Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

TL;DR

This paper investigates how convexity properties are preserved over time for Hamilton-Jacobi equations in geodesic spaces, showing that under certain conditions, solutions maintain initial convexity in various complex spaces.

Contribution

It establishes convexity preservation for Hamilton-Jacobi equations in Busemann spaces and extends results to more general geodesic spaces including lattice graphs.

Findings

Convexity is preserved in Busemann spaces for solutions of Hamilton-Jacobi equations.

Two different approaches are provided to prove convexity preservation.

Results are generalized to include spaces like lattice graphs.

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.