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arXiv · 2510.14364

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

Isaac Ohavi
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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.

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Isaac Ohavi. 2025-10-16. Viscosity solutions posed on star-shaped network with Kirchhoff's boundary condition: Well-posedness. https://arxiv.org/abs/2510.14364

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