Classification of nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations

TL;DR

This paper classifies nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations and establishes comparison principles, extending the understanding of boundary behaviors in nonconvex, noncoercive cases.

Contribution

It provides a classification of boundary conditions for nonconvex Hamilton-Jacobi equations and introduces comparison principles for complex boundary scenarios.

Findings

Classification of boundary conditions for nonconvex Hamiltonians

Comparison principle for noncoercive Hamiltonians with flat boundary parts

Extension of flux-limited formulation to nonconvex cases

Abstract

We study Hamilton-Jacobi equations in [0, +$\infty$) of evolution type with nonlinear boundary conditions of Neumann type in the case where the Hamiltonian is non necessarily convex with respect to the gradient variable. In this paper, we give two main results. First, we prove a classification of boundary condition result for a nonconvex, coercive Hamiltonian, in the spirit of the flux-limited formulation for quasi-convex Hamilton-Jacobi equations on networks recently introduced by Imbert and Monneau. Second, we give a comparison principle for a nonconvex and noncoercive Hamiltonian where the boundary condition can have flat parts.