Homogenization of Rough Boundaries and Interfaces

Homogenization is used to analyze a partial differential equation in a domain with a very rough boundary or interface, following the procedure of Kohler, Papanicolaou, and Varadhan [Boundary and interface problems in regions with very rough boundaries, in Multiple Scattering and Waves in Random Media, P. Chow, W. Kohler, and G. Papanicolaou, eds., North-Holland, Amsterdam, 1981, pp. 165--197] and Brizzi and Chalot [Homogénéisation de frontiére, Ph.D. thesis, Department of Mathematics, Université de Nice, Nice, France, 1978]. It is shown that such a boundary or interface can be replaced by an equivalent layer within which a modified differential equation holds. The coefficients in this new equation are certain "effective parameters" such as the effective conductivity, the effective dielectric constant, the effective refractive index, etc. These coefficients are determined by the solutions of certain special problems which involve the detailed shape of the boundary or interface. This analysis is applied to a second-order elliptic equation, and a similar analysis is applied to Maxwell's equations and to the equations of the linear theory of elasticity.