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Near radially symmetric solutions of an inverse problem in geometric optics. (English) Zbl 0658.35094
The following problem is studied: Light rays emanating from a point source reach a reflecting surface through a given aperture. The rays are reflected at the surface according to the laws of geometrical optics. The problem is to find the reflecting surface from the knowledge of the domain on \(S^ 2\) covered by the directional vectors of the reflected light rays, and from the knowledge of the light intensity distribution of the reflected field. This problem amounts to the solution of a nonlinear partial differential equation of Monge-Ampère type. In a previous publication [Lect. Notes Math. 1285, 361-374 (1987; Zbl 0645.35033)] the author and P. Waltman proved that for circular aperture and far field and radially symmetric intensity distribution solutions can be found. In the present paper they extend this result to intensity distributions differing from radially symmetric ones by a small amount in a Hölder norm. For the proof they use the inverse function theorem in Banach spaces.
Reviewer: H.D.Alber

MSC:
35R30 Inverse problems for PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
78A05 Geometric optics
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