Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case

TL;DR

This paper develops a multidimensional vertex test function for quasi-convex Hamilton-Jacobi equations on junctions, enabling viscosity solution analysis and comparison principles in complex multi-dimensional network structures.

Contribution

It constructs a new regular vertex test function G(x, y) for multi-dimensional junctions, extending previous network results to higher dimensions.

Findings

Constructed a regular vertex test function G(x, y) for multi-dimensional junctions.

Established compatibility conditions for gradients to replace quadratic penalization.

Extended the comparison principle to multi-dimensional quasi-convex Hamilton-Jacobi equations.

Abstract

A multi-dimensional junction is obtained by identifying the boundaries of a finite number of copies of an Euclidian half-space. The main contribution of this article is the construction of a multidimensional vertex test function G(x, y). First, such a function has to be sufficiently regular to be used as a test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi equations posed on a multi-dimensional junction. Second, its gradients have to satisfy appropriate compatibility conditions in order to replace the usual quadratic pe-nalization function |x -- y| 2 in the proof of strong uniqueness (comparison principle) by the celebrated doubling variable technique. This result extends a construction the authors previously achieved in the network setting. In the multi-dimensional setting, the construction is less explicit and more delicate. Mathematical Subject Classification: 35F21, 49L25, 35B51.