Ricerc@Sapienza

We study a class of weakly coupled systems of Hamilton–Jacobi equations at the critical level. We associate to it a family of scalar discounted equation. Using control-theoretic techniques we construct an algorithm which allows obtaining a critical solution to the system as limit of a monotonic sequence of subsolutions. We moreover get a characterization of isolated points of the Aubry set and establish semiconcavity properties for critical subsolutions.

We study the asymptotic behavior of the solutions to a family of discounted Hamilton-Jacobi equations, posed in RN, when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated with both the corresponding ergodic and discounted equations.