Stability for the Helmholtz equation in deterministic and random periodic structures

TL;DR

This paper establishes explicit stability estimates for the Helmholtz equation in both deterministic and random periodic structures, using variational methods, Fourier analysis, and stochastic regularity, under non-resonance conditions.

Contribution

It provides the first explicit stability estimates for Helmholtz equations in random periodic structures, extending deterministic results through a novel variable transform and stochastic analysis.

Findings

Stability estimates are explicit with respect to the wavenumber.

The variational approach applies to both deterministic and stochastic cases.

The method excludes resonances to ensure stability.

Abstract

Stability results for the Helmholtz equations in both deterministic and random periodic structures are proved in this paper. Under the assumption of excluding resonances, by a variational method and Fourier analysis in the energy space, the stability estimate for the Helmholtz equation in a deterministic periodic structure is established. For the stochastic case, by introducing a variable transform, the variational formulation of the scattering problem in a random domain is reduced to that in a definite domain with random medium. Combining the stability result for the deteministic case with regularity and stochastic regularity of the scattering surface, Pettis measurability theorem and Bochner's Theorem further yield the stability result for the scattering problem by random periodic structures. Both stability estimates are explicit with respect to the wavenumber.