Solutions Near Singular Points To The Eikonal And Related First Order Non-Linear Partial Differential Equations In Two Independent Variables

Abstract

. A detailed study of solutions to the rst order partial dierential equation H(x; y; z x ; z y ) = 0, with special emphasis on the eikonal equation z 2 x + z 2 y = h(x; y), is made near points where the equation becomes singular in the sense that dH = 0, in which case the method of characteristics does not apply. The main results are that there is a strong lack of uniqueness of solutions near such points and that solutions can be less regular than both the function H and the initial data of the problem, but that this loss of regularity only occurs along a pair of curves through the singular point. The main tools are symplectic geometry and the Sternberg normal form for Hamiltonian vector elds. 1. Introduction The eikonal equation in two independent variables for z = z(x; y) is z 2 x + z 2 y = h(x; y) (1.1) where h is a non-negative smooth function on the plane R 2 , and subscripts denote partial derivatives. Near points (x 0 ; y 0 ) where h 6= 0 all local solutions to th..