A priori estimates for the Helmholtz equation with electromagnetic potentials in exterior domains

TL;DR

This paper establishes a priori estimates for the Helmholtz equation with electromagnetic potentials in exterior domains, leading to results on the limiting absorption principle, absence of embedded eigenvalues, and zero-resonances.

Contribution

It introduces new multiplier techniques to derive a priori estimates for the Helmholtz equation with electromagnetic perturbations in exterior domains.

Findings

Proves a family of a priori estimates using Morawetz multiplier techniques.

Derives the limiting absorption principle from these estimates.

Provides conditions for absence of embedded eigenvalues and zero-resonances.

Abstract

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension $n\geq3$. We prove, by multiplier techniques in the sense of Morawetz, a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to the absence of embedded eigenvalues and zero-resonances, under explicit conditions on the potentials.