Uniform Resolvent Estimates for Subwavelength Resonators: The Minnaert Bubble Case

Abstract

Subwavelength resonators are small scaled objects that exhibit contrasting medium properties (eigher in intensity or sign) while compared to the ones of a uniform background. Such contrasts allow them to resonate at specific frequencies. There are two ways to mathematically define these resonances. First, as the frequencies for which the related system of integral equations is not injective. Second, as the frequencies for which the related resolvent operator of the natural Hamiltonian, given by the wave-operator, has a pole. In this work, we consider, as the subwavelength resonator, the Minneart bubble. We show that these two mentioned definitions are equivalent. Most importantly, 1. we derive the related resolvent estimates which are uniform in terms of the size/contrast of the resonators. As a by product, we show that the resolvent operators have no scattering resonances in the upper half complex plane while they exhibit two scattering resonances in the lower half plane which converge to the real axis, as the size of the bubble tends to zero. As these resonances are poles of the natural Hamiltonian, given by the wave-operator, and have the Minnaert frequency as their dominating real part, this justifies calling them Minnaert resonances. 2. we derive the asymptotic estimates of the generated scattered fields which are uniform in terms of the incident frequency and which are valid everywhere in space (i.e. inside or outside the bubble). The dominating parts, for both the resolvent operator and the scattered fields, are given by the ones of the point-scatterer supported at the location of the bubble. In particular, these dominant parts are non trivial (not the same as those of the background medium) if and only if the used incident frequency identifies with the Minnaert one.