Global structure and regularity of solutions to the Eikonal equation

2017/12/18 by Li, Tian-Hong, Wang, JingHua, Wen, HaiRui
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.06268

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 C¹ or C² 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.

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