Asymptotic Behavior of Solutions to the Helmholtz Equations with Sign Changing Coefficients

This paper is devoted to the study of the behavior of the unique solution u delta is an element of H-0(1)(Omega), as delta -> 0, to the equation div(s(delta)A del u(delta)) + k(2)s(0)Sigma u(delta) = s0 f in Omega, where Omega is a smooth connected bounded open subset of R-d with d = 2 or 3, f is an element of L-2(Omega), k is a non-negative constant, A is a uniformly elliptic matrixvalued function, Sigma is a real function bounded above and below by positive constants, and s(delta) is a complex function whose real part takes the values 1 and 1 and whose imaginary part is positive and converges to 0 as delta goes to 0. This is motivated from a result of Nicorovici, McPhedran, and Milton; another motivation is the concept of complementary media. After introducing the reflecting complementary media, complementary media generated by reflections, we characterize f for which parallel to u delta parallel to(Omega) remains bounded as goes to 0. For such an f, we also show that u(delta) converges weakly in H1(Q) and provide a formula to compute the limit.