Heat kernel estimates and related inequalities on metric graphs

2011/01/15 by Sebastian Haeseler, Haeseler, Sebastian · 3 citations
Mathematics · #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1101.3010

openalex publication_date 2011/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider metric graphs with Kirchhoff boundary conditions. We study the intrinsic metric, volume doubling and a Poincaré inequality. This enables us to prove a parabolic Harnack inequality. The proof involves various techniques from the theory of strongly local Dirichlet forms. Along our way we show Sobolev and Nash type inequalities and related heat kernel estimates.

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• Volume growth and stochastic completeness of graphs
• Extinction and propagation phenomena for semilinear parabolic equations on metric trees
• Threshold phenomena and existence results for NLS ground states on graphs

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