Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

Abstract

We study the convexity preserving property for a class of time-dependent Hamilton-Jacobi equations in a complete geodesic space. Assuming that the Hamiltonian is nondecreasing, we show that in a Busemann space the unique metric viscosity solution preserves the geodesic convexity of the initial value at any time. We provide two approaches and also discuss several generalizations for more general geodesic spaces including the lattice graph.

Equivalence of solutions of eikonal equation in metric spaces

“…In fact, we have (4.15) at each x 0 ∈ ∂Ω \ Σ g , which corresponds to the viscosity inequality (4.17). Such topological change was also observed in [42,Remark 5.10] for evolutionary Hamilton-Jacobi equations in metric spaces.…”

“…This was extended to the class of potentially nonconvex Hamiltonians H by Gangbo and Święch [20,21], who proposed a generalized notion of viscosity solutions via appropriate test classes and proved uniqueness and existence of the solutions to more general Hamilton-Jacobi equations in length spaces. Stability and convexity of such solutions are studied respectively in [35] and in [31]. Since this definition of solutions is based on the local slope, we shall call them slope-based solutions (or s-solutions for short) below.…”

Equivalence of solutions of eikonal equation in metric spaces

“…This was extended to the class of potentially nonconvex Hamiltonians H by Gangbo and Święch [20,21], who proposed a generalized notion of viscosity solutions via appropriate test classes and proved uniqueness and existence of the solutions to more general Hamilton-Jacobi equations in length spaces. Stability and convexity of such solutions are studied respectively in [34] and in [30]. Since this definition of solutions is based on the local slope, we shall call them slope-based solutions (or s-solutions for short) below.…”