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 and 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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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