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Sobolev type inequalities for compact metric graphs.

Muhammad Usman¹

¹Department of Mathematics, Syed Babar Ali School of Sciences and Engineering, Lahore University of Management Sciences, Lahore, Pakistan.

Journal of Inequalities and Applications

|October 27, 2018

Summary

This summary is machine-generated.

This study derives Sobolev inequalities for metric graphs, recovering the Cheeger inequality for the Laplace operator's first non-zero eigenvalue.

Keywords:

Isoperimetric inequalities Laplace operator Metric graphs Quantum graphs Sobolev inequalities

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Area of Science:

Analysis
Graph Theory
Differential Geometry

Background:

Sobolev inequalities are fundamental in analysis, relating function norms.
Metric graphs provide a framework for studying geometric and analytic properties.
The Laplace operator on graphs is crucial for spectral analysis.

Purpose of the Study:

To derive analogues of Sobolev inequalities specifically for compact and connected metric graphs.
To establish a connection between these new inequalities and existing spectral graph theory results.

Main Methods:

Derivation of generalized Sobolev inequalities tailored for the geometry of metric graphs.
Application of these inequalities to analyze the spectrum of the Laplace operator.

Main Results:

Novel Sobolev inequalities for compact, connected metric graphs are established.
The Cheeger inequality, a lower bound for the first non-zero eigenvalue, is recovered as a direct consequence.

Conclusions:

The derived Sobolev inequalities offer new tools for studying metric graphs.
This work reinforces the link between geometric inequalities and spectral properties on graphs.