Order, Coll. Math. Appl., 1987. | MR 934313 | Zbl 0619.35004

[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 1355840 | Zbl 0837.60026

[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 700525 | Zbl 0511.93077

[33] Mandras , Diseguanza di Harnack per sistemi elliptici debolmente accopiati, Boll. Un. Mat. Ital. A 14 (5) (1977) 313-332. | MR 499700 | Zbl 0357.35003

[34] Murray J.D., Mathematical Biology, Springer-Verlag, 1993. | MR 1239892 | Zbl 0682.92001

[35] Muscalu C., On the Harnack principle for strongly elliptic systems with nonsmooth coefficients, Comm. Pure Appl. Math. 52 (1999) 1213-1230. | MR 1699966 | Zbl 0949.35046

[36] Nicolescu M., Les fonctions polyharmoniques, Hermann & Cie, Paris, 1936. | Zbl 0015.15905

[37] Ovčarenko I.E., On multiply superharmonic functions, Uspekhi Math. Nauk 16 (3) (1961) 197-200, (in Russian). | MR 130389 | Zbl 0128.09901

[38] Serrin J., Local behaviour of solutions of quasilinear equations, Acta Math. 111 (1964) 247-302. | MR 170096 | Zbl 0128.09101

[39] Smyrnelis E.P., Une propriété de moyenne des fonctions biharmoniques, Bull. Sci. Math. (2) 109 (1985) 103-111. | MR 802527 | Zbl 0566.31004

[40] Sweers G., Strong positivity in C(Ω ¯) for elliptic systems, Math. Z. 209 (1992) 251-271. | Zbl 0727.35045

[41] Wang L., On the regularity theory of fully nonlinear parabolic equations: I, Comm. Pure Appl. Math. 45 (1) (1992) 27-76. | MR 1135923 | Zbl 0832.35025