A family of degenerate elliptic operators: Maximum principle and its consequences
Annales de l'I.H.P. Analyse non linéaire, Volume 35 (2018) no. 2, pp. 417-441.

In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.

DOI: 10.1016/j.anihpc.2017.05.003
Classification: 35J60, 35J70, 49L25
Keywords: Maximum principle, Fully nonlinear degenerate elliptic PDE, Eigenvalue problem
@article{AIHPC_2018__35_2_417_0,
     author = {Birindelli, Isabeau and Galise, Giulio and Ishii, Hitoshi},
     title = {A family of degenerate elliptic operators: {Maximum} principle and its consequences},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {417--441},
     publisher = {Elsevier},
     volume = {35},
     number = {2},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.05.003},
     mrnumber = {3765548},
     zbl = {1390.35079},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.05.003/}
}
TY  - JOUR
AU  - Birindelli, Isabeau
AU  - Galise, Giulio
AU  - Ishii, Hitoshi
TI  - A family of degenerate elliptic operators: Maximum principle and its consequences
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 417
EP  - 441
VL  - 35
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2017.05.003/
DO  - 10.1016/j.anihpc.2017.05.003
LA  - en
ID  - AIHPC_2018__35_2_417_0
ER  - 
%0 Journal Article
%A Birindelli, Isabeau
%A Galise, Giulio
%A Ishii, Hitoshi
%T A family of degenerate elliptic operators: Maximum principle and its consequences
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 417-441
%V 35
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2017.05.003/
%R 10.1016/j.anihpc.2017.05.003
%G en
%F AIHPC_2018__35_2_417_0
Birindelli, Isabeau; Galise, Giulio; Ishii, Hitoshi. A family of degenerate elliptic operators: Maximum principle and its consequences. Annales de l'I.H.P. Analyse non linéaire, Volume 35 (2018) no. 2, pp. 417-441. doi : 10.1016/j.anihpc.2017.05.003. http://www.numdam.org/articles/10.1016/j.anihpc.2017.05.003/

[1] Ambrosio, L.; Soner, H.M. Level set approach to mean curvature flow in arbitrary codimension, J. Differ. Geom., Volume 43 (1996), pp. 693–737 | DOI | MR | Zbl

[2] Amendola, M.E.; Galise, G.; Vitolo, A. Riesz capacity, maximum principle, and removable sets of fully nonlinear second-order elliptic operators, Differ. Integral Equ., Volume 26 (2013), pp. 845–866 | MR | Zbl

[3] Armstrong, S.N. Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differ. Equ., Volume 246 (2009), pp. 2958–2987 | DOI | MR | Zbl

[4] Berestycki, H.; Capuzzo Dolcetta, I.; Porretta, A.; Rossi, L. Maximum principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl. (9), Volume 103 (2015) no. 5, pp. 1276–1293 | DOI | MR | Zbl

[5] Berestycki, H.; Nirenberg, L.; Varadhan, S. The principle eigenvalue and maximum principle for second order elliptic operators in general domains, Commun. Pure Appl. Math., Volume 47 (1994) no. 1, pp. 47–92 | DOI | MR | Zbl

[6] Birindelli, I.; Demengel, F. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Commun. Pure Appl. Anal., Volume 6 (2007) no. 2, pp. 335–366 | MR | Zbl

[7] Birindelli, I.; Demengel, F. Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, Discrete Contin. Dyn. Syst. (2007) no. suppl., pp. 110–121 | MR | Zbl

[8] Birindelli, I.; Demengel, F. Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differ. Equ., Volume 249 (2010) no. 5, pp. 1089–1110 | DOI | MR | Zbl

[9] Birindelli, I.; Leoni, F. Symmetry minimizes the principal eigenvalue: an example for the Pucci's sup operator, Math. Res. Lett., Volume 21 (2014) no. 5, pp. 953–967 | DOI | MR | Zbl

[10] Bony, J.M. Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier, Volume 19 (1969) no. 1, pp. 277–304 | Numdam | MR | Zbl

