Steklov eigenvalue problem on subgraphs of integer lattices

Given a bounded smooth domain $\Omega$ in $\mathbb{R}^{n},$ we consider the Steklov (eigenvalue) problem on $\Omega.$ For a smooth function $\varphi$ on $\partial\Omega,$ we denote by $u_{\varphi}$ the harmonic extension of $\varphi$ into $\Omega$ which satisfies

$$\left\{\begin{array}[]{ll}\Delta u_{\varphi}(x)=0,&x\in\Omega,\\ u_{\varphi}|_{\partial\Omega}=\varphi.&\end{array}\right.$$

The Steklov problem on $\Omega$ is the following eigenvalue problem,

$$\Lambda\varphi:=\frac{\partial u_{\varphi}}{\partial n}=\lambda\varphi\quad\mathrm{on}\ \partial\Omega,$$

where $n$ denotes the unit outward normal of $\partial\Omega.$ The operator $\Lambda$ is called the Dirichlet-to-Neumann (DtN in short) operator, which is also known as the voltage-current map in physics, see e.g. Calderón’s problem [Cal06]. As a pseudo-differential operator, the eigenvalues of DtN operator $\Lambda,$ called Steklov eigenvalues, are discrete and can be ordered as

$$0=\sigma_{1}(\Omega)\leq\sigma_{2}(\Omega)\leq\cdots\leq\sigma_{k}(\Omega)\leq\cdots\uparrow\infty.$$

The Steklov problem has been intensively studied for domains in Euclidean spaces and Riemannian manifolds in the literature, see e.g. [Wei54, Esc97, Esc99, Bro01, HPS08, WX09, SU90, CSG11, Has11, GP12, FL14, KKP14, GLS16, FS16, Mat17, FW17, YY17].

For $n=2,$ Weinstock [Wei54] proved that for any planar simply-connected domain $\Omega$ with analytic boundary,

(1.1)

$$\sigma_{2}(\Omega)\leq\frac{2\pi}{\mathrm{L}(\partial\Omega)},$$

where $\mathrm{L}(\partial\Omega)$ denotes the length of the boundary $\partial\Omega.$ The statement was generalized to domains with Lipschitz boundary by Girouard and Polterovich[GP10]. The higher dimensional generalization was proved by Bucur, Ferone, Nitsch and Trombetti [BFNT17]: for any bounded convex $\Omega\subset{\mathbb{R}}^{n},$ $n\geq 3,$

$$\sigma_{2}(\Omega)(\mathrm{area}(\partial\Omega))^{\frac{1}{n-1}}\leq\sigma_{2}(B)(\mathrm{area}(\partial B))^{\frac{1}{n-1}},$$

where $B$ is the unit ball and $\mathrm{area}(\cdot)$ denotes the area of $(\cdot),$ and the equality holds only for balls, see also Fraser and Schoen [FS19] for related results.

The estimate (1.1) was improved by Hersch and Payne [HP68] to the following

(1.2)

$$\frac{1}{\sigma_{2}(\Omega)}+\frac{1}{\sigma_{3}(\Omega)}\geq\frac{\mathrm{L}(\partial\Omega)}{\pi}.$$

Concerning with the Steklov eigenvalues and the volume of the domain, Brock proved the following result.

In this paper, we study the Steklov problem on graphs and prove a discrete analog for Brock’s result. The DtN operator on a subgraph of a graph was introduced by [HHW17, HM17] independently, see e.g. [HHW18, Per19] for more results. A graph $G=(V,E)$ consists of the set of vertices $V$ and the set of edges $E.$ In this paper, we only consider simple, undirected graphs. Two vertices $x,y$ are called neighbours, denoted by $x\sim y$, if there is an edge $e$ connecting $x$ and $y,$ i.e. $e=\{x,y\}\in E.$ Integer lattice graphs are of particular interest which serve as the discrete counterparts of ${\mathbb{R}}^{n}.$ We denote by ${\mathbb{Z}}^{n}$ the set of integer $n$-tuples ${\mathbb{R}}^{n}.$ The $n$-dimensional integer lattice graph, still denoted by ${\mathbb{Z}}^{n},$ is the graph consisting of the set of vertices $V={\mathbb{Z}}^{n}$ and the set of edges

$$E=\left\{\{x,y\}:x,y\in{\mathbb{Z}}^{n},\sum_{i=1}^{n}|x_{i}-y_{i}|=1\right\}.$$

