Existence and uniqueness of the scattering solutions in the exterior of rough domains

TL;DR

This paper proves the existence and uniqueness of scattering solutions for a broad class of rough obstacles without relying on integral equations, using operator theory and limiting absorption principles.

Contribution

It extends the class of obstacles for which scattering solutions are known to exist and be unique, without requiring boundary smoothness for Dirichlet conditions.

Findings

Proves existence and uniqueness for rough obstacles larger than Lipschitz class.

Uses operator representation and limiting absorption principles instead of integral equations.

No boundary smoothness assumptions needed for Dirichlet boundary condition.

Abstract

Existence and uniqueness of the scattering solutions is proved for a class of bounded rough obstacles which is much larger than the class of Lipschitz obstacles. Integral equations method is not used. The approach is based on the representation theorem for selfadjoint operators via closed symmetric quadratic forms and on the limiting absorption principles. For the Dirichlet boundary condition no assumptions on the smoothness of the boundary are needed, for the Neumann boundary condition and the Robin boundary condition some assumptions on the smoothness of the boundary are needed.