Global structure and regularity of solutions to the Eikonal equation

TL;DR

This paper investigates the structure and regularity of solutions to the time-dependent Eikonal equation, revealing new phenomena like contact discontinuities due to its weak regularizing effect and analyzing the singularity set for smooth initial data.

Contribution

It provides a detailed analysis of the singularity set and regularity properties of solutions to the Eikonal equation with non-smooth Hamiltonian, highlighting new phenomena.

Findings

Singularity points form a set with countable connected components.

Solutions are regular outside the singularity set.

New phenomena such as contact discontinuities are identified.

Abstract

The time dependent Eikonal equation is a Hamilton-Jacobi equation with Hamiltonian $H(P)=|P|$, which is not strictly convex nor smooth. The regularizing effect of Hamiltonian for the Eikonal equation is much weaker than that of strictly convex Hamiltonians, therefore leading to new phenomena such as the appearance of "contact discontinuity". In this paper, we study the set of singularity points of solutions in the upper half space for $C^1$ or $C^2$ initial data, with emphasis on the countability of connected components of the set. The regularity of solutions in the complement of the set of singularity points is also obtained.