Sharp all-times heat-content comparison and concentration on compact metric graphs
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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 L^2 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. We also give a separate convolution formula for an asymmetric three-edge pitchfork. The proofs adapt established graph coarea, symmetrization and resolvent-to-semigroup methods.
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory
- Keywords
- metric graphs, heat content, heat concentration, Faber–Krahn inequality, symmetrization
- Manuscript date
- 2026-10-20
- Reader reuse
- No additional permission granted
Scope
Sharp integrated concentration comparison for nonnegative heat data on the stated compact metric graphs, an all-times heat-content bound for datum one and single-positive-time equality classification, with an explicit deficit representation.
Disclosures
OpenAI Codex assisted with literature comparison, proof checking and manuscript preparation. The author takes responsibility for the mathematics and accompanying files.
Reproduction
Eleven-page analytical manuscript and fourteen-member package with all eight TeX source files, portable fixed-date Tectonic build, reader instructions, canonical CFF/BibTeX and checksums. The supplied PDF reproduces exactly from the package using the documented build. The argument is analytic, with integrated resolvent comparison, continuous-time positive defect and graph topology proofs. Equality at a single positive time concerns initial datum one. Archive edition 2.
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