Calculus of Variations and Geometric Measure Theory

D. Castorina - A. Cesaroni - L. Rossi

On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary

created by cesaroni on 13 Oct 2015
modified by castorina on 31 Oct 2018


Published Paper

Inserted: 13 oct 2015
Last Updated: 31 oct 2018

Journal: Comm. Pure App. Anal.
Volume: 15
Number: 4
Pages: 1251-1263
Year: 2016
Doi: 10.3934/cpaa.2016.15.1251

ArXiv: 1509.00177 PDF


We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in \cite{bcr}. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive problem converge uniformly, in the reference frame moving with constant velocity $c$, to a unique steady state solving a suitable ergodic problem.