Harmonic fields on the extended projective disc and a problem in optics

TL;DR

This paper investigates harmonic fields on the extended projective disc model of hyperbolic space, establishing existence results for weak solutions of Hodge equations that change type, with applications to optics and wave motion near caustics.

Contribution

It introduces new existence theorems for weakly harmonic 1-fields on the extended projective disc, addressing equations that change type and formulating elliptic-hyperbolic boundary-value problems.

Findings

Existence of weak solutions for harmonic 1-fields with changing type.

Solutions can be obtained via small perturbations of the equations.

Applications to wave motion near caustics in optics.

Abstract

The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for weakly harmonic 1-fields, changing type on the unit circle, is derived under Dirichlet conditions imposed on the non-characteristic portion of the boundary. A similar system arises in the analysis of wave motion near a caustic. A class of elliptic-hyperbolic boundary-value problems is formulated for those equations as well. For both classes of boundary-value problems, an arbitrarily small lower-order perturbation of the equations is shown to yield solutions which are strong in the sense of Friedrichs.