Kettunen, Henrik; Lassas, Matti; Ola, Petri
(2018)
This paper considers transmission problems for the Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials in the model tends to zero, the fields may blow up. When the speed of the blow up is suitable, this is called the anomalous localized resonance (ALR). In this paper we study this phenomenon and formulate a new condition, the weak anomalous resonance (w-AR), where the speed of the blow up of fields may be slower. Using this concept, we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi-static approximation. We give simple geometric conditions under which w-AR or ALR may or may not appear. In particular, we show that in a case of a curved layer of negative material with a strictly convex boundary, neither ALR nor w-AR appears with nonzero frequencies (i.e., in the dynamic range) in dimensions d >= 3. In the case when the boundary of the negative material contains a flat subset, we show that w-AR always happens with some point sources in dimensions d >= 2.