Homogenization of a transmission problem with Hamilton–Jacobi equations and a two-scale interface. Effective transmission conditions

Abstract

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of ε while the period is of the order of ε2. As ε→0, the interfaces tend to a straight line Γ. We study the asymptotic behavior of the value function as ε→0. We prove that the value function tends to the solution of Hamilton–Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations.