M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces
Illya M. Karabash
TL;DR
This work characterizes when an acoustic operator with generalized impedance boundary conditions in a bounded Lipschitz domain has a compact resolvent, tying this property to the impedance operator’s compactness from $H^{1/2}(\partial\Omega)$ to $H^{-1/2}(\partial\Omega)$. The authors develop an abstract operator framework based on m-boundary tuples and boundary triples, proving that m-dissipativity together with compact resolvent is equivalent to a compact impedance operator, which in turn implies a purely discrete spectrum. The paper further connects second- and first-order acoustic formulations via unitary reductions and provides concrete results for impedance BCs with bounded and singular impedance coefficients, including $\zeta$ in $L^q(\partial\Omega)$ and multipliers in $M^s(\partial\Omega)$. These results advance the spectral theory of lossy resonators and open cavities, with implications for damping modeling and energy leakage in wave equations. The methods, rooted in boundary tuple theory, are adaptable to other wave equations and support rigorous analysis of absorbing boundary conditions and their discretization.
Abstract
The paper studies properties of acoustic operators in bounded Lipschitz domains $Ω$ with m-dissipative generalized impedance boundary conditions. We prove that such acoustic operators have a compact resolvent if and only if the impedance operator from the trace space $H^{1/2} (\partial Ω)$ to the other trace space $H^{-1/2} (\partial Ω)$ is compact. This result is applied to the question of the discreteness of the spectrum and to the particular cases of damping and impedance boundary conditions. The method of the paper is based on abstract results written in terms of boundary tuples and is applicable to other types of wave equations.