July  2021, 41(7): 3021-3029. doi: 10.3934/dcds.2020395

Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem

1. 

Dipartimento di Matematica G. Castelnuovo, Sapienza Università di Roma, P.le A. Moro 2, 00185 Roma, Italy

2. 

UMR 20-88, CY Paris University, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France

* Corresponding author

Received  May 2020 Revised  October 2020 Published  July 2021 Early access  December 2020

We prove gradient boundary blow up rates for ergodic functions in bounded domains related to fully nonlinear degenerate/singular elliptic operators. As a consequence, we deduce the uniqueness, up to constants, of the ergodic functions. The results are obtained by means of a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators.

Citation: Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395
References:
[1]

S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-8137-3.

[2]

G. BarlesS. KoikeO. Ley and E. Topp, Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians, Calc. Var., 54 (2015), 539-572.  doi: 10.1007/s00526-014-0794-x.

[3]

B. BarriosL. Del PezzoJ. García-Melián and A. Quaas, Symmetry results in the halfspace for a semilinear fractional Laplace equation, Annali di Matematica, 197 (2018), 1385-1416.  doi: 10.1007/s10231-018-0729-9.

[4]

H. BerestyckiF. Hamel and R. Monneau, One dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.  doi: 10.1215/S0012-7094-00-10331-6.

[5]

I. Birindelli and F. Demengel, Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient, Nonlinear Differ. Equ. Appl., 23 (2016), Art. 41, 17 pp. doi: 10.1007/s00030-016-0392-z.

[6]

I. Birindelli, F. Demengel and F. Leoni, Ergodic pairs for singular or degenerate fully nonlinear operators, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 75, 28 pp. doi: 10.1051/cocv/2018070.

[7]

I. Birindelli, F. Demengel and F. Leoni, Dirichlet Problems for Fully Nonlinear Equations with "Subquadratic" Hamiltonians, in Contemporary Research in Elliptic PDEs and Related Topics (Ed. S. Dipierro), Springer INdAM Series 33 2019,107–127. doi: 10.1007/978-3-030-18921-1.

[8]

I. Birindelli, F. Demengel and F. Leoni, $\mathcal{C}^{1, \gamma}$ regularity for singular or degenerate fully nonlinear equations and applications, Nonlinear Differ. Equ. Appl., 26 (2019), Paper No. 40. doi: 10.1007/s00030-019-0586-2.

[9]

I. Capuzzo DolcettaF. Leoni and A. Porretta, Hölder estimates for degenerate elliptic equations with coercive Hamiltonians, Trans. Amer. Math. Soc., 362 (2010), 4511-4536.  doi: 10.1090/S0002-9947-10-04807-5.

[10]

F. Demengel and I. Birindelli, One-dimensional symmetry for solutions of Allen Cahn fully nonlinear equations, in Symmetry for elliptic PDEs, (A. Farina and E. Valdinci Eds.), Amer. Math. Soc., Providence, RI (2010), 1–15. doi: 10.1090/conm/528.

[11]

A. FarinaL. MontoroG. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1-22.  doi: 10.1016/j.anihpc.2013.09.005.

[12]

R. FilippucciP. Pucci and P. Souplet, A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations, Comm. Partial Differential Equations, 45 (2020), 321-349.  doi: 10.1080/03605302.2019.1684941.

[13]

Y. Giga and M. Ohnuma, On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations, Int. J. Pure Appl. Math., 22 (2005), 165-184. 

[14]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of degenerate fully nonlinear elliptic equations, Adv. Math., 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.

[15]

T. Kilpeläinen, H. Shahgholian and X. Zhong, Growth estimates through scaling for quasilinear partial differential equation, Ann. Acad. Sci. Fenn. Math., 32 (2007), no. 2,595–599.

[16]

J.-M. Lasry and P.-L. Lions, Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints. Ⅰ. The model problem, Math. Ann., 283 (1989), 583-630.  doi: 10.1007/BF01442856.

[17]

T. Leonori and A. Porretta, Gradient bounds for elliptic problems singular at the boundary, Arch. Ration. Mech. Anal., 202 (2011), 663-705.  doi: 10.1007/s00205-011-0436-9.

[18]

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

show all references

References:
[1]

S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-8137-3.

[2]

G. BarlesS. KoikeO. Ley and E. Topp, Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians, Calc. Var., 54 (2015), 539-572.  doi: 10.1007/s00526-014-0794-x.

[3]

B. BarriosL. Del PezzoJ. García-Melián and A. Quaas, Symmetry results in the halfspace for a semilinear fractional Laplace equation, Annali di Matematica, 197 (2018), 1385-1416.  doi: 10.1007/s10231-018-0729-9.

[4]

H. BerestyckiF. Hamel and R. Monneau, One dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.  doi: 10.1215/S0012-7094-00-10331-6.

[5]

I. Birindelli and F. Demengel, Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient, Nonlinear Differ. Equ. Appl., 23 (2016), Art. 41, 17 pp. doi: 10.1007/s00030-016-0392-z.

[6]

I. Birindelli, F. Demengel and F. Leoni, Ergodic pairs for singular or degenerate fully nonlinear operators, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 75, 28 pp. doi: 10.1051/cocv/2018070.

[7]

I. Birindelli, F. Demengel and F. Leoni, Dirichlet Problems for Fully Nonlinear Equations with "Subquadratic" Hamiltonians, in Contemporary Research in Elliptic PDEs and Related Topics (Ed. S. Dipierro), Springer INdAM Series 33 2019,107–127. doi: 10.1007/978-3-030-18921-1.

[8]

I. Birindelli, F. Demengel and F. Leoni, $\mathcal{C}^{1, \gamma}$ regularity for singular or degenerate fully nonlinear equations and applications, Nonlinear Differ. Equ. Appl., 26 (2019), Paper No. 40. doi: 10.1007/s00030-019-0586-2.

[9]

I. Capuzzo DolcettaF. Leoni and A. Porretta, Hölder estimates for degenerate elliptic equations with coercive Hamiltonians, Trans. Amer. Math. Soc., 362 (2010), 4511-4536.  doi: 10.1090/S0002-9947-10-04807-5.

[10]

F. Demengel and I. Birindelli, One-dimensional symmetry for solutions of Allen Cahn fully nonlinear equations, in Symmetry for elliptic PDEs, (A. Farina and E. Valdinci Eds.), Amer. Math. Soc., Providence, RI (2010), 1–15. doi: 10.1090/conm/528.

[11]

A. FarinaL. MontoroG. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1-22.  doi: 10.1016/j.anihpc.2013.09.005.

[12]

R. FilippucciP. Pucci and P. Souplet, A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations, Comm. Partial Differential Equations, 45 (2020), 321-349.  doi: 10.1080/03605302.2019.1684941.

[13]

Y. Giga and M. Ohnuma, On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations, Int. J. Pure Appl. Math., 22 (2005), 165-184. 

[14]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of degenerate fully nonlinear elliptic equations, Adv. Math., 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.

[15]

T. Kilpeläinen, H. Shahgholian and X. Zhong, Growth estimates through scaling for quasilinear partial differential equation, Ann. Acad. Sci. Fenn. Math., 32 (2007), no. 2,595–599.

[16]

J.-M. Lasry and P.-L. Lions, Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints. Ⅰ. The model problem, Math. Ann., 283 (1989), 583-630.  doi: 10.1007/BF01442856.

[17]

T. Leonori and A. Porretta, Gradient bounds for elliptic problems singular at the boundary, Arch. Ration. Mech. Anal., 202 (2011), 663-705.  doi: 10.1007/s00205-011-0436-9.

[18]

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

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