Resolutive ideal boundaries of nonlinear resistive networks

Abstract

In this paper, we deal with nonlinear resistive networks in the framework of modular sequence spaces, introduced by De Michele and Soardi. We consider ideal boundaries of a network and investigate Dirichlet boundary value problems for solutions of Poisson equations.

Appendix

Let (G, r) be a nonparabolic network. We are given a family Q of bounded functions in L(G, r), and consider the compactification $\mathcal{K}_{Q}$ of G relative to the family. In this appendix, we deal with the boundary behavior of functions in L(G, r) and Perron solutions.

Definition

We say a nonnegative function $\sigma$ on the set of edges E is an r-pseudo metric if

$$\begin{aligned} W^*\left( \frac{\sigma }{t}\right) := \sum _{B \in E} M^*_B\left( \frac{\sigma (B)}{t}\right) <+\,\infty \end{aligned}$$

for some $t>0$. Let $\mathcal{M}(G,r)$ be the set of r-pseudo metrics and let

$$\begin{aligned} \Vert \sigma \Vert _*:= \inf \left\{ t>0 ~|~ W^*\left( \frac{\sigma }{t}\right) \le 1 \right\} \end{aligned}$$

for $\sigma \in \mathcal{M}(G,r)$. For instance, a 1-cochain df, where $f \in L(G,r)$, gives an r-pseudo metric |df|.

A finite path $c=\{ c(i) ~|~ i=0,1,\ldots ,n\}$ in G is by definition a sequence of vertices c(i) such that $[c(i), c(i+1)] \in E^o$ for $i=0,1,\ldots ,n-1$. For two subsets A, B of V, we denote by $\mathcal{P}_{A,B}$ the set of finite paths $c=\{ c(i) ~|~ i=0,1, \ldots ,n\}$ such that $c(0) \in A$ and $c(n) \in B$. An infinite path $c=\{ c(i)\}$ in G is by definition an infinite sequence of vertices c(i) such that $[c(i),c(i+1)] \in E^o$ for $i=0,1,2, \ldots$. We say that an infinite path $c=\{ c(i)\}$tends to infinity if, for any finite subset U, there exists a number n such that c(i) lies outside U for any $i \ge n$. For a subset U of V, we denote by $\mathcal{P}_{U,\infty }$ the set of infinite paths which start at verticies in U and tend to infinity, and set $\mathcal{P}_{\infty }=\cup \{ \mathcal{P}_{x,\infty } ~|~ x \in V\}$.

We are given an r-pseudo metric $\sigma$. For a finite path $c=\{ c(i) ~|~ i=0,1,\ldots ,n\}$ of length n in G, we let

$$\begin{aligned} L_{\sigma }c=\sum _{i=0}^{n-1} \sigma ([c(i),c(i+1)]) (< +\,\infty ). \end{aligned}$$

For an infinite path $c=\{ c(i) ~|~ i=0,1,2,\ldots \}$, we let

$$\begin{aligned} L_{\sigma }c=\sum _{i=0}^{\infty } \sigma ([c(i),c(i+1)]) (\le +\,\infty ). \end{aligned}$$

Definition

The extremal length of a family $\mathcal{P}$ of finite or infinite paths is defined by

$$\begin{aligned} \lambda (\mathcal{P}) = \sup \left\{ \frac{ \inf \{ L_{\sigma }c ~|~ c \in \mathcal{P} \} }{ \Vert \sigma \Vert _{*} } ~|~ \sigma \in \mathcal{M}(G,r) \right\} (\le +\,\infty ). \end{aligned}$$

We say that a property holds for almost all paths in $\mathcal{P}$ if the subset of all paths for which the property is not true has extremal length $\infty$.

We mention several propositions on extremal length.

