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