Solutions near singular points to the eikonal and related first-order nonlinear partial differential equations in two independent variables

Abstract

A detailed study of solutions to the first-order partial differential equation $H(x,y,z_x,z_y)=0$, with special emphasis on the eikonal equation $z_x^2+z_y^2=h(x,y)$, 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 fields.