• (i) For countable families $\{ \mathcal{P}_n \}$ of paths, it holds that

  $$\begin{aligned} \frac{1}{\lambda (\cup _n \mathcal{P}_n) } \le \sum _{n} \frac{1}{\lambda (\mathcal{P}_n)}~~~ (\le +\,\infty ); \end{aligned}$$

  in particular, $\lambda (\cup _n \mathcal{P}_n)=+\,\infty$ if and only if $\lambda (\mathcal{P}_n)=+\,\infty$ for all n (see Lemma 27 in [15]).

• (ii) Given a vertex x of V, we have

  $$\begin{aligned} \lambda (\mathcal{P}_{x,\infty })= \sup \left\{ \frac{|g(x)|}{\Vert dg \Vert _*} ~|~ g \in L_0(G,r) \right\} \ (\le +\,\infty ), \end{aligned}$$

  which is the reciprocal of $\mathcal{C}_r(x)$, and given a pair of vertices x, y of V, we have

  $$\begin{aligned} \lambda (\mathcal{P}_{x,y})= \sup \left\{ \frac{|u(x)-u(y)|}{\Vert du \Vert _*} ~|~ u\in L(G,r) \right\} \ (< +\,\infty ) \end{aligned}$$

  (see Proposition 28 in [15]). We note that

  $$\begin{aligned} \lambda (\mathcal{P}_{x,\infty })\le \lambda (\mathcal{P}_{x,y})+\lambda (\mathcal{P}_{y,\infty }) \end{aligned}$$

  and that $\lambda (\mathcal{P}_{x,y})$ gives a distance between a pair of vertices $x, y \in V$.

• (iii) (G, r) is nonparabolic if and only if $\lambda (\mathcal{P}_{x,\infty })<+\,\infty$ for some $x \in V$ (see Proposition 30 in [15]).

• (iv) For a function $u \in L(G,r)$, u(c(i)) converges as $i\rightarrow \infty$ for almost all $c \in \mathcal{P}_{\infty }$, and $u \in L_0(G,r)$ if and only if $\lim _{i\rightarrow \infty }u(c(i))=0$ for almost all $c \in \mathcal{P}_{\infty }$ (see Theorem 38 and Corollary 39 in [15]).

For a path $c=\{c(i) ~|~ i=0,1,2,\ldots \} \in \mathcal{P}_{\infty }$, we let

$$\begin{aligned} E_{Q}(c)= \overline{\{ c(i) ~|~ i=0,1,2,\ldots \} }\cap \partial \mathcal{K}_{Q} \end{aligned}$$

and if $E_{Q}(c)$ consists of a single point, that is, c converges to a point of the boundary, we denote it by $c(\infty )$. Given a nonempty subset J of $\partial \mathcal{K}_{Q}$, we let $\mathcal{Q}_{x,J}=\{ c \in \mathcal{P}_{x,\infty } ~|~ E_{Q}(c) \subset J \}$ and $\mathcal{P}_{x,J} =\{ c \in \mathcal{P}_{x,\infty } ~|~ c(\infty ) \in J \}$. Obviously $\mathcal{P}_{x,J} \subset \mathcal{Q}_{x,J}$, so that $\lambda (\mathcal{Q}_{x,J})\le \lambda (\mathcal{P}_{x,J})$.

Lemma A.1

Let J be a closed subset of $\partial \mathcal{K}_{Q}$ such hat $J \cap \varDelta _{Q} =\emptyset$. Then we have

$$\begin{aligned} \lambda (\cup _{x \in V} \mathcal{Q}_{x,J})=+\,\infty \end{aligned}$$

Proof

  We take a function $g \in L_0(G,r)$ such that g(x) tends to infinity as $x \in V \rightarrow J$ (see Lemma 3.2 (iii)). Then $L_{|dg|}(c)=+\,\infty$ for all $c \in \cup _{x \in V} \mathcal{Q}_{x,J}$, since we have $L_{|dg|}(c)=\sum _{i=0}^{\infty } |g(c(i)-g(c(i+1))| \ge |g(c(n))-g(c(o))|$ for all n and $\lim _{n \rightarrow \infty } g(c(n)) = +\,\infty$. Thus $\lambda ( \cup _{x \in V} \mathcal{Q}_{x,J}) = +\,\infty$. This completes the proof of Lemma A.1. $\square$

