Sobolev type inequalities for compact metric graphs

Muhammad Usman

doi:10.1186/s13660-018-1872-y

Abstract

In this paper analogues of Sobolev inequalities for compact and connected metric graphs are derived. As a consequence of these inequalities, a lower bound, commonly known as Cheeger inequality, on the first non-zero eigenvalue of the Laplace operator with standard vertex conditions is recovered.

Highlights

We define them as follows: Let G denote a graph with vertex set V (G)
For a compactly supported smooth function h on Rn the classical Sobolev inequality [30] and Gagliardo–Nirenberg inequality [14, 26] state the existence of positive constants C and C such that n–1 |∇h| dx ≥ C(n) |h| n n–1 dx n, n>1 (1) Rn Rn and n–2 |∇h|2 dx ≥ C(n) 2n n–2 dx, n>2
For a connected graph G, the discrete analogue of the Sobolev inequalities state the existence of positive constants C1

Summary

Introduction

We define them as follows: Let G denote a graph with vertex set V (G). For a connected graph G, the discrete analogue of the Sobolev inequalities state the existence of positive constants C1 The lower bound on the kth eigenvalue λk of the discrete Laplacian on a connected graph G is obtained by using inequality (5) as λk ≥ C3 k Vol(G)

Results

Conclusion