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On Hamilton Jacobi equations with time measurable Hamiltonians posed on a 1-dimensional junction

Ariela Briani

Abstract

In this paper, we study evolutive Hamilton Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. We introduce a notion of (flux-limited) viscosity solution for Hamiltonians H=H(t,x,p) that are assumed to be only measurable in time t. The flux limiter, A=A(t), acting at the junction, is not required to be continuous but only in L infinity. In the case of convex Hamiltonians, we prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalisations to the nonconvex case and to more general networks are also discussed.

On Hamilton Jacobi equations with time measurable Hamiltonians posed on a 1-dimensional junction

Abstract

In this paper, we study evolutive Hamilton Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. We introduce a notion of (flux-limited) viscosity solution for Hamiltonians H=H(t,x,p) that are assumed to be only measurable in time t. The flux limiter, A=A(t), acting at the junction, is not required to be continuous but only in L infinity. In the case of convex Hamiltonians, we prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalisations to the nonconvex case and to more general networks are also discussed.
Paper Structure (5 sections, 121 equations)

This paper contains 5 sections, 121 equations.

Table of Contents

  1. Introduction
  2. The definition of Flux limited $t$-measurable solution
  3. On comparison results
  4. An existence result via optimal control problems
  5. On possible generalizations