Sobolev multipliers and fractional Gaussian fields on Lipschitz boundaries with applications to deterministic and random acoustic systems

Abstract: Motivated by Applied Physics and Photonics studies of random resonators, we study in the stochastic part of this paper random acoustic operators in non-smooth bounded domains $G \subset \mathbb{R}^d$ and introduce m-dissipative impedance boundary conditions containing fractional Gaussian fields (FGFs). The deterministic part of the paper constructs and studies the spaces of pointwise multipliers on Lipschitz continuous boundaries $\partial G$, as well as the spaces of Sobolev (distribution-type) multipliers on boundaries $\partial G$ of better regularity. These multipliers are used as generalized impedance coefficients $ζ(x)$, $x \in \partial G$, in impedance boundary conditions accompanying the first order acoustic system. The main efforts are aimed on the m-dissipativity of associated acoustic operators and the discreteness of the related spectra under weakest possible assumptions on the regularity of $ζ$. In order to connect the deterministic results with the randomization, we introduce FGFs on Lipschitz boundaries $\partial G$ and study their regularity. To this end, we prove that a rough Weyl-type asymptotics takes place for the Laplace-Beltrami eigenvalues on arbitrary compact boundary $\partial G$ of $C^{0,1}$-regularity.