On complex-valued solutions to a two-dimensional eikonal equation. II. Existence theorems

摘要

The equation w(x)(2) + w(y)(2) + n(2)(x, y) = 0, which arises in generalizations of geometrical optics, is investigated from a theoretical point of view. Here x and y denote rectangular coordinates in the Euclidean plane, and n is real-valued and strictly positive. A framework is set up that involves a Backlund transformation relating Re(w) and Im(w), second-order partial differential equations in divergence and nondivergence form governing Re(w), a variational integral, and related free boundary problems, boundary value problems, and viscosity solutions. The present paper is a continuation of a preceding one [R. Magnanini and G. Talenti, Contemp. Math. 283, AMS, Providence, RI, 1999, pp. 203-229], where qualitative properties of smooth solutions are offered. Here the existence of the real part of solutions, which need not be smooth, is derived. [References: 34]