Calculus of Variations and Geometric Measure Theory

N. Shanmugalingam - Q. Liu - X. Zhou

Equivalence of solutions of eikonal equation in metric spaces

created by shanmugal on 27 May 2020



Inserted: 27 may 2020
Last Updated: 27 may 2020

Year: 2020


In this paper we prove the equivalence between some known notions of solutions to the eikonal equation and more general analogs of the Hamilton-Jacobi equations in complete and rectifiably connected metric spaces. The notions considered are that of curve-based viscosity solutions, slope-based viscosity solutions, and Monge solutions. By using the induced intrinsic (path) metric, we reduce the metric space to a length space and show the equivalence of these solutions to the associated Dirichlet boundary problem. Without utilizing the boundary data, we also localize our argument and directly prove the equivalence for the definitions of solutions. Regularity of solutions related to the Euclidean semi-concavity is discussed as well.

Tags: GeMeThNES