Lemma A.2

Suppose that a given family Q of bounded functions in L(G, r) admits a countable subfamily which is dense in Q with respect to the uniform topology. Then we have

$$\begin{aligned} \lambda ( \mathcal{P}_{x,\partial \mathcal{K}_{Q} {\setminus } \varDelta _{Q}}) = +\,\infty \end{aligned}$$

for any $x \in V$; in particular,

$$\begin{aligned} \lambda (\mathcal{P}_{x,\partial \mathcal{K}_{Q}}) =\lambda (\mathcal{P}_{x,\varDelta _{Q}}). \end{aligned}$$

Proof

  Since $\mathcal{K}_{Q}$ is metrizable, we can take a countable family of compact sets $J_n$ in $\partial \mathcal{K}_{Q} {\setminus } \varDelta _{Q}$ such that $\cup _n J_n=\partial \mathcal{K}_{Q} {\setminus } \varDelta _{Q}$. Then it follows from Lemma A.1 that $\lambda (\mathcal{P}_{x,J_n})=+\,\infty$ for each n, and hence $\lambda (\mathcal{P}_{x,\cup _n J_n})=+\,\infty$. This completes the proof of Lemma A.2. $\square$

Lemma A.3

Let $F:V \rightarrow (X,d_X)$ be a map from V to a metric space $(X,d_X)$, and define a nonnegative function |dF| on the set of edges E by

$$\begin{aligned} |dF|(\{ x,y \})=d_X( F(x),F(y)) \end{aligned}$$

for $\{ x,y\} \in E$. Suppose that $|dF| \in \mathcal{M}(G,r)$, that is,

$$\begin{aligned} W^*(|dF|)= \sum _{\{ x,y\} \in E} M_{\{x,y\}}^*(d_X(F(x),F(y)) <+\,\infty . \end{aligned}$$

Then the pullback $F^*v(=v \circ F)$ of a Lipschitz continuous function v on X with Lipschitz constant $\gamma$ belongs to L(G, r) and $\Vert d(F^*v)\Vert _* \le \gamma \Vert dF\Vert _*.$

Proof

   Let v be a Lipschitz continuous function on X with Lipschitz constant $\gamma$. Then $|d(v \circ F)(\{ x,y\})|=|v(F(x))-v(F(y))| \le \gamma d_X(F(x),F(y))$ for all $\{x,y\} \in E$, so that we get

$$\begin{aligned} W^* \left( \frac{d(v \circ F)}{t \gamma } \right)= & {} \frac{1}{2} \sum _{\{x,y\} \in E} M_{\{x,y\}}^* \left( \frac{ d(v \circ F)(\{x,y\})}{t\gamma } \right) \\\le & {} \frac{1}{2} \sum _{\{x,y\} \in E} M_{\{ x,y\}}^* \left( \frac{d_X(F(x),F(y))}{t}\right) \\= & {} W^*\left( \frac{ |dF| }{ t } \right) \end{aligned}$$

for any $t>0$. This shows that $v\circ F$ belongs to L(G, r) and it holds that $\Vert d(v \circ F) \Vert _* \le \gamma \Vert |dF| \Vert _*$. $\square$

Let $F:V \rightarrow (X,d_X)$ be as in Lemma A.3. We suppose that X is compact. Let

$$\begin{aligned} K_F:=\{ \xi \in X ~|~ \lim _{ x_n\rightarrow \xi } F(x_n) =\xi \ \text{ for } \text{ some } \text{ divergent } \text{ sequence } \ \{ x_n \} \subset V \} \end{aligned}$$

Then we get a compactification $\overline{V}^{F}=V \cup K_F$ of V by adding the points of $K_F$ to V in such a way that a sequence $\{ x_n \}$ of V converges to a point $\xi$ of $K_F$ if $F(x_n)$ tends to $\xi$ as $n \rightarrow \infty$. Let $F^*C^{0,1}(X)=\{ v \circ F ~|~ v \in C^{0,1}(X) \}$. Then $F^*C^{0,1}(X) \subset BL(G,r)$ and $F^*C^{0,1}(X)$-compactification coincides with the above $\overline{V}^{F}$. We note that $F^*C^{0,1}(X)$ admits a countable subfamily which is dense in $F^*C^{0,1}(X)$ with respect to the uniform topology.

