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On the reflector shape design. (English) Zbl 1201.53007

The authors study a reflector system which consists of a light source at the origin \(O\), a reflecting surface \(\Gamma\), and a bounded smooth object \(\Sigma\) to be illuminated. First of all, the authors derive the fact that the equation for the reflector system is a fully nonlinear partial differential equation of Monge-Ampère type, subject to a non-linear second boundary condition. For instance, if the reflector \(\Gamma\) is a radial graph over a domain in the unit sphere given by the radial function \(\rho\) and if \(u = \frac{1}{\rho}\), then the reflecting equation is given by \(\det D^2 u = h(x, u, Du)\) for some function \(h\). The authors prove existence and regularity of a reflector \(\Gamma\) such that the light from \(O\) is reflected off to the object \(\Sigma\) and the density of reflected light on \(\Sigma\) is equal to a given non-negative function.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35J96 Monge-Ampère equations
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