All-times Faber–Krahn inequality for heat content on quantum graphs

Establish whether, for every time t > 0, the Rayleigh–Faber–Krahn inequality for the heat content holds on compact connected metric graphs: specifically, determine whether among all graphs of fixed total length |G| (volume), the heat content Qt(G; VD) is maximized by the path graph P|G| (an interval of length |G|) equipped with a Dirichlet condition at one endpoint and Kirchhoff–Neumann conditions elsewhere, for all t > 0.

Background

The paper proves that path graphs maximize the heat content among compact metric graphs of equal volume in two extremal regimes: for sufficiently small times and for sufficiently large times. The authors use two distinct methods: a probabilistic random-walk expansion via the Feynman–Kac formula for small times, and a spectral-theoretic approach (Mercer’s theorem) for large times.

Despite these results, the general, all-times version of the Faber–Krahn inequality for heat content on metric graphs remains unresolved. The domain case (with Dirichlet boundary) is known: balls maximize the heat content at all times, but metric graphs differ significantly from domains, making a direct transfer of those methods unclear.