Homogenization of some periodic Hamilton-Jacobi equations with defects

We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.

Automated summary

We study the homogenization problem for a certain class of stationary Hamilton-Jacobi equations. By perturbing a periodic Hamiltonian near the origin, we prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, and a Dirichlet condition at the origin to account for the perturbation. Various comments and extensions are provided.