# A priori bounds for elastic scattering by deterministic and random unbounded rough surfaces

## TL;DR

This paper establishes explicit frequency-dependent bounds for elastic wave scattering by both deterministic and random unbounded rough surfaces, ensuring well-posedness and providing foundational estimates for such scattering problems.

## Contribution

It introduces new a priori bounds for elastic scattering on unbounded rough surfaces, extending deterministic results to the random case with rigorous mathematical proofs.

## Key findings

- Derived explicit frequency-dependent bounds for deterministic rough surface scattering.
- Proved well-posedness of the scattering problem with random rough surfaces.
- Extended bounds to the random case using measure and integrability theorems.

## Abstract

This paper investigates the elastic scattering by unbounded deterministic and random rough surfaces, which both are assumed to be graphs of Lipschitz continuous functions. For the deterministic case, an a priori bound explicitly dependent on frequencies is derived by the variational approach. For the scattering by random rough surfaces with a random source, well-posedness of the corresponding variation problem is proved. Moreover, a similar bound with explicit dependence on frequencies for the random case is also established based upon the deterministic result, Pettis measurability theorem and Bochner's integrability Theorem.