Mathematical Models and Methods in Applied Sciences Vol. 10, No. 4 (2000) 615–628 c  World Scientific Publishing Company ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO MAXWELL’S EQUATIONS IN BOUNDED DOMAINS WITH APPLICATION TO MAGNETOTELLURICS JUAN E. SANTOS * Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA and CONICET, Observatorio Astron´ omico, Universidad Nacional de La Plata, 1900 La Plata, Argentina DONGWOO SHEEN † Department of Mathematics, Seoul National University, Seoul 151-742, Korea Communicated by F. Brezzi Received 13 January 1998 Revised 10 September 1999 We analyze the solution of the time-harmonic Maxwell equations with vanishing elec- tric permittivity in bounded domains and subject to absorbing boundary conditions. The problem arises naturally in magnetotellurics when considering the propagation of electromagnetic waves within the earth’s interior. Existence and uniqueness are shown under the assumption that the source functions are square integrable. In this case, the electric and magnetic fields belong to H(curl; Ω). If, in addition, the divergences of the source functions are square integrable and the coefficients are Lipschitz-continuous, a stronger regularity result is obtained. A decomposition of the space of square integrable vector functions and a new compact imbedding result are exploited. 1. Introduction The magnetotelluric method is used to infer distribution of the earth’s electric con- ductivity from measurements of natural electric and magnetic fields on the earth’s surface (see Refs. 3, 15, 18 and 21). Applications of the magnetotelluric method in- clude petroleum exploration in regions where the seismic reflection method is very expensive or impossible to perform. The aim of this paper is to derive existence and uniqueness results for a mathematical model arising from magnetotellurics. * E-mail: santos@math.purdue.edu † E-mail: sheen@math.snu.ac.kr 615