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

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

2004-12-20

arxiv.org/abs/math-ph/0412072v4

Journal of Mathematical Physics 46, 113501 (2005), 21 pp

Physics

Mathematical Physics

30 pages; Section 3.3 has been revised

Scientific paper

10.1063/1.2098529

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.

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