Example

We consider sphere packings in the sphere ${S}^n$ of Euclidean space ${R}^{n+1}$ of dimension $n+1 (\ge 3)$. Let $\mathcal{B}=\{ \mathcal{B}_x ~|~ x\in V\}$ be a collection of closed balls indexed by a countably infinite set V with disjoint interiors in ${S}^n$. Associated to the collection, we have a graph $G=(V,E)$ with the set of vertices V and the set of edges E defined by $\{ x,y\} \in E$ if and only if $\mathcal{B}_x$ and $\mathcal{B}_y$ are tangent. We assume that G is connected. Assigning to a vertex $x \in V$ the center of $\mathcal{B}_x$, we obtain a map $F:V \rightarrow {S}^{n}$. We consider conductance functions $r^{-1}=\{ r_B^{-1} ~|~ B \in E\}$ satisfying

$$\begin{aligned} r_B^{-1}(t) \le t^{n-1}, \quad 0 \le t \le a(B) \end{aligned}$$

for $B \in E$. Then noting that the sum of the volume of balls $\mathcal{B}_x$ in $\mathcal{B}$ is finite and the condition on $r_B^{-1}$ just mentioned, we see that $|dF| \in \mathcal{M}(G,r)$.

Now we assume that the given family Q of bounded functions in L(G, r) admits a countable subfamily S which is dense in Q with respect to the uniform topology. Then we can introduce a distance on $\mathcal{K}_{Q}$ as follows: we choose a sequence $\{ a_u ~|~ u \in S \}$ of positive numbers such that

$$\begin{aligned} \sum _{u \in S} a_u \le 1; \ \sum _{u \in S} a_u (\sup _{x \in V} |u(x)| + W^*(du)) <+\,\infty . \end{aligned}$$

For any $x, y \in V$, we let

$$\begin{aligned} \delta (x,y)= \sum _{u \in \varLambda } a_u |u(x)-u(y)|. \end{aligned}$$

(A.1)

Then $\delta$ extends to a distance on $\mathcal{K}_{Q}$ which induces the topology of $\mathcal{K}_{Q}$. We denote by $\iota$ the inclusion map of V into the compact metric space $\mathcal{K}_{Q}$ endowed with the distance $\delta$, and define a nonnegative function $|d\iota |$ on the set of edges E by

$$\begin{aligned} |d\iota |(\{x,y\})= \delta (x,y), \quad \{x,y\} \in E. \end{aligned}$$

(A.2)

Then $|d\iota | \in \mathcal{M}(G,r)$. In fact, we have

$$\begin{aligned} W^*(|d\iota |)= & {} \frac{1}{2} \sum _{[x,y] \in E^o} M_{[x,y]}^*( \delta (x,y))\\\le & {} \frac{1}{2} \sum _{[x,y] \in E^o} \sum _{u \in \varLambda } a_u M^*_{[x,y]}(du([x,y])) \\= & {} \sum _{u \in \varLambda } a_u W^*(du) < +\,\infty . \end{aligned}$$

Theorem A.4

Let (G, r) be a nonparabolic network and q a function as in Theorem 4.3 or Theorem 4.5. Suppose that a given family Q of bounded functions in L(G, r) admits a countable subfamily which is dense in Q with respect to the uniform topology.

