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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).