×

Ergodic pairs for degenerate pseudo Pucci’s fully nonlinear operators. (English) Zbl 1466.35206

Summary: We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when at least one of the components of the gradient vanishes. We extend here the results in [C. Dolcetta et al., Trans. Am. Math. Soc. 362, No. 9, 4511–4536 (2010; Zbl 1198.35110); I. Birindelli et al., ESAIM, Control Optim. Calc. Var. 25, Paper No. 75, 28 p. (2019; Zbl 1437.35370); T. Leonori and A. Porretta, Commun. Partial Differ. Equations 41, No. 6, 952–998 (2016; Zbl 1348.35083)].

MSC:

35J70 Degenerate elliptic equations
35J75 Singular elliptic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] G. Barles; J. Busca, Existence and comparison results for fully non linear degenerate elliptic equations without zeroth order terms, Communications in Partial Differential Equations, 26, 2323-2337 (2001) · Zbl 0997.35023 · doi:10.1081/PDE-100107824
[2] G. Barles; E. Chasseigne; C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc., 13, 1-26 (2011) · Zbl 1207.35277 · doi:10.4171/JEMS/242
[3] G. Barles; F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Archive for Rational Mechanics and Analysis, 133, 77-101 (1995) · Zbl 0859.35031 · doi:10.1007/BF00375351
[4] G. Barles; A. Porretta, Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equation, Ann. Scuola Norm. Sup Pisa, Cl Sci, 5, 107-136 (2006) · Zbl 1150.35030
[5] G. Barles; A. Porretta; T. Tabet Tchamba, On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations, Journal de Mathématique Pures et Appliquées, 94, 497-519 (2010) · Zbl 1209.37069 · doi:10.1016/j.matpur.2010.03.006
[6] I. Birindelli and F. Demengel, First eigenvalue and Maximum principle for fully nonlinear singular operators, Advances in Differential Equations, Vol 11 (2006), 91-119. · Zbl 1132.35427
[7] I. Birindelli; F. Demengel, Existence and regularity results for fully nonlinear operators on the model of the pseudo Pucci’s operators, J. Elliptic Parabol. Equ., 2, 171-187 (2016) · Zbl 1396.35013 · doi:10.1007/BF03377400
[8] I. Birindelli; F. Demengel, \( \mathcal{C}^{1, \beta}\) regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var., 20, 1009-1024 (2014) · Zbl 1315.35108 · doi:10.1051/cocv/2014005
[9] I. Birindelli, F. Demengel and F. Leoni, Dirichlet problems for fully nonlinear equations with “subquadratic” Hamiltonians, Contemporary research in elliptic PDEs and related topics, Springer INdAM Ser., 33, Springer, Cham, 2019, 107-127. · Zbl 1433.35125
[10] I. Birindelli, F. Demengel and F. Leoni, Ergodic pairs for singular or degenerate fully nonlinear operators, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 75, 28 pp. · Zbl 1437.35370
[11] I. Birindelli, F. Demengel and F. Leoni, On the \(\mathcal{C}^{1, \gamma}\) regularity for Fully non linear singular or degenerate equations with a subquadratic hamiltonian, NoDEA Nonlinear Differential Equations Appl., 26 (2019). · Zbl 1428.35132
[12] P. Bousquet; L. Brasco, \( \mathcal{C}^1\) regularity of orthotropic p-harmonic functions in the plane, Anal. PDE, 11, 813-854 (2018) · Zbl 1387.35293 · doi:10.2140/apde.2018.11.813
[13] P. Bousquet and L. Brasco, Lipschitz regularity for orthotropic functionals with non standard growth conditions, Rev. Mat. Iberoam., 36 (2020), 1989-2032. arXiv: 1810.03837v, et · Zbl 1460.35162
[14] P. Bousquet; L. Brasco; V. Julin, Lipschitz regularity for local minimizers of some widely degenerate problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16, 1235-1274 (2016) · Zbl 1366.35047
[15] L. Brasco; G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds, Adv. Calc. Var., 7, 379-407 (2014) · Zbl 1305.49067 · doi:10.1515/acv-2013-0007
[16] I. Capuzzo Dolcetta; F. Leoni; A. Porretta, Hölder’s estimates for degenerate elliptic equations with coercive Hamiltonian, Transactions of the American Society, 362, 4511-4536 (2010) · Zbl 1198.35110 · doi:10.1090/S0002-9947-10-04807-5
[17] M. G. Crandall; H. Ishii; P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1-67 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[18] F. Demengel, Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo \(p\)-Laplacian Equation, Advances in Differential Equations, 21, 373-400 (2016) · Zbl 1339.35132
[19] F. Demengel, Regularity properties of Viscosity Solutions for Fully Non linear Equations on the model of the anisotropic \(\vec p\)-Laplacian., Asymptotic Analysis, 105, 27-43 (2017) · Zbl 1390.35081 · doi:10.3233/ASY-171433
[20] I. Fonseca; N. Fusco; P. Marcellini, An existence result for a non convex variational problem via regularity, ESAIM: Control, Optimisation and Calculus of Variations, 7, 69-95 (2002) · Zbl 1044.49011 · doi:10.1051/cocv:2002004
[21] H. Ishii, Viscosity solutions of Nonlinear fully nonlinear equations, Sugaku Expositions, Vol 9, number 2, December 1996. · Zbl 0846.49011
[22] H. Ishii; P.-L. Lions, Viscosity solutions of Fully-Nonlinear Second Order Elliptic Partial Differential Equations, J. Differential Equations, 83, 26-78 (1990) · Zbl 0708.35031 · doi:10.1016/0022-0396(90)90068-Z
[23] J.-M. Lasry and P.-L. Lions, Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints,, Math. Ann., 283, (1989), 583-630. · Zbl 0688.49026
[24] T. Leonori; A. Porretta, Large solutions and gradient bounds for quasilinear elliptic equations, Comm. in Partial Differential Equations, 41, 952-998 (2016) · Zbl 1348.35083 · doi:10.1080/03605302.2016.1169286
[25] T. Leonori; A. Porretta; G. Riey, Comparison principles for p-Laplace equations with lower order terms, Annali di Matematica Pura ed Applicata, 196, 877-903 (2017) · Zbl 1376.35081 · doi:10.1007/s10231-016-0600-9
[26] P. Lindqvist and D. Ricciotti, Regularity for an anisotropic equation in the plane, Non Linear Analysis, 177, (2018), 628-636. · Zbl 1454.35171
[27] A. Porretta, The ergodic limit for a viscous Hamilton- Jacobi equation with Dirichlet conditions, Rend. Lincei Mat. Appl., 21, 59-78 (2010) · Zbl 1189.35102 · doi:10.4171/RLM/561
[28] N. Uraltseva; N. Urdaletova, The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations, Vest. Leningr. UniV. Math, 16, 263-270 (1984) · Zbl 0569.35029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.