Sobolev type inequalities for compact metric graphs

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.


Introduction
For a compactly supported smooth function h on R n the classical Sobolev inequality [30] and Gagliardo-Nirenberg inequality [14,26] state the existence of positive constants C and C such that , n > 2 (2) hold. Inequalities (1) and (2) follow from a more general Gagliardo-Nirenberg-Sobolev (GNS) inequality, where 1 ≤ 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 δ depending on the isoperimetric dimension δ if the number of edges between every subset Z of the vertex set V (G) and its complimentZ, denoted by |E(Z,Z)|, satisfies Here Vol(Z) denotes the sum of the valencies of all vertices in Z. Let h : V (G) → R be an arbitrary function and τ h is the largest value such that For a connected graph G, the discrete analogue of the Sobolev inequalities state the existence of positive constants C 1 = C δ (δ-1) holds. Here η = 2δ δ-2 and u ∼ 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 λ k of the discrete Laplacian on a connected graph G is obtained by using inequality (5) as 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 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, . . . , N .
In the Hilbert space L 2 ( ) := N j=1 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 The extended normal derivative is ∂h( 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][21][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 {λ j : λ j /π 2 / ∈ Z} of the Laplace operator (6) with vertex conditions (7) is related to the set of eigenvalues μ j of the normalized discrete Laplacian as 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)

Main results
Let be a compact and connected metric graph. We say that has an isoperimetric constant C γ depending on the isoperimetric dimension γ if for every subgraph Z of , the number of edges that depart from Z, denoted by |∂Z|, satisfies HereZ denotes the complement of Z. For a non-negative function h on the metric graph , we define and Here t ≥ 0. It is easy to observe that there always exists a non-negative number t h such that Let V denote the set of vertices of . We say that h ∈ C 1 ( ) if h| e j ∈ C 1 (e j ) and h satisfies the standard matching conditions (7) at all the vertices. The following two theorems are our main results: where Z + h (0) = {x ∈ : h(x) > 0}.

Proofs of the main results
In this section we prove our main results. We closely follow the argument of Cheeger [7] and Maz'ya [23,24]. A similar argument was used in the case of discrete graphs [8,9]. We will need the following lemma.
Lemma 3.1 For γ > 1, the following inequalities hold: Proof For simplicity we put p = γ γ -1 and, using the definition of Lebesgue integral, we write As Vol(Z + h (t)) is a monotonically decreasing function of t, we have and hence Using inequality (17) and by putting g(t) = t 0 Vol(Z + h (t h + τ )) 1 p dτ , equation (16) becomes which is the desired inequality (14).
To prove the second inequality, we consider As Now Vol(Zh (t)) is a monotonically increasing function of t and therefore Using inequality (20) and by putting q(t) = This implies that

Proof of Theorem 2.1
Define the set Z h (t) as We know that Vol(Z + h (t)) ≤ Vol(Zh (t)) = Vol(Z + h (t)) for all t > t h and Vol(Zh (t)) ≤ Vol(Z + h (t)) = Vol(Zh (t)) for all t ≤ t h . Therefore, and In order to prove the theorem, we need the co-area formula Equation (23), along with inequalities (21) and (22). implies that which, due to Lemma 3.1, gives

Proof of Theorem 2.3
We apply estimate (12) to h q assuming h ≥ 0, h ∈ C 1 ( ) and q > 1 is an integer, which gives We choose p such that qγ γ -1 = p(q-1) p-1 , that is, With this choice of p, the above inequality becomes where C γ ,q =

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 of a metric graph is defined as 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 = lim γ →∞ C γ , where C γ is the isoperimetric constant of the graph defined in (8). Using this observation, one can rewrite the inequalities obtained in where Z + h (0) = {x ∈ : h(x) > 0}.
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].

Corollary 4.4
Let λ 1 be the lowest non-zero eigenvalue of the Laplace operator H given by (6) with the standard or Kirchhoff vertex conditions (7) defined in L 2 ( ), where is a compact and connected metric graph. Then we have Proof The lowest eigenvalue of the operator H is zero and corresponding eigenfunction is a constant. Let ψ 1 denote the eigenfunction corresponding to the lowest non-zero eigenvalue λ 1 . We can assume ψ 1 to be real-valued. We define + := x ∈ | ψ 1 (x) > 0 and ψ 1 := 1 + ψ 1 .
Clearly, ψ 1 disappears on the boundary of + . By min-max principle, we have Here we used integration by parts and applied inequality (30) to ψ 1 with q = 2 along with the fact that the mean value of the eigenfunction is 0. Note that ( ψ 1 , 2, ) = 0.

Conclusions
Metric graphs are locally one-dimensional objects. We first defined isoperimetric dimension of these graphs which may be greater than one and may not be an integer. We obtained Sobolev type inequalities for such graphs. These inequalities depend on the isoperimetric dimension of the graph. Moreover, we gave versions of these inequalities that involve graph's Cheeger constant. Functional inequalities are important tools for the study of spectral properties of differential operators. We demonstrated this connection by obtaining a previously known lower bound on the first non-zero eigenvalue of the Laplace operator defined on a metric graph.