1 Introduction
For a compactly supported smooth function
h on
\(\mathbb{R}^{n}\) the classical Sobolev inequality [
30] and Gagliardo–Nirenberg inequality [
14,
26] state the existence of positive constants
C and
C̃ such that
$$ \int_{\mathbb{R}^{n}} \vert \nabla h \vert \,dx \geq C(n) \biggl( \int_{\mathbb{R}^{n}} \vert h \vert ^{\frac{n}{n-1}} \,dx \biggr)^{\frac{n-1}{n}} ,\quad n>1 $$
(1)
and
$$ \biggl( \int_{\mathbb{R}^{n}} \vert \nabla h \vert ^{2} \,dx \biggr)^{\frac{1}{2}}\geq \widetilde{C}(n) \biggl( \int_{\mathbb{R}^{n}} \vert h \vert ^{\frac{2n}{n-2}} \,dx \biggr)^{\frac{n-2}{2n}} ,\quad n>2 $$
(2)
hold. Inequalities (
1) and (
2) follow from a more general Gagliardo–Nirenberg–Sobolev (GNS) inequality,
$$ \biggl( \int_{\mathbb{R}^{n}} \vert \nabla h \vert ^{p}\,dx \biggr)^{\frac{1}{p}}\geq C(n,p) \biggl( \int_{\mathbb{R}^{n}} \vert h \vert ^{\frac{np}{n-p}} \,dx \biggr)^{\frac{n-p}{np}}, $$
(3)
where
\(1\leq p < n\). Inequalities (
1) and (
2) are obtained by choosing
\(p=1\) and
\(p=2\), respectively, in (
3). These inequalities are important in the study of PDEs, heat kernel and spectral estimates (see, e.g., [
10,
11]).
On the other hand, the corresponding versions of these inequalities for discrete graphs were obtained by Chung and Yau (see [
8,
9]). The discrete analogue of (
1) and (
2) depend on a parameter associated with a graph, which we call the isoperimetric dimension of the graph, and a constant, which we call the isoperimetric constant of the graph. We define them as follows:
Let
G denote a graph with vertex set
\(V(G)\). The graph
G has an
isoperimetric constant
\(C_{\delta}\) depending on the
isoperimetric dimension
δ if the number of edges between every subset
Z of the vertex set
\(V(G)\) and its compliment
Z̄, denoted by
\(|E(Z, \bar{Z})|\), satisfies
$$ \bigl\vert E(Z, \bar{Z}) \bigr\vert \geq C_{\delta}\bigl( \operatorname{Vol}(Z)\bigr)^{\frac{{\delta -1}}{{\delta}}}, \quad \mbox{whenever } \operatorname{Vol}(Z)\leq \operatorname{Vol}(\bar{Z}). $$
Here
\(\operatorname{Vol}(Z)\) denotes the sum of the valencies of all vertices in
Z. Let
\(h: V(G)\rightarrow\mathbb{R}\) be an arbitrary function and
\(\tau_{h}\) is the largest value such that
$$\sum_{h(v)< \tau_{h}}d_{v}\leq\sum _{h(u)\geq\tau_{h}} d_{u} $$
with
\(d_{u}\) being the valency of the vertex
u.
For a connected graph
G, the discrete analogue of the Sobolev inequalities state the existence of positive constants
\(C_{1}=\frac {C_{\delta}(\delta-1)}{\delta}\) and
\(C_{2}=\frac{\delta-1}{2\delta C_{\delta}^{-1/2}}\) such that
$$ \sum_{u\sim v} \bigl\vert h(u)-h(v) \bigr\vert \geq C_{1} \biggl(\sum_{v} \bigl\vert h(v)-\tau_{h} \bigr\vert ^{\frac{\delta }{\delta-1}} d_{v} \biggr)^{\frac{\delta-1}{\delta}}, \quad\delta>1 $$
(4)
and
$$ \biggl(\sum_{u\sim v} \bigl\vert h(u)-h(v) \bigr\vert ^{2} \biggr)^{\frac{1}{2}}\geq C_{2} \min_{\tau _{h}} \biggl(\sum _{v} \bigl\vert h(v)-\tau_{h} \bigr\vert ^{\eta}d_{v} \biggr)^{\frac{1}{\eta}}, \quad\delta>2 $$
(5)
holds. Here
\(\eta=\frac{2\delta}{\delta-2}\) and
\(u\sim v\) means
u and
v are neighbors.
Inequalities such as (
4) and (
5) play an important role in the study of heat kernel and spectral estimates of discrete Laplacian on graphs. For instance, the lower bound on the
kth eigenvalue
\(\lambda_{k}\) of the discrete Laplacian on a connected graph
G is obtained by using inequality (
5) as
$$\lambda_{k} \geq C_{3} \biggl(\frac{k}{\operatorname{Vol}(G)} \biggr)^{\frac{2}{\delta}}, $$
where the constant
\(C_{3}\) depends on
δ.
One can consider a generalization of discrete graphs by identifying each edge by an interval of the real line instead of an ordered pair of vertices. In this way, one can define a distance function on such graphs, which can be the smallest path length between two points on the graph. This new object, with a metric defined on it, is called a metric graph. In addition, we can define ordinary differential operators on each edge with certain boundary conditions at the vertices. Boundary conditions or vertex conditions are chosen in a way which makes the overall operator self-adjoint on the graph.
