The transmission problem on a three-dimensional wedge

TL;DR

This paper investigates the spectral properties and well-posedness of the transmission problem for Laplace's equation on a 3D wedge, using layer potentials and harmonic analysis of a non-commutative group.

Contribution

It characterizes the spectrum and well-posedness conditions for the problem in two different formulations, revealing the infinite multiplicity nature of the spectrum.

Findings

Identifies complex parameters for well-posedness

Characterizes the spectrum's infinite multiplicity

Differentiates spectral pictures in two formulations

Abstract

We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge.