A note on Kazdan-Warner equation on networks

TL;DR

This paper extends the Kazdan-Warner equation theory to network structures, analyzing differential equations on edges with Kirchhoff conditions at vertices, including the critical case.

Contribution

It generalizes the Kazdan-Warner equation to networks, establishing existence and properties of solutions with Kirchhoff boundary conditions.

Findings

Theory extends to network settings

Analysis of critical case solutions

Kirchhoff conditions are key to the extension

Abstract

We investigate the Kazdan-Warner equation on a network. In this case, the differential equation is defined on each edge, while appropriate transition conditions of Kirchhoff type are prescribed at the vertices. We show that the Kazdan-Warner theory extends to the present setting and we study also the critical case.