Let $\Omega$ be a finite subset of ${\mathbb{Z}}^{n}.$ We denote by

$$\delta\Omega:=\{x\in{\mathbb{Z}}^{n}\setminus\Omega:\exists\ y\in\Omega,s.t.\ x\sim y\}$$

the vertex boundary of $\Omega.$ Analogous to the continuous setting, one can define the DtN operator on $\Omega,$

$$\Lambda:{\mathbb{R}}^{\delta\Omega}\to{\mathbb{R}}^{\delta\Omega},$$

where ${\mathbb{R}}^{\delta\Omega}$ denotes the set of functions on $\delta\Omega,$ see Section 2 for details. In this paper, we denote by $|\cdot|$ the cardinality of a set. Note that $\Lambda$ can be written as a symmetric matrix whose eigenvalues are ordered as

(1.3)

$$0=\lambda_{1}(\Omega)\leq\lambda_{2}(\Omega)\leq\cdots\leq\lambda_{N}(\Omega),\ \text{where}~N=|\delta\Omega|.$$

The following is our main result, which is a discrete analog of [Bro01, Theorem 3].

The main ingredient of the proof of Brock’s result is a weighted isoperimetric inequality, see Lemma 3.5 below, which depends on the rotational symmetry of the Euclidean spaces. As is well-known in the discrete theory, the symmetrization approaches do not work on ${\mathbb{Z}}^{n}$ due to the lack of rotational symmetry. In this paper, we follow Brock’s idea to bound the Steklov eigenvalues by the geometric quantities of the subset in the lattice ${\mathbb{Z}}^{n},$ and then estimate these discrete quantities by their counterparts in the Euclidean space ${\mathbb{R}}^{n}$, for which we can apply the symmetrization approach. Through this process we obtain the quantitative estimate of discrete Steklov eigenvalues, but lose the sharpness of the constants $C_{1}$ and $C_{2}$ in our result.

As a corollary, we obtain the estimate for the first non-trivial eigenvalue of the DtN operator.

This yields an interesting consequence that for any sequence of finite subsets in ${\mathbb{Z}}^{n},$ $\{\Omega_{i}\}_{i=1}^{\infty},$ satisfying $|\Omega_{i}|\to\infty,$

(1.6)

$$\lambda_{2}(\Omega_{i})\to 0,\quad i\to\infty.$$

This is a rather special property for integer lattices, which does not hold for general infinite graphs. At the end of the paper, we give an example of an infinite graph of bounded degree, Example 3.7, in which there exist subsets $\Omega_{i}$ with $|\Omega_{i}|\to\infty$ such that

$$\lambda_{2}(\Omega_{i})\geq c>0,\quad\forall i.$$

Note that this graph locally behaves similarly as a homogeneous tree of degree $3,$ which is a discrete hyperbolic model.

The paper is organized as follows: In the next section, we recall some facts on DtN operators on graphs. In Section 3, we prove Theorem 1.2.

Let $G=(V,E)$ be a simple, undirected graph. For two subsets $A,B$ of $V$, we define

$$E(A,B):=\{\{x,y\}\in E:~x\in A,~y\in B,\ \mathrm{or\ vice\ versa}\},$$

which is a set of edges connecting a vertex in $A$ to another vertex in $B.$ For a subset $\Omega$ of $V$, we write $\Omega^{c}:=V\setminus\Omega.$ The edge boundary of $\Omega$ is defined as $\partial\Omega:=E(\Omega,\Omega^{c}),$ and we set $\bar{\Omega}:=\Omega\cup\delta\Omega$.

For a finite subset $S$ of $V,$ we denote by ${\mathbb{R}}^{S}$ the set of functions on $S,$ by $\ell^{2}(S)$ the Hilbert space of functions on $S$ equipped with the inner product

$$\langle\varphi,\psi\rangle_{S}=\sum_{x\in S}\varphi(x)\psi(x),\quad\varphi,\psi\in\mathbb{R}^{S}.$$

In the following, we will take $S=\Omega,$ $\delta\Omega,$ or $\bar{\Omega},$ respectively. For $u\in{\mathbb{R}}^{\bar{\Omega}},$ the Dirichlet energy of $u$ is defined as

$$D_{\Omega}(u):=\sum_{\{x,y\}\in E(\Omega,\bar{\Omega})}(u(x)-u(y))^{2}.$$

The polarization of the Dirichlet energy is given by

(2.1)

$$D_{\Omega}(u,v):=\sum_{\{x,y\}\in E(\Omega,\bar{\Omega})}(u(x)-u(y))(v(x)-v(y)),\ \forall u,v\in{\mathbb{R}}^{\bar{\Omega}}.$$

The Laplace operator on $\Omega$ is defined as

	$\displaystyle\Delta:{\mathbb{R}}^{\bar{\Omega}}$	$\displaystyle\rightarrow{\mathbb{R}}^{\Omega}$
	$\displaystyle u$	$\displaystyle\mapsto\Delta u(x):=\sum_{y\in V:y\sim x}(u(y)-u(x)).$

