Differential and Integral Equations

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

Emil Cornea, Ralph Howard, and Per-Gunnar Martinsson

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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.

Article information

Source
Differential Integral Equations, Volume 14, Number 12 (2001), 1441-1468.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123005

Mathematical Reviews number (MathSciNet)
MR1859916

Zentralblatt MATH identifier
1161.35336

Subjects
Primary: 35F25: Initial value problems for nonlinear first-order equations
Secondary: 35B38: Critical points 35F20: Nonlinear first-order equations

Citation

Cornea, Emil; Howard, Ralph; Martinsson, Per-Gunnar. Solutions near singular points to the eikonal and related first-order nonlinear partial differential equations in two independent variables. Differential Integral Equations 14 (2001), no. 12, 1441--1468. https://projecteuclid.org/euclid.die/1356123005


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