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Viscosity solutions posed on star-shaped network with Kirchhoff's boundary condition: Well-posedness

Isaac Ohavi

TL;DR

This work studies fully nonlinear Hamilton-Jacobi-Bellman systems on a star-shaped network with a nonlinear Kirchhoff boundary at the vertex and potential degeneracy. The authors design vertex test functions by solving a networked Eikonal system and show that generalized Kirchhoff viscosity solutions coincide with Kirchhoff viscosity solutions, effectively removing the need to evaluate Hamiltonians at the vertex. A comparison principle for discontinuous viscosity solutions and a Perron-based existence/uniqueness theory are established for both first- and second-order formulations. The results provide a robust well-posedness framework for PDEs on networks, with potential implications for stochastic control problems on junctions.

Abstract

The aim of this work is to establish the well-posedness of fully nonlinear partial differential equations (PDE) posed on a star-shaped network, having nonlinear Kirchhoff's boundary condition at the vertex, and possibly degenerate. We obtain a comparison theorem, for discontinuous viscosity solutions, following the recent ideas obtained by Ohavi for second order problems, building test functions at the vertex solutions of Eikonal equations with well-designed coefficients. Another strong result obtained in this contribution is to show that any generalized Kirchhoff's viscosity solution introduced by Lions-Souganidis, is indeed a Kirchhoff's viscosity solution. In other terms, the values of the Hamiltonians are not required at the vertex in the analysis of these types of PDE systems.

Viscosity solutions posed on star-shaped network with Kirchhoff's boundary condition: Well-posedness

TL;DR

This work studies fully nonlinear Hamilton-Jacobi-Bellman systems on a star-shaped network with a nonlinear Kirchhoff boundary at the vertex and potential degeneracy. The authors design vertex test functions by solving a networked Eikonal system and show that generalized Kirchhoff viscosity solutions coincide with Kirchhoff viscosity solutions, effectively removing the need to evaluate Hamiltonians at the vertex. A comparison principle for discontinuous viscosity solutions and a Perron-based existence/uniqueness theory are established for both first- and second-order formulations. The results provide a robust well-posedness framework for PDEs on networks, with potential implications for stochastic control problems on junctions.

Abstract

The aim of this work is to establish the well-posedness of fully nonlinear partial differential equations (PDE) posed on a star-shaped network, having nonlinear Kirchhoff's boundary condition at the vertex, and possibly degenerate. We obtain a comparison theorem, for discontinuous viscosity solutions, following the recent ideas obtained by Ohavi for second order problems, building test functions at the vertex solutions of Eikonal equations with well-designed coefficients. Another strong result obtained in this contribution is to show that any generalized Kirchhoff's viscosity solution introduced by Lions-Souganidis, is indeed a Kirchhoff's viscosity solution. In other terms, the values of the Hamiltonians are not required at the vertex in the analysis of these types of PDE systems.
Paper Structure (4 sections, 6 theorems, 69 equations)

This paper contains 4 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Notations, Definitions and Main results
  3. The First order case
  4. The Second Order Case

Key Result

Theorem 2.3

Assume assumption $(\mathcal{H})$. Let $u \in \mathcal{L}_{sc}^b(\mathcal{N}_R,\mathbb R)$ (resp. $v \in \mathcal{U}_{sc}^b(\mathcal{N}_R,\mathbb R)$ be a viscosity generalized Kirchhoff's super (resp. sub) solution in the sense of Definition def : weak sur/sous solutions of system eq PDE 0. Then $u

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 4 more