Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls
Ikoma, Norihisa ; Ishii, Hitoshi
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 783-812 / Harvested from Numdam
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We study the eigenvalue problem for positively homogeneous, of degree one, elliptic ODE on finite intervals and PDE on balls. We establish the existence and completeness results for principal and higher eigenpairs, i.e., pairs of an eigenvalue and its corresponding eigenfunction.

@article{AIHPC_2012__29_5_783_0,
     author = {Ikoma, Norihisa and Ishii, Hitoshi},
     title = {Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {783-812},
     doi = {10.1016/j.anihpc.2012.04.004},
     mrnumber = {2971031},
     zbl = {1254.35166},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_5_783_0}
}

Ikoma, Norihisa; Ishii, Hitoshi. Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 783-812. doi : 10.1016/j.anihpc.2012.04.004. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_5_783_0/

[1] S.N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations 246 no. 7 (2009), 2958-2987 | MR 2503031 | Zbl 1171.35093

[2] H. Berestycki, On some nonlinear Sturm–Liouville problems, J. Differential Equations 26 no. 3 (1977), 375-390 | MR 481230 | Zbl 0331.34020

[3] I. Birindelli, F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differential Equations 11 no. 1 (2006), 91-119 | MR 2192416 | Zbl 1132.35427

[4] J. Busca, M.J. Esteban, A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucciʼs operators, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 2 (2005), 187-206 | Numdam | MR 2124162 | Zbl 1205.35087

[5] L.A. Caffarelli, Interior W 2,p estimates for solutions of the Monge–Ampere equation, Ann. of Math. (2) 131 no. 1 (1990), 135-150 | MR 1038360 | Zbl 0704.35044

[6] L.A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence (1995) | MR 1351007 | Zbl 0834.35002

[7] L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 no. 3–4 (1985), 261-301 | MR 806416 | Zbl 0654.35031

[8] E.N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 no. 4 (1976/77), 283-300 | MR 499709 | Zbl 0351.35037

[9] L. Escauriaza, W 2,n a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J. 42 (1993), 413-423 | MR 1237053 | Zbl 0792.35020

[10] M.J. Esteban, P. Felmer, A. Quaas, Eigenvalues for radially symmetric fully nonlinear operators, Comm. Partial Differential Equations 35 no. 9 (2010), 1716-1737 | MR 2754061 | Zbl 1206.35186

[11] P. Fok, Some maximum principles and continuity estimates for fully nonlinear elliptic equations of second order, PhD thesis, UCSB, 1996, 82 pp. | MR 2694855

[12] S. Fučik, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat. 101 no. 1 (1976), 69-87 | MR 447688 | Zbl 0332.34016

[13] H. Ishii, Y. Yoshimura, Demi-eigenvalues for uniformly elliptic Isaacs operators, preprint.

[14] S. Koike, A. ŚwiȩCh, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan 61 no. 3 (2009), 723-755 | MR 2552914 | Zbl 1228.35104

[15] P.-L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal. 7 no. 2 (1983), 177-207 | MR 688774

[16] N. Nadirashvili, S. Vlăduţ, On axially symmetric solutions of fully nonlinear elliptic equations, Math. Z. 270 (2012), 331-336 | MR 2875836 | Zbl 1244.35055

[17] S. Patrizi, Principal eigenvalues for Isaacs operators with Neumann boundary conditions, NoDEA Nonlinear Differential Equations Appl. 16 no. 1 (2009), 79-107 | MR 2486349 | Zbl 1178.35180

[18] A. Quaas, B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math. 218 no. 1 (2008), 105-135 | MR 2409410 | Zbl 1143.35077

[19] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal. 195 no. 2 (2010), 579-607 | MR 2592289 | Zbl 1193.35054

[20] N.S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 no. 2 (1995), 151-164 | MR 1368245 | Zbl 0887.35061