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 (eigenfunction) fractional Gaussian fields. The deterministic part of the paper constructs and studies the spaces of pointwise multipliers on Lipschitz 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 $\zeta$ in impedance boundary conditions $\zeta p = \mathbf{n} \cdot \mathbf{v}$ accompanying the 1st order acoustic system. We study the m-dissipativity of associated acoustic operators and the discreteness of their spectra aiming the main efforts on the weakest possible assumptions on the regularity of $\zeta$. In order to pass from deterministic results to the randomization, we introduce fractional Gaussian fields (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 Lipschitz boundary $\partial G$.