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_δ ∈ H₀¹ (Ω), as δ → 0, to the equation div(s_δA∇u_δ) + k²s₀Σu_δ = s₀ f in Ω, where Ω is a smooth connected bounded open subset of ℝ^d with d = 2 or 3, f ∈ L²(Ω), k is a non-negative constant, A is a uniformly elliptic matrixvalued function, Σ is a real function bounded above and below by positive constants, and s_δ is a complex function whose real part takes the values 1 and −1 and whose imaginary part is positive and converges to 0 as δ 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 ||u_δ|| H¹(Ω) remains bounded as δ goes to 0. For such an f, we also show that u_δ converges weakly in H¹(Ω) and provide a formula to compute the limit. [ABSTRACT FROM AUTHOR]