Matthias Täufer (Hagen)
Abstract: We speak about the heat content at time t on metric graphs. This quantity measures how much mass of a constant initial configuration remains in the graph at time t under the action of the heat semigroup. It has been known that its integral over all times satisfies a Faber-Krahn property: it is maximized by path graphs - the analogon of balls.
We prove that surprisingly, the heat content radically contradicts the Faber-Krahn property at some times, but not for others: Intervals are no longer minimizers for small times, but are asymptotically optimal for large times. Consequently, when designing an optimal topology for a battery-like network under leakage, one must take the timescale into account.
Our proofs rely on probabilistic arguments: the Feynman-Kac formula and results from the theory of discrete random walks. This is joint work with Delio Mugnolo and Patrizio Bifulco (both Hagen).