Let Γ denote a metric graph with
N being the number of edges and denote the
jth edge by
\(e_{j}\) which is identified by an interval of the real line, i.e.,
\(e_{j}=(x_{2j-1}, x_{2j})\),
\(j=1,2,\dots, N\). In the Hilbert space
\(L_{2}(\Gamma):=\bigoplus\sum _{j=1}^{N} L_{2}(e_{j})\), the Laplace operator (
6) with standard conditions (
7), which are also known as Kirchhoff, Neumann or free conditions, is a self-adjoint operator
$$\begin{aligned}& H h:= -h'' , \end{aligned}$$
(6)
$$\begin{aligned}& \textstyle\begin{cases} h \mbox{ is continuous at the vertex V},\\ \sum _{x_{j}\in V} \partial h(x_{j})=0. \end{cases}\displaystyle \end{aligned}$$
(7)
The extended normal derivative is
\(\partial h(x_{j}):=\lim_{x\rightarrow x_{j}}\frac{d}{dx}h(x)\) if
\(x_{j}\) is the left-end point and
\(\partial h(x_{j}):=\lim_{x\rightarrow x_{j}}-\frac{d}{dx}h(x)\) if
\(x_{j}\) is the right-end point. For the description of all possible vertex conditions for which (
6) is self-adjoint, see [
16,
18]. The pair of a metric graph and a self-adjoint differential operator is called a
quantum graph. Such objects naturally arise in different areas of mathematics, science and engineering when analyzing various processes in systems which, locally, look like a thin neighborhood of a graph. In the last two decades, quantum graphs have evolved as an interesting branch of mathematical physics and have found many useful applications in physics, particularly in quantum chaos [
15,
19] and mesoscopic physics [
12,
13,
18]. For a detailed study of quantum graphs, we refer the reader to [
3,
20‐
22,
29].
Although a quantum graph is fundamentally a different object from a discrete graph, in some special situations their spectra are related to each other. For example, it is well known (see [
2,
4‐
6,
27,
31]) that if all edges of a compact metric graph are of the same length then the set of eigenvalues
\(\{ \lambda_{j}: \lambda_{j}/\pi^{2} \notin\mathbb{Z}\}\) of the Laplace operator (
6) with vertex conditions (
7) is related to the set of eigenvalues
\(\mu_{j}\) of the normalized discrete Laplacian as
$$1-\cos (\sqrt{\lambda_{j}} )=\mu_{j},\quad \mu_{j} \neq0,2. $$
Therefore, it is a natural question to ask whether it is possible to derive some functional inequalities of the type discussed above for metric graphs and whether one can obtain some estimates on the spectrum of the related quantum graphs? In this paper, we try to answer this question by deriving analogues of (
3) for compact and connected metric graphs. A different version of GNS inequalities for non-compact metric graphs has been used by Adami, Serra and Tilli [
1] to study ground states of certain NLSE.
The plan of the paper is as follows. In the next section we fix the notation and state our main results, Theorems
2.1 and
2.3. Section
3 contains proofs of the main results. Functional inequalities involving the graph’s Cheeger constant is the theme of Sect.
4. As a consequence of Theorem
4.3, we recover the well known Cheeger inequality for quantum graphs, which gives a lower bound on the lowest non-zero eigenvalue of the Laplacian (
6) with standard conditions (
7) on the vertices. For quantum graphs the same lower bound, Corollary
4.4, was first obtained by Nicaise [
25, Theorem 3.2] and also by Post [
28, Theorem 6.1].
4 Sobolev inequalities involving Cheeger constant
Finding an optimal cut of a graph into two or more disjoint subsets is one of the fundamental problems in graph theory. Cheeger cut, which we define below, among several kinds of balanced graph cut, is the most widely used tool to obtain optimal partitioning of a graph. The
Cheeger constant
\(C_{\Gamma}\) of a metric graph Γ is defined as
$$C_{\Gamma}:= \inf\frac{ \vert \partial Z \vert }{\min\{\operatorname{Vol}(Z), \operatorname{Vol}(\bar{Z})\}} $$
where inf is taken over all Lebesgue measurable open subsets
Z of the metric graph. The partition
\((Z, \bar{Z})\) of Γ is called a
Cheeger cut if
$$\frac{ \vert \partial Z \vert }{\min\{\operatorname{Vol}(Z), \operatorname {Vol}(\bar{Z})\}} = C_{\Gamma}. $$
In order to find the Cheeger cut of a metric graph, one would really need to know the value of the associated Cheeger constant or at least an approximation of it. The most well-known technique to approximate the Cheeger constant is via lowest non-zero eigenvalue of the standard Laplace operator. Precisely, if
\(\lambda_{1}\) denotes the lowest non-zero eigenvalue of the Laplace operator (
6) subject to standard conditions (
7) at the vertices, then the Cheeger inequality
$$\lambda_{1} \geq\frac{C_{\Gamma}^{2}}{4} $$
provides an upper bound on the Cheeger constant. For more details on the theory of Cheeger constants for quantum graph, we refer to the review article [
17].
It is easy to see that
\(C_{\Gamma}=\lim_{\gamma\rightarrow\infty} C_{\gamma}\), where
\(C_{\gamma}\) is the isoperimetric constant of the graph defined in (
8). Using this observation, one can rewrite the inequalities obtained in Theorems
2.1 and
2.3 in terms of the graph Cheeger constant. The obtained inequalities in the following theorems could be used as an alternative to estimate the Cheeger constant.
The following result follows from Theorem
4.3 and is commonly known as Cheeger inequality. It was first proved by Nicaise [
25, Theorem 3.2].
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