We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub-solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.
Mots-clés : Fully nonlinear equations, degeneracy, ergodic pairs, explosive solutions
@article{COCV_2019__25__A75_0,
author = {Birindelli, Isabeau and Demengel, Fran\c{c}oise and Leoni, Fabiana},
title = {Ergodic pairs for singular or degenerate fully nonlinear operators},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018070},
zbl = {1437.35370},
mrnumber = {4039137},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2018070/}
}
TY - JOUR AU - Birindelli, Isabeau AU - Demengel, Françoise AU - Leoni, Fabiana TI - Ergodic pairs for singular or degenerate fully nonlinear operators JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2018070/ DO - 10.1051/cocv/2018070 LA - en ID - COCV_2019__25__A75_0 ER -
%0 Journal Article %A Birindelli, Isabeau %A Demengel, Françoise %A Leoni, Fabiana %T Ergodic pairs for singular or degenerate fully nonlinear operators %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2018070/ %R 10.1051/cocv/2018070 %G en %F COCV_2019__25__A75_0
Birindelli, Isabeau; Demengel, Françoise; Leoni, Fabiana. Ergodic pairs for singular or degenerate fully nonlinear operators. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 75. doi : 10.1051/cocv/2018070. http://archive.numdam.org/articles/10.1051/cocv/2018070/
[1] and , Large viscosity solutions for some fully nonlinear equations. Nonlinear Differ. Equ. App. 20 (2013) 1453–1472. | DOI | MR | Zbl
[2] , Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sr. I Math. 305 (1987) 725–728. | MR | Zbl
[3] and , Existence and comparison results for fully non linear degenerate elliptic equations without zeroth order terms. Commun. Part. Diff. Eq. 26 (2001) 2323–2337. | DOI | MR | Zbl
[4] and , Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Ration. Mech. Anal. 133 (1995) 77–101. | DOI | MR | Zbl
[5] and , Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equation. Ann. Scuola Norm. Sup Pisa, Cl Sci 5 (2006) 107–136. | Numdam | MR | Zbl
[6] , and On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations. J. Math. Pures Appl. 94 (2010) 497–519. | DOI | MR | Zbl
[7] and , First eigenvalue and Maximum principle for fully nonlinear singular operators. Adv. Differ. Equ. 11 (2006) 91–119. | MR | Zbl
[8] and , Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators. J. Differ. Equ. 249 (2010) 1089–1110. | DOI | MR | Zbl
[9] and , C1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations. ESAIM: COCV 20 (2014) 1009–1024. | MR | Zbl
[10] , and , Dirichlet Problems for Fully Nonlinear Equations with “Subquadratic” Hamiltonians. Springer Indam Series. Preprint [math.AP] (2018). | arXiv | MR
[11] , and , On the C1,γ regularity for fully non linear singular or degenerate equations, in progress.
[12] , and , Nonlinear eigenvalues and bifurcation problems for Pucci’s operators. Ann. Inst. Henri Poincaré Anal. Non Linéaire 22 (2005) 187–206. | DOI | Numdam | MR | Zbl
[13] , and , Hölder’s estimates for degenerate elliptic equations with coercive Hamiltonian. Trans. Am. Math. Soc. 362 (2010) 4511–4536. | DOI | MR | Zbl
[14] , and , Entire subsolutions of fully nonlinear degenerate elliptic equations. Bull. Inst. Math. Acad. Sin. (New Ser.) 9 (2014) 147–161. | MR | Zbl
[15] , and , On the inequality F(x, D2u) ≥ f(u) + g(u)|Du|q. Math. Ann. 365 (2016) 423–448. | MR | Zbl
[16] , and , User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27 (1992) 1–67. | DOI | MR | Zbl
[17] and , Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Commun. Pure Appl. Anal. 12 (2013) 621–645. | MR | Zbl
[18] , and , Super-linear elliptic equation for fully nonlinear operators without growth restrictions for the data. Proc. Roy. Soc. Edinburgh 53 (2010) 125–141. | DOI | MR | Zbl
[19] , , and , Isoperimetric inequalities for an ergodic stochastic control problem. Calc. Var. 46 (2013) 749–768. | DOI | MR | Zbl
[20] , Viscosity solutions of fully nonlinear equations. Sugaku Expositions 9 (1996) 135–152. | MR | Zbl
[21] and , Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints. Math. Ann. 283 (1989) 583–630. | DOI | MR | Zbl
[22] and , Large solutions and gradient bounds for quasilinear elliptic equations. Commun. Part. Diff. Eq. 41 (2016) 952–998. | DOI | MR | Zbl
[23] , and , Comparison principles for p-Laplace equations with lower order terms. Ann. Mat. Pura Appl. 196 (2017) 877–903. | DOI | MR | Zbl
[24] , The ergodic limit for a viscous Hamilton–Jacobi equation with Dirichlet conditions. Rend. Lincei Mat. Appl. 21 (2010) 59–78. | MR | Zbl
Cité par Sources :
