The refractor problem with loss of energy
type: Item
instance of: scholarly article
title: The refractor problem with loss of energy
zbMATH Open document ID: 1267.78008
author: Cristian E. Gutiérrez
author: Henok Mawi
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2013-04-22T00:00:00Z
review text: 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.
reviewed by: Luisa Consiglieri
Mathematics Subject Classification ID: 78A40
zbMATH DE Number: 6156225
zbMATH Keywords: refractor
zbMATH Keywords: Fresnel formulas
zbMATH Keywords: Snell's law
zbMATH Keywords: Monge-Ampère equation
MaRDI profile type: Publication
full work available at URL: j.na.2012.11.024
OpenAlex ID: W2000511647
DOI: 10.1016/J.NA.2012.11.024
description: scientific article; zbMATH DE number 6156225