Equivalence of solutions of eikonal equation in metric spaces

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