Preprint Mathematics
Sharp all-times heat-content comparison and concentration on compact metric graphs
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Overview
Abstract
A sharp integrated concentration comparison is proved for heat flow on finite compact connected metric graphs with a nonempty Dirichlet vertex set and standard Kirchhoff conditions elsewhere. The comparison with the mixed interval of equal total length applies to every nonnegative square-integrable initial datum. For initial datum one, it yields an all-times heat-content bound and characterizes equality at a single positive time. An exact positive-kernel deficit representation and a separate convolution formula for an asymmetric three-edge pitchfork accompany the proofs.
Keywords metric graphs · heat content · heat concentration · Faber–Krahn inequality · symmetrization
Scope of the work
Finite compact connected unweighted metric graphs with positive edge lengths, a nonempty Dirichlet vertex set, and standard Kirchhoff conditions at remaining vertices. Equality is classified for initial datum one.
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Scott Schubr. (2026). Sharp all-times heat-content comparison and concentration on compact metric graphs (Version 1). Luminary Research Archive. https://luminaryoddities.com/researcharchive/sharp-all-times-heat-content-comparison-concentration-compact-metric-graphs/v/1
BibTeX
@techreport{luminary_sharp_all_times_heat_content_comparison_concentration_compact_metric_graphs_v1,
title = {Sharp all-times heat-content comparison and concentration on compact metric graphs},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Preprint},
note = {Version 1},
url = {https://luminaryoddities.com/researcharchive/sharp-all-times-heat-content-comparison-concentration-compact-metric-graphs/v/1}
}Reader correspondence
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