Viscosity solutions for junctions: well posedness and stability

TL;DR

This paper introduces a new framework for viscosity solutions at junctions in Hamilton-Jacobi equations, establishing well-posedness, stability, and approximation methods, with extensions to more complex problems.

Contribution

It develops a novel notion of state-constraint viscosity solutions for junction problems with non-convex Hamiltonians, and analyzes their well-posedness and stability.

Findings

Viscosity approximations select the state-constraint solution or converge to a unique limit.

Fattening the domain provides an alternative approximation method.

Connections with convex cases and extensions to time-dependent/multi-dimensional problems are discussed.

Abstract

We introduce a notion of state-constraint viscosity solutions for one dimensional \junction"-type problems for Hamilton-Jacobi equations with non convex coercive Hamiltonians and study its well- posedness and stability properties. We show that viscosity approximations either select the state- constraint solution or have a unique limit. We also introduce another type of approximation by fattening the domain. We also make connections with existing results for convex equations and discuss extensions to time dependent and/or multi-dimensional problems.