Adaptation of domain-based methods to metric graphs for heat content maximization

Determine whether the methods used to prove that balls maximize the heat content for all times on domains with Dirichlet boundary (Burchard–Schmuckenschläger, 2001) can be adapted to compact metric graphs (quantum graphs) with Dirichlet vertices to establish an analogous Faber–Krahn inequality for heat content.

Background

On domains with Dirichlet boundary, it is known that balls maximize the heat content at all times. The authors point out fundamental differences between metric graphs and domains (e.g., structural and geometric properties), which hinder a straightforward adaptation of those techniques.

This uncertainty about method transfer underscores the difficulty of proving an all-times Faber–Krahn inequality for heat content on metric graphs and motivates the development of graph-specific approaches.

References

However, since metric graphs behave fundamentally differ- ent from domains, for instance among the class of all graphs of a given volume, there is no polynomial bound on e-balls, it is not clear how to transfer these methods to metric graphs.

— Faber-Krahn inequality for the heat content on quantum graphs via random walk expansion (2501.09693 - Bifulco et al., 16 Jan 2025) in Section 1. Introduction