Effective transmission conditions for Hamilton–Jacobi equations defined on two domains separated by an oscillatory interface

Abstract

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Γε. The oscillations of the interface have small period and amplitude, both of the order of ε, and the interfaces Γε 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 Γε.