Sharp all-times heat-content comparison and concentration on compact metric graphs

Abstract

We prove a sharp concentration comparison for the heat equation on a finite compact connected metric graph with a nonempty Dirichlet vertex set and standard Kirchhoff conditions elsewhere. For every nonnegative L² initial datum, the decreasing rearrangement of its heat evolution is dominated in integrated mass by the heat evolution of its rearrangement on the interval of the same total length, with a reflecting endpoint and an absorbing endpoint. For initial datum one this yields the sharp heat-content bound at every positive time. For this datum, equality at even one positive time occurs precisely for the mixed interval, up to non-Dirichlet degree-two subdivisions. An exact positive-kernel representation of the deficit separates the contributions of multiple level crossings and unequal crossing slopes. If every bridge has Dirichlet vertices on both sides, the comparison sharpens to the double-Dirichlet interval of the same length. Equality at one positive time is then equivalent to the known sharper torsion extremality, including balanced two-edge pumpkin chains. We also give a convolution formula for an asymmetric three-edge pitchfork. The proofs adapt established graph coarea, symmetrization and resolvent-to-semigroup methods.