Eikonal equations on ramified spaces

TL;DR

This paper extends the theory of Hamilton-Jacobi equations to higher-dimensional ramified spaces, introducing new geometric structures and establishing a viscosity solutions framework with proven existence and uniqueness.

Contribution

It introduces ramified manifolds and LEP spaces, and develops a viscosity solutions theory for Hamilton-Jacobi equations on these complex structures.

Findings

Established existence of viscosity solutions on LEP spaces

Proved uniqueness of solutions in the ramified setting

Extended previous results to higher-dimensional ramified spaces

Abstract

We generalize the results in [16] to higher dimensional ramified spaces. For this purpose we introduce ramified manifolds and, as special cases, locally elementary polygonal ramified spaces (LEP spaces). On LEP spaces we develop a theory of viscosity solutions for Hamilton-Jacobi equations, providing existence and uniqueness results.