• (i) For almost all $c \in \mathcal{P}_{\infty }$, c(i) tends to a point $c(\infty )$ in $\varDelta _{Q}(=\pi _{Q}(\varDelta (G,r)))$ as $i\rightarrow \infty$

• (ii) For a continuous function $\phi :\partial \mathcal{K}_{Q} \rightarrow {R}$, we have

  $$\begin{aligned} \lim _{i \rightarrow \infty } \mathcal{H}_q\phi (c(i))= \phi (c(\infty )) \end{aligned}$$

  for almost all $c \in \mathcal{P}_{\infty }$.

Proof

Defining a distance $\delta$ on $\mathcal{K}_{Q}$ by (A.1), we obtain the element $|d \iota |$ of $\mathcal{M}(G,r)$ defined by (A.2), so that the extremal length of the set of paths c such that $L_{|d\iota |}c=\infty$ is not finite. We also observe that if $L_{|d\iota |}c=\sum _{i=0}^{\infty } \delta (c(i),c(i+1)) < \infty$ for an infinite path c, then $\{ c(i)\}$ is a Cauchy sequence in $(\mathcal{K}_{Q},\delta )$, which converges to a point $c(\infty )$ in $\partial \mathcal{K}_{Q}$ as $i \rightarrow \infty$. This shows the first assertion.

Let $\psi =tr(f)$, where $f \in Q$. Then $\mathcal{H}_q \psi -f \in L_0(G,r)$, so that we have $\lim _{i \rightarrow \infty } \mathcal{H}_q \psi (c(i))=\lim _{i \rightarrow \infty } f(c(i)) =\psi (c(\infty ))$ for almost all paths $c \in \mathcal{P}_{\infty }$. For a continuous function $\phi$ on $\partial \mathcal{K}_{Q}$, we take a sequence $\{ \phi _n\}$ in tr(Q) converging uniformly to $\phi$ as $n \rightarrow \infty$. Let $\varepsilon (n)= \sup \{ |\phi (\xi )-\phi _n(\xi )| ~|~ \xi \in \partial \mathcal{K}_{Q} \}$. Then $|\mathcal{H}_q\phi -\mathcal{H}_q\phi _n| \le \varepsilon (n)$ on V, so that we have

$$\begin{aligned} |\mathcal{H}_q\phi (c(i))-\phi (c(\infty ))|\le & {} |\mathcal{H}_q\phi (c(i))-\mathcal{H}_q\phi _n(c(i)))| + |\mathcal{H}_q\phi _n (c(i))-\phi _n(c(\infty ))| \\&+\, | \phi _n(c(\infty ))-\phi (c(\infty ))|\\\le & {} 2\varepsilon (n) +|\mathcal{H}_q\phi _n(c(i))-\phi _n(c(\infty ))|. \end{aligned}$$

This shows that $\mathcal{H}_q\phi (c(i))$ tends to $\phi (c(\infty ))$ as $i \rightarrow \infty$ for almost all $c \in \mathcal{P}_{\infty }$, since $\mathcal{H}_q\phi _n(c(i))$ goes to $\phi _n(c(\infty ))$ as $i \rightarrow \infty$ for almost all $c \in \mathcal{P}_{\infty }$. $\square$

Lemma A.5

Let $\eta$ be a point of the Royden boundary $\partial \mathcal{R}(G,r)$ and $\{ x_n\}$ a sequence in V which converges to $\eta$ as $n \rightarrow \infty$. Then the following are mutually equivalent:

• (i) $\eta \in \varDelta (G,r)$;

• (ii) for any $g \in L_0(G,r)$, $|g(x_n)|$ is bounded as $n \rightarrow \infty$;

• (iii) $\lambda (\mathcal{P}_{x_n,\infty })$ is bounded as $n \rightarrow \infty$.

