Uniqueness in reflector mappings and the Monge-Ampère equation

1981 ◽  
Vol 378 (1775) ◽  
pp. 529-538 ◽  
Keyword(s):  
Angular Distribution ◽  
Boundary Value Problem ◽  
Point Source ◽  
Uniqueness Theorem ◽  
Nonlinear Boundary ◽  
Far Field ◽  
Convexity Condition ◽  
Practical Application ◽  
Reflector Surface ◽  
Monge Ampere

A uniqueness theorem is presented for reflector mappings in the complex plane. These arise in geometrical optics, in the synthesis of a reflector surface to produce a ray beam with specified angular distribution of energy when illuminated by a non-isotropic point-source. The mappings may be represented as solutions of a nonlinear boundary-value problem involving the Monge-Ampère equation. When the equation is elliptic there are at most two solutions, provided that either the incident or far-field ray cone satisfies a convexity condition, which is always the case if one cone is circular. The result has practical application to the design of single and offset dual reflectors used for radar and communications purposes.

scholarly journals The field from an SH-point source in a continuously layered inhomogeneous medium: I. The field in a layer of finite depth

1966 ◽  
Vol 56 (3) ◽  
pp. 715-724
Author(s):  
N. J. Vlaar
Keyword(s):  
Boundary Value Problem ◽  
Point Source ◽  
Finite Number ◽  
Finite Depth ◽  
Far Field ◽  
Source Depth ◽  
Fourier Synthesis ◽  
Periodic Disturbance ◽  
Sturm Liouville ◽  
Heterogeneous Layer

abstract Expressions are derived for the field from an SH point source in a stratified heterogeneous layer of finite depth. It is found, that for a periodic disturbance, the contribution to the far field is mainly due to at most a finite number of unattenuated normal Love modes. The transient response of the medium is obtained by a Fourier synthesis. The final expressions are of a simple form, involving the eigenfunctions of a Sturm-Liouville boundary value problem. The excitation of a certain mode as a function of frequency and source depth is formulated in a concise form.

Singular perturbation of a nonlinear problem with multiple solutions

1985 ◽  
Vol 100 (3-4) ◽  
pp. 327-341
Author(s):  
Anne-Marie Lefevere
Keyword(s):  
Free Boundary ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Critical Value ◽  
Free Boundary Value Problem ◽  
Singular Limits ◽  
Natural Extensions ◽  
Barrier Parameter

SynopsisA nonlinear boundary value problem (P) having positive parameters L and a is considered. We associate with it a family of perturbed problems () affected by the presence of a barrier parameter γ related to L and a. There is a critical value L*(a) of the parameter L such that for L >L*(a), (P) has no regular solution. Then some natural extensions of (P), solutions of a free boundary value problem, arise as singular limits of ().

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