Calculus of Variations and Geometric Measure Theory

M. Bardi - A. Cesaroni - L. Rossi

Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control

created by bardi on 07 Jan 2015
modified by cesaroni on 23 Aug 2016


Published Paper

Inserted: 7 jan 2015
Last Updated: 23 aug 2016

Journal: ESAIM Control Optim. Calc. Var.
Volume: 22
Number: 3
Pages: 842-861
Year: 2016


For a class of Bellman equations in bounded domains we prove that sub- and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with ergodic control with state constraints. In this context, our condition on the behavior of the operator near the boundary ensures some invariance property of the domain for the associated controlled diffusion process.