Proof

If $\eta$ does not belong to the harmonic boundary $\varDelta (G,r)$, then there exists a function $g \in L_0(G,r)$ such that $\overline{g}(\eta )=+\,\infty$, and hence $g(x_n)$ diverges to $+\,\infty$ as $n \rightarrow \infty$ (see Lemma 46 in [15]). Since

$$\begin{aligned} \frac{|g(x_n)|}{\Vert dg \Vert _*} \le \lambda (\mathcal{P}_{x_n,\infty }), \end{aligned}$$

we see that $\lambda (\mathcal{P}_{x_n,\infty })$ diverges as $n \rightarrow \infty$. Thus (ii) or (iii) implies (i). (i) obviously implies (ii). To show that (iii) follows (i), we suppose that $\lambda (\mathcal{P}_{x_n,\infty })$ tends to $+\,\infty$ as $n \rightarrow \infty$. Then we can find a sequence $\{ g_n \}$ in $L_0(G,r)$ such that $0 \le g_n \le 1$, $g_n(x_n)= 1$, and $\lim _{n \rightarrow \infty }\Vert g_n \Vert _*=0$. Taking a subsequence if necessary, we may assume that $\Vert dg_n \Vert _* \le 2^{-n}$. Let $g=\sum _{n=1}^{\infty } g_n$. Then g belongs to $L_0(G,r)$ and satisfies that $g(x_n) \ge g_n(x_n) =1$ for all n. Therefore $\overline{g}(\eta )=\lim _{n \rightarrow \infty } g(x_n) \ge 1$. This shows that $\eta$ does not belong to $\varDelta (G,r)$. This completes the proof of Lemma A.5. $\square$

Using Lemma A.5, we can deduce the following

Theorem A.6

Let (G, r) be a nonparabolic network and J a closed subset of $\partial \mathcal{K}_{Q}$. Then the following are mutually equivalent:

• (i) $\pi _{\varGamma }^{-1}(J) \subset \varDelta (G,r)$;

• (ii) for any $g \in L_0(G,r)$, $\lim _{x \in V\rightarrow J}|g(x)|=0$;

• (iii) for any $g \in L_0(G,r)$, $\limsup _{x \in V\rightarrow J} |g(x)| <+\,\infty$;

• (iv) $\limsup _{x \in V \rightarrow J} \lambda (\mathcal{P}_{x,\infty })< +\,\infty$.

Moreover if theses are the cases, it holds that

$$\begin{aligned} \lim _{x \in V \rightarrow \xi } \mathcal{H}_q\phi (x)=\phi (\xi ) \end{aligned}$$

for all $\phi \in C(\partial \mathcal{K}_{Q})$ and $\xi \in J$, where q is a function as in Theorem 4.3 or Theorem 4.5.

Proof

  The equivalence of the above conditions are clear from Lemma A.5. It remains to prove the last statement. For a continuous function $\phi \in C(\partial \mathcal{K}_{Q})$, we take a sequence of $\{ \phi _n=tr(f_n) ~|~ f_n \in Q \}$ which converges uniformly to $\phi$ as $n \rightarrow \infty$. Let $\varepsilon (n)=\sup |\phi -\phi _n|$. Then we have

$$\begin{aligned} |\mathcal{H}_q\phi (x)-\phi (\xi )|\le & {} |\mathcal{H}_q \phi (x)-\mathcal{H}_q\phi _n(x)| + |\mathcal{H}_q \phi _n(x)-f_n(x)| \\&\quad + \, |f_n(x)-\phi _n(\xi )| +|\phi _n(\xi )-\phi (\xi )|\\\le & {} 2\varepsilon (n) + | H_q f_n(x)-f_n(x)| + |f_n(x)-\phi _n(\xi )|. \end{aligned}$$

Since $H_qf-f_n \in L_0(G,r)$, in view of (ii), we see that $\lim _{x \in V \rightarrow \xi }H_qf_n(x)-f_n(\xi ))=0$. Moreover since $f_n \in Q$, $\lim _{x \in V \rightarrow \xi } f_n(x)=\phi _n(\xi )$. This completes the proof of Theorem A.6. $\square$

Taking $J=\partial \mathcal{K}_{Q}$ in the theorem, we obtain the following

Corollary A.7

Let (G, r) be a nonparabolic network. Then $\sup _{x \in V} \lambda (\mathcal{P}_{x,\infty }) <+\,\infty$, or $\inf _{x \in V} \mathcal{C}_r(x) >0$ if and only if $\partial \mathcal{R}(G,r)=\varDelta (G,r)$.

