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.
Keywords: Fully nonlinear equations, degeneracy, ergodic pairs, explosive solutions
Birindelli, Isabeau 1 ; Demengel, Françoise 1 ; Leoni, Fabiana 1
@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},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018070},
zbl = {1437.35370},
mrnumber = {4039137},
language = {en},
url = {https://www.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 - https://www.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 https://www.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
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