Asymptotic behavior of Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface.

Summary We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Γ ε. The oscillations of the interface have small period and amplitude, both of the order of ε, and the interfaces Γ ε tend to a straight line Γ. We study the asymptotic behavior 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 of Γ ε .