Proposition A.8

Let (G, r) be a nonparabolic network. Let Q be a family of bounded functions in L(G, r) such that Q is dense in the Banach space. Suppose that $\sup _{x,y \in V}\lambda (\mathcal{P}_{x,y})<+\,\infty$. Then $\mathcal{K}_{Q}=\mathcal{R}(G,r)$ and $\partial \mathcal{K}_{Q}=\partial \mathcal{R}(G,r)=\varDelta (G,r)$.

Proof

We observe that

$$\begin{aligned} |u(x)-u(y)| \le \lambda (\mathcal{P}_{x,y}) \Vert du\Vert _* \le \sup _{x,y \in V}\lambda (\mathcal{P}_{x,y}) \Vert du\Vert _* \end{aligned}$$

for all $u \in L(G,r)$. This implies that if Q is dense in L(G, r), then it is also dense there with respect to the uniform norm. Therefore $\partial \mathcal{K}_{Q}$ must coincide with the Royden boundary. $\square$

Given a subset U of V. Let $\mathcal{V}(U)=\{ f \in L(G,r) ~|~ f=0 \ \text{ on } \ U \}$. Since $d\mathcal{V}(U)$ is a closed convex subspace in dL(G, r), we see that for any $f \in L(G,r)$, there exists a unique function $f_U \in L(G,r)$ such that $f_U=f$ on U and $W^*(df_U)\le W^{*}(df_U +dv)$ for all $v \in \mathcal{V}(U)$. Let

$$\begin{aligned} K(G,U)=\{ f_U \in L(G,r) ~|~ f \in L(G,r)\}. \end{aligned}$$

We note that $K(G,U) \subset K(G,W)$ if $U \subset W$.

Definition

The compactification relative to the family

$$\begin{aligned} Q_K:= \cup \{ K(G,U) ~|~ U \ \text{ is } \text{ a } \text{ finite } \text{ subset } \text{ of } \ V \} \end{aligned}$$

is called the Kuramochi compactification of a network (G, r), and denoted by $\mathcal{K}(G,r)$.

We remark (see [15], Theorems 22, 54) that

• (i) $Q_K$ is dense in L(G, r);

• (ii) $Q_K$ admits a countable subfamily which is dense in $Q_K$ with respect to the uniform topology.

We also note that when the graph G under consideration is a locally finite tree, then the Kuramochi compactification $\mathcal{K}(G,r)$ topologically coincides with the end compactification of the tree. In fact, a function in K(G, U) is constant on each connected component of $V {\setminus } U$.

Now we introduce an intrinsic distance of the network (G, r). For a finite path $c=(c(0),c(1),\ldots ,c(n))$, we let

$$\begin{aligned} \ell (c) = \sup \left\{ \sum _{i=0}^{n-1} s_i ~|~ 0\le s_i ( i=0,1,\ldots ,n-1), \sum _{i=0}^{n-1} M^*_{[c(i),c(i+1)]}(s_i) \le 1 \right\} . \end{aligned}$$

and

$$\begin{aligned} d_r(x,y)= \inf \{ \ell (c) ~|~ c \in \mathcal{P}_{x,y} \} \end{aligned}$$

for a pair of vertices x, y of V. Then it is easy to see that $d_r$ induces a distance on V. Moreover for any $u \in L(G,r)$ and every $c \in \mathcal{P}_{x,y}$, we have by the definition of the norm $\Vert du\Vert _*$

$$\begin{aligned} \frac{|u(x)-u(y)|}{\Vert du\Vert _*} \le \sum _{i=0}^{n-1} \frac{ |u(c(i))-u(c(i+1))|}{\Vert du\Vert _*} \le \ell (c), \end{aligned}$$

which shows that

$$\begin{aligned} \lambda (\mathcal{P}_{x,y}) \le d_r(x,y) \end{aligned}$$

(A.3)

for $x, y \in V$.

