The refractor problem with loss of energy (Q1947400)

For prescribed emitting illumination intensity \(f\) on \(\Omega\) and refracted illumination intensity \(\mu\) on \(\Omega^*\), a refractor \(\mathcal R\) is found such that solves \(\int_{\mathcal{T}_\mathcal{R}(\omega)}f(x) t_\mathcal{R}(x)dx \geq \mu(\omega)\), for every Borel subset \(\omega\) of \(\Omega^*\). Here, \(\Omega\) and \(\Omega^*\) are two domains of the unit sphere of \(\mathbb{R}^n\) such that obey a condition on the angles of incidence and refraction, \(\mathcal{T}_\mathcal{R}\) is the tracing map of \(\mathcal{R}\), and \(t_\mathcal{R}\) denotes the percentage of energy transmitted which is expressed by the Fresnel formulas and the Snell's law. The solvability of \(\mathcal{R}\) is proved first by considering \(\mu\) to be equal to a linear combination of delta measures, and then by approximation when \(\mu\) is a finite Radon measure. Finally, it is shown that the surface solution to the refractor problem satisfies an inequality involving a Monge-Ampère type operator.