The exterior normal derivative on vertex boundary set $\delta\Omega$ is defined as

	$\displaystyle\frac{\partial}{\partial n}:{\mathbb{R}}^{\bar{\Omega}}$	$\displaystyle\rightarrow{\mathbb{R}}^{\delta\Omega}$
	$\displaystyle u$	$\displaystyle\mapsto\frac{\partial u}{\partial n}(x):=\sum_{y\in\Omega:y\sim x}(u(x)-u(y)),\ x\in\delta\Omega.$

The Dirichlet-to-Neumann (DtN in short) operator is defined as

	$\displaystyle\Lambda:{\mathbb{R}}^{\delta\Omega}$	$\displaystyle\rightarrow{\mathbb{R}}^{\delta\Omega}$
	$\displaystyle\varphi$	$\displaystyle\mapsto\Lambda\varphi:=\frac{\partial u_{\varphi}}{\partial n},$

where $u_{\varphi}$ is the harmonic function on $\Omega$ whose Dirichlet boundary condition on $\delta\Omega$ is given by $\varphi.$ We recall the well-known Green formula.

Let $\lambda$ be an eigenvalue of the DtN operator $\Lambda$ satisfying

$$\frac{\partial u_{\varphi}}{\partial n}=\lambda u_{\varphi}\ \mathrm{on\ \delta\Omega}.$$

Then, by the Green formula (2.2) we get

$$D_{\Omega}(u_{\varphi})=\lambda\langle u_{\varphi},u_{\varphi}\rangle_{\delta\Omega}=\lambda\sum_{z\in\delta\Omega}{\varphi}^{2}(z).$$

The following variational principle is useful, see e.g. [Ban80, p.99].

In this section, we prove the main theorem. Let ${\mathbb{Z}}^{n}$ be the $n$-dimensional integer lattice graph, and

$$e_{i}:=(0,\cdots,\underset{i-1}{0},\underset{i}{1},\underset{i+1}{0},\cdots 0),\quad 1\leq i\leq n,$$

be the standard basis of ${\mathbb{R}}^{n}.$ We denote by

$$Q=\{x=(x_{1},x_{2},\cdots,x_{n})\in{\mathbb{R}}^{n}:|x_{i}|\leq\frac{1}{2},\ \forall 1\leq i\leq n\}$$

the unit cube centered at the origin in ${\mathbb{R}}^{n},$ and by

$$Q_{i}:=Q\cap\{x\in{\mathbb{R}}^{n}:x_{i}=0\},\quad 1\leq i\leq n,$$

the $(n-1)$-dimensional unit cube in $x_{i}$-hyperplane. For any edge $\tau=\{x,y\}$ in ${\mathbb{Z}}^{n},$ there is a unique $e_{i}$ such that $y-x=e_{i}$ or $-e_{i},$ and we define

$$\tau^{\perp}:=\frac{1}{2}(x+y)+Q_{i}.$$

That is, $\tau^{\perp}$ is an $(n-1)$-dimensional unit cube centered at the point $\frac{1}{2}(x+y)$ parallel to $x_{i}$-hyperplane. Note that such a cube $\tau^{\perp}$ is one-to-one corresponding to $\tau.$

For any finite subset $\Omega\subset{\mathbb{Z}}^{n},$ we denote

$$\partial\Omega^{\perp}:=\{\tau^{\perp}:\tau\in\partial\Omega\}.$$

Note that for any $\tau\in\partial\Omega,$ there is a unique end-vertex of the edge $\tau$ belonging to $\delta\Omega.$ We define a mapping

	$\displaystyle P:\partial\Omega^{\perp}$	$\displaystyle\rightarrow\delta\Omega$
	$\displaystyle\tau^{\perp}$	$\displaystyle\mapsto P(\tau^{\perp}),$

where $P(\tau^{\perp})$ is the end-vertex of $\tau$ in $\delta\Omega.$

For any $r>0,$ we denote by

$$Q_{r}(x):=\{y\in{\mathbb{R}}^{n}:\max_{1\leq i\leq n}|y_{i}-x_{i}|\leq\frac{r}{2}\}$$

the cube of side length $r$ centered at $x.$ For any subset $\Omega\subset{\mathbb{Z}}^{n},$ we construct a domain $\hat{\Omega}$ in ${\mathbb{R}}^{n}$ as follows

$$\hat{\Omega}:=\bigcup_{x\in\Omega}Q_{1}(x).$$

Note that the topological boundary of $\hat{\Omega},$ denoted by $\partial(\hat{\Omega}),$ is one-to-one corresponding to $\partial\Omega^{\perp},$ i.e.

