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

TL;DR

This paper investigates Sobolev multipliers and fractional Gaussian fields on Lipschitz boundaries, applying these concepts to analyze deterministic and random acoustic systems with impedance boundary conditions.

Contribution

It introduces new function spaces for multipliers on Lipschitz boundaries and studies their application to acoustic operators with random and deterministic boundary conditions.

Findings

Established m-dissipativity of acoustic operators with generalized impedance boundary conditions.

Proved Weyl-type asymptotics for Laplace-Beltrami eigenvalues on Lipschitz boundaries.

Developed regularity results for fractional Gaussian fields on Lipschitz boundaries.

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 $\partial G$.