Geometric two-scale convergence on forms and its applications to Maxwell's equations

Abstract

We develop the geometric two-scale convergence on forms in order to describe the homogenization of partial differential equations with random variables on non-flat domain. We prove the compactness theorem and some two-scale behaviours for differential forms. For its applications, we investigate the limiting equations of the n-dimensional Maxwell equations with random coefficients, with given initial and boundary conditions, where are symmetric positive-definite matrices for x ∈ M, and M is an n-dimensional compact oriented Riemannian manifold with smooth boundary. The limiting system of n-dimensional Maxwell equations turns out to be degenerate and it is proven to be well-posed. The homogenized coefficients affected by the geometry of the domain are presented, and compared with the homogenized coefficient of the second order elliptic equation. We present the convergence theorem in order to explain the convergence of the solutions of Maxwell system as a parabolic partial differential equation.