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

摘要

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 be comes 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.