Random acoustic boundary conditions and Weyl's law for Laplace-Beltrami operators on non-smooth boundaries

Abstract

Motivated by engineering and photonics research on resonators in random or uncertain environments, we study rigorous randomizations of boundary conditions for wave equations of the acoustic-type in Lipschitz domains $\mathcal{O}$. First, a parametrization of essentially all m-dissipative boundary condition by contraction operators in the boundary $L^2$-space is constructed with the use of m-boundary tuples (boundary value spaces). We consider randomizations of these contraction operators that lead to acoustic operators random in the resolvent sense. To this end, the use of Neumann-to-Dirichlet maps and Krein-type resolvent formulae is crucial. We give a description of random m-dissipative boundary conditions that produce acoustic operators with almost surely (a.s.) compact resolvents, and so, also with a.s. discrete spectra. For each particular applied model, one can choose a specific boundary condition from the constructed class either by means of optimization, or on the base of empirical observations. A mathematically convenient randomization is constructed in terms of eigenfunctions of the Laplace-Beltrami operator $\Delta^{\partial \mathcal{O}}$ on the boundary $\mathcal{O}$ of the domain. We show that for this randomization the compactness of the resolvent is connected with the Weyl-type asymptotics for the eigenvalues of $\Delta^{\partial \mathcal{O}}$.