Research Article · Preprint Mathematics
Sharp all-times heat-content comparison and concentration on compact metric graphs
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords metric graphs · heat content · heat concentration · Faber–Krahn inequality · symmetrization
Scope of the work
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.
Reproduction notes
The paper has 11 pages. The editable-source archive contains 12 files. Compile main.tex with its seven included section files; the complete analytic argument and references are present.
The manuscript
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Cite this edition
Scott Schubr. (2026). Sharp all-times heat-content comparison and concentration on compact metric graphs (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-all-times-heat-content-comparison-concentration-compact-metric-graphs/v/2
BibTeX
@techreport{luminary_sharp_all_times_heat_content_comparison_concentration_compact_metric_graphs_v2,
title = {Sharp all-times heat-content comparison and concentration on compact metric graphs},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/sharp-all-times-heat-content-comparison-concentration-compact-metric-graphs/v/2}
}Loading discussion…