Viscosity solutions posed on star-shaped network with Kirchhoff's boundary condition: Well-posedness
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.
Keywords
Cite
@article{arxiv.2510.14364,
title = {Viscosity solutions posed on star-shaped network with Kirchhoff's boundary condition: Well-posedness},
author = {Isaac Ohavi},
journal= {arXiv preprint arXiv:2510.14364},
year = {2025}
}