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Applicable Analysis
An International Journal
Volume 85, 2006 - Issue 1-3: In Memory of Carlo Pucci. Guest Editors: Rolando Magnanini and Giorgio Talenti
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Original Articles

On complex-valued solutions to a 2D Eikonal equation. Part three: analysis of a Bäcklund transformation

Rolando Magnanini Università di Firenze, Dipartimento di Matematica U. Dini, viale Morgagni 67/A, 50134 Firenze, ItalyCorrespondence[email protected]
&
Giorgio Talenti Università di Firenze, Dipartimento di Matematica U. Dini, viale Morgagni 67/A, 50134 Firenze, Italy; Consiglio Nazionale delle Ricerche, Istituto per le Applicazioni, del Calcolo – Sezione di Firenze, via Madonna del Piano, 50019 Sesto Fiorentino (FI), Italy
Pages 249-276 | Received 26 Jul 2004, Accepted 11 Oct 2004, Published online: 25 Jan 2007
 
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Abstract

The following equation and related partial differential equations and systems, which arise in a generalization of geometrical optics, are investigated from a theoretical point of view. Here x and y denote rectangular coordinates in the Euclidean plane, n is real-valued, strictly positive and smooth enough. Qualitative properties of smooth solutions were derived in Magnanini and Talenti (1999, On complex-valued solutions to a 2D Eikonal equation. Part One: qualitative properties. Contemporary Mathematics, 283, 203–229). Partial differential equations governing Re(w) were treated in Magnanini and Talenti (2002, On Complex-Valued Solutions to a 2D Eikonal Equation. Part Two: Existence Theorems. SIAM Journal on Mathematical Analysis, 34, 805–835). Here we put to use viscosity and variational methods, and a Bäcklund transformation relating Re(w) and Im (w).

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