(3.1)

$$\partial(\hat{\Omega})=\bigcup_{\tau\in\partial\Omega}\tau^{\perp}.$$

For our purposes, we will use the geometry of $\hat{\Omega}$, which is a domain in ${\mathbb{R}}^{n}$, to estimate that of $\Omega,$ a subset in ${\mathbb{Z}}^{n}.$

We give the definition of “bad” boundary vertices on $\delta\Omega.$

In the following, we give an example to present the role of “bad” boundary vertices in our estimates, which motivates us to distinguish boundary vertices. In the proof of Theorem 1.2, following Brock [Bro01] we reduce the estimate of Steklov eigenvalues to the lower bound estimate of $\sum_{z\in\delta\Omega}\sum_{\omega\in\delta\Omega}|z-\omega|^{2},$ see (3.8) below. The strategy is to find its counterpart in the continuous setting for deriving such a lower bound, which can be estimated by isoperimetric inequalities in ${\mathbb{R}}^{n}.$ A natural choice is the quantity

$$\int_{\partial(\hat{\Omega})}\int_{\partial(\hat{\Omega})}|s-t|^{2}d{\mathcal{H}}^{n-1}(s)d{\mathcal{H}}^{n-1}(t)=2{\mathcal{H}}^{n-1}(\partial(\hat{\Omega}))\int_{\partial(\hat{\Omega})}|s-c|^{2}d{\mathcal{H}}^{n-1}(s),$$

where $c=\frac{1}{{\mathcal{H}}^{n-1}(\partial(\hat{\Omega}))}\int_{\partial(\hat{\Omega})}sd{\mathcal{H}}^{n-1}(s)$ is the barycenter of $\partial(\hat{\Omega})$ and the right hand side is the derivation of the position vector of $\partial(\hat{\Omega})$ with respect to the measure ${\mathcal{H}}^{n-1}.$ For the desired estimate

(3.2)

$$\sum_{z\in\delta\Omega}\sum_{\omega\in\delta\Omega}|z-\omega|^{2}\geq C\int_{\partial(\hat{\Omega})}\int_{\partial(\hat{\Omega})}|s-t|^{2}d{\mathcal{H}}^{n-1}(s)d{\mathcal{H}}^{n-1}(t)+\cdots,$$

we give the following example to indicate the difficulty.

The next example shows that the above estimate is sharp.

For any subset $\Omega\subset{\mathbb{Z}}^{n},$ any vertex in $V\setminus\bar{\Omega}$ is called an exterior vertex of $\Omega.$ For simplicity, we write $\Omega^{e}:=V\setminus\bar{\Omega}$ for the set of exterior vertices of $\Omega.$ This yields the decomposition

(3.3)

$${\mathbb{Z}}^{n}=\Omega\sqcup\delta\Omega\sqcup\Omega^{e},$$

where $\sqcup$ denotes the disjoint union.

For our purposes, we classify pairs of boundary edges, $\partial\Omega\times\partial\Omega,$ into “good” pairs and “bad” pairs. The motivation is similar to that for “bad” boundary vertices, see Example 3.2.

By the definition,

(3.5)

$$\partial\Omega\times\partial\Omega=(\partial\Omega)^{2}_{g}\sqcup(\partial\Omega)^{2}_{b}$$

and

$$(\partial\Omega)^{2}_{b}=\{(\tau_{1},\tau_{2})\in\partial\Omega\times\partial\Omega~|P(\tau^{\perp}_{1})=P(\tau^{\perp}_{2})\in\delta^{\prime}\Omega\}.$$

We estimate the number of “bad” pairs of boundary vertices by the number of “bad” vertices.

We define a mapping on“good” pairs of boundary vertices as

(3.6)		$\displaystyle f:(\partial\Omega)^{2}_{g}$	$\displaystyle\rightarrow\delta\Omega\times\delta\Omega$
	$\displaystyle(\tau_{1},\tau_{2})$	$\displaystyle\mapsto f(\tau_{1},\tau_{2})=(f_{1}(\tau_{1},\tau_{2}),f_{2}(\tau_{1},\tau_{2})):=\begin{cases}(P(\tau_{1}^{\perp}),P(\tau_{2}^{\perp})),&\text{if }P(\tau_{1}^{\perp})\neq P(\tau_{2}^{\perp})\\ (P(\tau_{1}^{\perp}),z),&\text{if }P(\tau_{1}^{\perp})=P(\tau_{2}^{\perp})\not\in\delta^{\prime}\Omega,\end{cases}$

where $z\in Q_{2}(P(\tau_{1}^{\perp}))\cap\delta\Omega$ such that $z\neq P(\tau_{1}^{\perp}).$