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

Summary 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 ε 2. 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.