July  2021, 41(7): 3465-3488. doi: 10.3934/dcds.2021004

Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators

UMR 8088, CY Cergy Paris University, 2 avenue Adolphe Chauvain, Cergy, France

* Corresponding author: Françoise Demengel

Received  March 2020 Revised  September 2020 Published  July 2021 Early access  January 2021

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 [16], [10], [24].

Citation: Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3465-3488. doi: 10.3934/dcds.2021004
References:
[1]

G. Barles and J. Busca, Existence and comparison results for fully non linear degenerate elliptic equations without zeroth order terms, Communications in Partial Differential Equations, 26 (2001), 2323-2337.  doi: 10.1081/PDE-100107824.  Google Scholar

[2]

G. BarlesE. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc., 13 (2011), 1-26.  doi: 10.4171/JEMS/242.  Google Scholar

[3]

G. Barles and F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Archive for Rational Mechanics and Analysis, 133 (1995), 77-101.  doi: 10.1007/BF00375351.  Google Scholar

[4]

G. Barles and A. Porretta, Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equation, Ann. Scuola Norm. Sup Pisa, Cl Sci, 5 (2006), 107-136.   Google Scholar

[5]

G. BarlesA. Porretta and 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 (2010), 497-519.  doi: 10.1016/j.matpur.2010.03.006.  Google Scholar

[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.  Google Scholar

[7]

I. Birindelli and F. Demengel, Existence and regularity results for fully nonlinear operators on the model of the pseudo Pucci's operators, J. Elliptic Parabol. Equ., 2 (2016), 171-187.  doi: 10.1007/BF03377400.  Google Scholar

[8]

I. Birindelli and F. Demengel, $\mathcal{C}^{1, \beta} $ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var., 20 (2014), 1009-1024.  doi: 10.1051/cocv/2014005.  Google Scholar

[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.  Google Scholar

[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. doi: 10.1051/cocv/2018070.  Google Scholar

[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). doi: 10.1007/s00030-019-0586-2.  Google Scholar

[12]

P. Bousquet and L. Brasco, $\mathcal{C}^1$ regularity of orthotropic p-harmonic functions in the plane, Anal. PDE, 11 (2018), 813-854.  doi: 10.2140/apde.2018.11.813.  Google Scholar

[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 doi: 10.4171/rmi/1189.  Google Scholar

[14]

P. BousquetL. Brasco and V. Julin, Lipschitz regularity for local minimizers of some widely degenerate problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 1235-1274.   Google Scholar

[15]

L. Brasco and G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds, Adv. Calc. Var., 7 (2014), 379-407.  doi: 10.1515/acv-2013-0007.  Google Scholar

[16]

I. Capuzzo DolcettaF. Leoni and A. Porretta, Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian, Transactions of the American Society, 362 (2010), 4511-4536.  doi: 10.1090/S0002-9947-10-04807-5.  Google Scholar

[17]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[18]

F. Demengel, Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian Equation, Advances in Differential Equations, 21 (2016), 373-400.   Google Scholar

[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 (2017), 27-43.  doi: 10.3233/ASY-171433.  Google Scholar

[20]

I. FonsecaN. Fusco and P. Marcellini, An existence result for a non convex variational problem via regularity, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 69-95.  doi: 10.1051/cocv:2002004.  Google Scholar

[21]

H. Ishii, Viscosity solutions of Nonlinear fully nonlinear equations, Sugaku Expositions, Vol 9, number 2, December 1996.  Google Scholar

[22]

H. Ishii and P.-L. Lions, Viscosity solutions of Fully-Nonlinear Second Order Elliptic Partial Differential Equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[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. doi: 10.1007/BF01442856.  Google Scholar

[24]

T. Leonori and A. Porretta, Large solutions and gradient bounds for quasilinear elliptic equations, Comm. in Partial Differential Equations, 41 (2016), 952-998.  doi: 10.1080/03605302.2016.1169286.  Google Scholar

[25]

T. LeonoriA. Porretta and G. Riey, Comparison principles for p-Laplace equations with lower order terms, Annali di Matematica Pura ed Applicata, 196 (2017), 877-903.  doi: 10.1007/s10231-016-0600-9.  Google Scholar

[26]

P. Lindqvist and D. Ricciotti, Regularity for an anisotropic equation in the plane, Non Linear Analysis, 177, (2018), 628–636. doi: 10.1016/j.na.2018.02.002.  Google Scholar

[27]

A. Porretta, The ergodic limit for a viscous Hamilton- Jacobi equation with Dirichlet conditions, Rend. Lincei Mat. Appl., 21 (2010), 59-78.  doi: 10.4171/RLM/561.  Google Scholar

[28]

N. Uraltseva and N. Urdaletova, The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations, Vest. Leningr. UniV. Math, 16 (1984), 263-270.   Google Scholar

show all references

References:
[1]

G. Barles and J. Busca, Existence and comparison results for fully non linear degenerate elliptic equations without zeroth order terms, Communications in Partial Differential Equations, 26 (2001), 2323-2337.  doi: 10.1081/PDE-100107824.  Google Scholar

[2]

G. BarlesE. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc., 13 (2011), 1-26.  doi: 10.4171/JEMS/242.  Google Scholar

[3]

G. Barles and F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Archive for Rational Mechanics and Analysis, 133 (1995), 77-101.  doi: 10.1007/BF00375351.  Google Scholar

[4]

G. Barles and A. Porretta, Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equation, Ann. Scuola Norm. Sup Pisa, Cl Sci, 5 (2006), 107-136.   Google Scholar

[5]

G. BarlesA. Porretta and 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 (2010), 497-519.  doi: 10.1016/j.matpur.2010.03.006.  Google Scholar

[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.  Google Scholar

[7]

I. Birindelli and F. Demengel, Existence and regularity results for fully nonlinear operators on the model of the pseudo Pucci's operators, J. Elliptic Parabol. Equ., 2 (2016), 171-187.  doi: 10.1007/BF03377400.  Google Scholar

[8]

I. Birindelli and F. Demengel, $\mathcal{C}^{1, \beta} $ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var., 20 (2014), 1009-1024.  doi: 10.1051/cocv/2014005.  Google Scholar

[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.  Google Scholar

[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. doi: 10.1051/cocv/2018070.  Google Scholar

[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). doi: 10.1007/s00030-019-0586-2.  Google Scholar

[12]

P. Bousquet and L. Brasco, $\mathcal{C}^1$ regularity of orthotropic p-harmonic functions in the plane, Anal. PDE, 11 (2018), 813-854.  doi: 10.2140/apde.2018.11.813.  Google Scholar

[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 doi: 10.4171/rmi/1189.  Google Scholar

[14]

P. BousquetL. Brasco and V. Julin, Lipschitz regularity for local minimizers of some widely degenerate problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 1235-1274.   Google Scholar

[15]

L. Brasco and G. Carlier, On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds, Adv. Calc. Var., 7 (2014), 379-407.  doi: 10.1515/acv-2013-0007.  Google Scholar

[16]

I. Capuzzo DolcettaF. Leoni and A. Porretta, Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian, Transactions of the American Society, 362 (2010), 4511-4536.  doi: 10.1090/S0002-9947-10-04807-5.  Google Scholar

[17]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[18]

F. Demengel, Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian Equation, Advances in Differential Equations, 21 (2016), 373-400.   Google Scholar

[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 (2017), 27-43.  doi: 10.3233/ASY-171433.  Google Scholar

[20]

I. FonsecaN. Fusco and P. Marcellini, An existence result for a non convex variational problem via regularity, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 69-95.  doi: 10.1051/cocv:2002004.  Google Scholar

[21]

H. Ishii, Viscosity solutions of Nonlinear fully nonlinear equations, Sugaku Expositions, Vol 9, number 2, December 1996.  Google Scholar

[22]

H. Ishii and P.-L. Lions, Viscosity solutions of Fully-Nonlinear Second Order Elliptic Partial Differential Equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[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. doi: 10.1007/BF01442856.  Google Scholar

[24]

T. Leonori and A. Porretta, Large solutions and gradient bounds for quasilinear elliptic equations, Comm. in Partial Differential Equations, 41 (2016), 952-998.  doi: 10.1080/03605302.2016.1169286.  Google Scholar

[25]

T. LeonoriA. Porretta and G. Riey, Comparison principles for p-Laplace equations with lower order terms, Annali di Matematica Pura ed Applicata, 196 (2017), 877-903.  doi: 10.1007/s10231-016-0600-9.  Google Scholar

[26]

P. Lindqvist and D. Ricciotti, Regularity for an anisotropic equation in the plane, Non Linear Analysis, 177, (2018), 628–636. doi: 10.1016/j.na.2018.02.002.  Google Scholar

[27]

A. Porretta, The ergodic limit for a viscous Hamilton- Jacobi equation with Dirichlet conditions, Rend. Lincei Mat. Appl., 21 (2010), 59-78.  doi: 10.4171/RLM/561.  Google Scholar

[28]

N. Uraltseva and N. Urdaletova, The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations, Vest. Leningr. UniV. Math, 16 (1984), 263-270.   Google Scholar

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