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 first order partial differential equation H (x,y,z(x), z(y)) = 0, with special emphasis on the eikonal equation z2/x + z2/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 the 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 21, 2000
- Accession Number
- ADA640692
Entities
People
- Emil Cornea
- Per-gunnar Martinsson
- Ralph Howard
Organizations
- University of South Carolina