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

Illya M. Karabash

TL;DR

The article develops a rigorous framework for both deterministic and random acoustic operators on Lipschitz boundaries by leveraging Sobolev and pointwise multiplier theory to define generalized impedance coefficients. It establishes deep connections between boundary multipliers, m-dissipativity, and spectral discreteness, and introduces fractional Gaussian fields on $\partial G$ to randomize impedance while preserving well-posedness. A rough Weyl-type law for Laplace-Beltrami eigenvalues on $\partial G$ enables controlled randomization, and Friedrichs-type extensions provide explicit nonnegative boundary realizations. The results yield a robust model for deterministic and random resonators with energy leakage and offer explicit conditions (including $L^q$-coefficient criteria) under which the spectrum remains discrete, informing applications in acoustic and photonic 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.

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

TL;DR

The article develops a rigorous framework for both deterministic and random acoustic operators on Lipschitz boundaries by leveraging Sobolev and pointwise multiplier theory to define generalized impedance coefficients. It establishes deep connections between boundary multipliers, m-dissipativity, and spectral discreteness, and introduces fractional Gaussian fields on to randomize impedance while preserving well-posedness. A rough Weyl-type law for Laplace-Beltrami eigenvalues on enables controlled randomization, and Friedrichs-type extensions provide explicit nonnegative boundary realizations. The results yield a robust model for deterministic and random resonators with energy leakage and offer explicit conditions (including -coefficient criteria) under which the spectrum remains discrete, informing applications in acoustic and photonic 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 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 , as well as the spaces of Sobolev (distribution-type) multipliers on boundaries of better regularity. These multipliers are used as generalized impedance coefficients , , 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 . In order to connect the deterministic results with the randomization, we introduce FGFs on Lipschitz boundaries 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 of -regularity.
Paper Structure (32 sections, 35 theorems, 89 equations)

This paper contains 32 sections, 35 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Acoustic operators in bounded Lipschitz domains
  3. Acoustic systems, trace maps, and functional spaces
  4. M-dissipativity and generalized impedance conditions
  5. Laplace-Beltrami operators and m-dissipative GIBCs.
  6. Pointwise multipliers on ${\partial G}$ and m-dissipativity
  7. Pointwise multipliers on Lipschitz boundaries
  8. M-dissipativity and restricted multiplication operators
  9. A bit of operator theory for restricted multipliers
  10. Proofs of Theorems \ref{['t:Inter']} and \ref{['t:Macc']}
  11. Distributions as multipliers on ${\partial G}$ and m-dissipativity
  12. Sobolev multipliers on $C^{k-1,1}$-boundaries ${\partial G}$
  13. Generalized functions as impedance coefficient
  14. Multiplicative positivity and proofs of Theorems \ref{['t:InterD']} and \ref{['t:Hacc']}.
  15. Friedrichs-type extensions of nonnegative measures
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $d=2$. Let us fix arbitrary constants $s>0$ and $c\in \mathbb{R}$. We define the (generalized) random impedance coefficient $\zeta$ by Then $\zeta$ is a random vector in the space $\mathbb{M}^{\mathfrak{S}_\infty} (H^{1/2} ({\partial G}) { \mathrel{ \mkern2mu \clipbox{{.3} 0 0 0}{$$} }} H^{-1/2} ({\partial G}))$, and the randomized acoustic operator $\mathcal{A}:\omega \mapsto \mathcal{A}

Theorems & Definitions (82)

  • Theorem 1.1
  • Definition 2.1: GIBCs, see HJN05PT24K26K25
  • Definition 2.2
  • Theorem 2.1: cf. EK22K25
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Definition 3.2: restricted multiplication operators
  • Theorem 3.3
  • Theorem 3.4
  • ...and 72 more