• Previous Article
    Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case
  • DCDS Home
  • This Issue
  • Next Article
    The relationship between word complexity and computational complexity in subshifts
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  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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[2]

Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046

[3]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[4]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[5]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[6]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[7]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[8]

Anton A. Kutsenko. Isomorphism between one-dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270

[9]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[10]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408

[11]

Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020291

[12]

Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314

[13]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227

[14]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[15]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

[16]

João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321

[17]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

[18]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006

[19]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439

[20]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354

2019 Impact Factor: 1.338

Article outline

[Back to Top]