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Open AccessArticle

Approximations of Metric Graphs by Thick Graphs and Their Laplacians

Fachbereich 4—Mathematik, Universität Trier, 54286 Trier, Germany
Symmetry 2019, 11(3), 369; https://doi.org/10.3390/sym11030369
Received: 30 January 2019 / Revised: 7 March 2019 / Accepted: 8 March 2019 / Published: 12 March 2019
(This article belongs to the Special Issue Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks)
The main purpose of this article is two-fold: first, to justify the choice of Kirchhoff vertex conditions on a metric graph as they appear naturally as a limit of Neumann Laplacians on a family of open sets shrinking to the metric graph (“thick graphs”) in a self-contained presentation; second, to show that the metric graph example is close to a physically more realistic model where the edges have a thin, but positive thickness. The tool used is a generalization of norm resolvent convergence to the case when the underlying spaces vary. Finally, we give some hints about how to extend these convergence results to some mild non-linear operators. View Full-Text
Keywords: metric graphs; open sets converging to metric graphs; Laplacians; norm convergence of operators; convergence of spectra metric graphs; open sets converging to metric graphs; Laplacians; norm convergence of operators; convergence of spectra
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Post, O. Approximations of Metric Graphs by Thick Graphs and Their Laplacians. Symmetry 2019, 11, 369.

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