[11] Busca, J.; Esteban, M.J.; Quaas, A. Nonlinear eigenvalues and bifurcation problems for Pucci's operator, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005), pp. 187–206 | Numdam | MR | Zbl

[12] Caffarelli, L.; Li, Y.Y.; Nirenberg, L. Some remarks on singular solutions of nonlinear elliptic equations. I, J. Fixed Point Theory Appl., Volume 5 (2009), pp. 353–395 | DOI | MR | Zbl

[13] Caffarelli, L.; Li, Y.Y.; Nirenberg, L. Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators, Commun. Pure Appl. Math., Volume 66 (2013), pp. 109–143 | DOI | MR | Zbl

[14] Capuzzo Dolcetta, I.; Leoni, F.; Vitolo, A. On the inequality F(x,D2u)f(u)+g(u)|Du|q , Math. Ann., Volume 365 (2016) no. 1–2, pp. 423–448 | MR | Zbl

[15] Crandall, M.G. Viscosity Solutions and Applications, Lecture Notes in Math., Volume vol. 1660, Springer, Berlin (1997), pp. 1–43 (Montecatini Terme, 1995) | DOI | MR | Zbl

[16] Crandall, M.G.; Ishii, H.; Lions, P.-L. User's guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc. (N. S.), Volume 27 (1992) no. 1, pp. 1–67 | DOI | MR | Zbl

[17] Cutrì, A.; Leoni, F. On the Liouville property for fully nonlinear equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 17 (2000) no. 2, pp. 219–245 | DOI | Numdam | MR | Zbl

[18] Galise, G.; Vitolo, A. Removable singularities for degenerate elliptic Pucci operators, Adv. Differ. Equ., Volume 22 (2017) no. 1–2, pp. 77–100 | MR | Zbl

[19] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., vol. 224, Springer-Verlag, Berlin–New York, 1983 | MR | Zbl

[20] Harvey, F.R.; Lawson, H.B. Jr. Dirichlet duality and the nonlinear Dirichlet problem, Commun. Pure Appl. Math., Volume 62 (2009), pp. 396–443 | DOI | MR | Zbl

[21] Harvey, F.R.; Lawson, H.B. Jr. p-convexity, p-plurisubharmonicity and the Levi problem, Indiana Univ. Math. J., Volume 62 (2013), pp. 149–169 | DOI | MR | Zbl

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

[23] Juutinen, P. Principal eigenvalue of a very badly degenerate operator and applications, J. Differ. Equ., Volume 236 (2007) no. 2, pp. 532–550 | DOI | MR | Zbl

[24] Kim, I.C. A free boundary problem arising in flame propagation, J. Differ. Equ., Volume 191 (2003), pp. 470–489 | MR | Zbl

[25] Kohn, J.J.; Nirenberg, L. Degenerate elliptic–parabolic equations of second order, Commun. Pure Appl. Math., Volume 20 (1967), pp. 797–872 | MR | Zbl

[26] Lions, P.-L. Bifurcation and optimal stochastic control, Nonlinear Anal., Volume 7 (1983), pp. 177–207 | MR

[27] Oberman, A.M.; Silvestre, L. The Dirichlet problem for the convex envelope, Trans. Am. Math. Soc., Volume 363 (2011), pp. 5871–5886 | DOI | MR | Zbl

[28] Polovinkin, E.S. Strongly convex analysis, Sb. Math., Volume 187 (1996), pp. 259–286 | DOI | MR | Zbl

[29] Pucci, C. Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Am. Math. Soc., Volume 17 (1966), pp. 788–795 | DOI | MR | Zbl

[30] Quaas, A.; Sirakov, B. Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., Volume 218 (2008) no. 1, pp. 105–135 | DOI | MR | Zbl

[31] Sha, J.P. Handlebodies and p-convexity, J. Differ. Geom., Volume 25 (1987), pp. 353–361 | MR | Zbl

[32] Wu, H. Manifolds of partially positive curvature, Indiana Univ. Math. J., Volume 36 (1987), pp. 525–548 | MR | Zbl

Cited by Sources: