##
**On the design of a reflector antenna. II.**
*(English)*
Zbl 1065.78013

The author investigates a reflector antenna design system consisting of a detector located at the origin \({\mathcal O}\), a reflecting surface \(\Gamma\) which is a radial graph over a domain \(\Omega\) in the north hemisphere \(\{x=(x_1,x_2,x_3) \in S^2: x_3>0 \}\),
\[
\Gamma =\{x \rho(x);\;x \in \Omega\}, \quad \rho>0,
\]
and a target area in the outer space from which the signals are received, where \(S^2=\{x\in {\mathbb R}^3 : | x| =1\}\) is the unit sphere. The target area is identified with a domain \(\Omega^* \subset S^2\) in such a way that a ray from the target area is regarded as a point in \(\Omega^*\). This problem requires to solve a second boundary value problem for an extremely complicated, fully nonlinear partial differential equation of Monge-Ampère type, for which the traditional discretization methods fail to create a satisfactory discretization scheme, except for some special cases, such as the radially symmetric case, which reduces the problem to an ordinary differential equation.

In this study, the author shows that the reflector antenna design problem is indeed an optimal transportation problem, and can be reduced to that of finding a minimizer or a maximizer of a linear functional subject to a linear constraint. Therefore it becomes a linear optimization problem and the solution is a maximizer or minimizer of a linear functional.

Part I, cf. the author, Inverse Probl. 12, No. 3, 351–375 (1996; Zbl 0858.35142).

In this study, the author shows that the reflector antenna design problem is indeed an optimal transportation problem, and can be reduced to that of finding a minimizer or a maximizer of a linear functional subject to a linear constraint. Therefore it becomes a linear optimization problem and the solution is a maximizer or minimizer of a linear functional.

Part I, cf. the author, Inverse Probl. 12, No. 3, 351–375 (1996; Zbl 0858.35142).

Reviewer: Ömer Kavaklioglu (Washington)

### MSC:

78A50 | Antennas, waveguides in optics and electromagnetic theory |

78A05 | Geometric optics |

28A50 | Integration and disintegration of measures |

49Q20 | Variational problems in a geometric measure-theoretic setting |

90C05 | Linear programming |