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 procedu re of Kohler, Papanicolaou, and Varadhan [Boundary and interface probl ems 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 Chalet [Ho mogeneisation de frontiere, Ph.D. thesis, Department of Mathematics, U niversite de Nice, Nice, France, 1978]. It is shown that such a bounda ry or interface can be replaced by an equivalent layer within which a modified differential equation holds. The coefficients in this new equ ation are certain ''effective parameters'' such as the effective condu ctivity, the effective dielectric constant, the effective refractive i ndex, etc. These coefficients are determined by the solutions of certa in special problems which involve the detailed shape of the boundary o r interface. This analysis is applied to a second-order elliptic equat ion, and a similar analysis is applied to Maxwell's equations and to t he equations of the linear theory of elasticity.