We remark that $\lambda (\mathcal{P}_{x,y}) = d_r(x,y)$ for all $x, y \in V$ when the graph G is a tree.

Corollary A.9

Let (G, r) be a nonparabolic network. Suppose that the diameter of G relative to the distance $d_r$, $\sup _{x,y \in V} d_r(x,y)$, is finite. Let Q be a family of bounded functions in L(G, r) such that Q is dense in the Banach space. Then $\mathcal{K}_{Q}=\mathcal{R}(G,r)$ and $\partial \mathcal{K}_{Q}=\partial \mathcal{R}(G,r)=\varDelta (G,r)$.

Now a Rayleigh’s monotonicity law is stated in the following

Theorem A.10

Consider two conductance functions $r_1^{-1}=\{ r_{1:B}^{-1}(t) ~|~ B \in E\}$ and $r_2^{-1}=\{ r_{2:B}^{-1}(t) ~|~ B \in E\}$ of $G=(V,E)$ satisfying conditions (a)–(e), and suppose that

$$\begin{aligned} r_{1;B}^{-1}(t) \le r_{2:B}^{-1}(t), \quad 0 \le t \le a_{1}^*(B). \end{aligned}$$

Then $L(G,r_2)$ (resp. $L_0(G,r_2)$) is included in $L(G,r_1)$ (resp. $L_0(G,r_1)$), and there is a constant $\gamma \ge 1$ such that

$$\begin{aligned} \Vert df \Vert _{1:*} \le \gamma \Vert df\Vert _{2:*} \end{aligned}$$

for all $f \in L(G,r_2)$. In particular, it holds that

$$\begin{aligned} \mathcal{C}_{r_1}(x) \le \gamma \ \mathcal{C}_{r_2}(x) \end{aligned}$$

that is,

$$\begin{aligned} \lambda _{r_1}(\mathcal{P}_{x\infty }) \ge \frac{1}{\gamma \ } \lambda _{r_2} (\mathcal{P}_{x \infty }) \end{aligned}$$

for all $x \in V$, and

$$\begin{aligned} \lambda _{r_1}(\mathcal{P}_{x,y}) \ge \frac{1}{\gamma } \ \lambda _{r_2} (\mathcal{P}_{x,y}) \end{aligned}$$

for all $x,y \in V$.

Proof

   We define conductance functions $r_3^{-1} =\{ r_{3:B}^{-1}(t) ~|~ B \in E\}$ by

$$\begin{aligned} r_{3:B}^{-1}(t) =\left\{ \begin{array}{ll} \min \{ r_{1:B}^{-1}(t),r_{2:B}^{-1}(t)\}, &{}\quad t \ge 0;\\ - \min \{ r_{1:B}^{-1}(-t),r_{2:B}^{-1}(-t)\}, &{}\quad t \le 0. \end{array} \right. \end{aligned}$$

Since we have

$$\begin{aligned} M_{3:B}^*(df) \le M_{1:B}^*(df) \end{aligned}$$

for all functions f on V, we see that $L(G,r_1) \subset L(G,r_3)$ and $\Vert df \Vert _{3:*} \le \Vert df\Vert _{1:*}$ for $f \in L(G,r_1)$. We note that $a^*_{3}(B)=a^*_{1}(B)$ for all $B \in E$ and if $|df(B)|=|f(x)-f(y)|\le a^*_3(B)(=a_1^*(B))$ for $B =\{ x, y \} \in E$, then $M_{1:B}^*(|df(B)|) = M_{3:B}^*(|df(B)|)$. This implies that for $f \in L(G,r_3)$, letting

$$\begin{aligned} E^{\prime }=\{ B \in E~|~ |df(B)| > a_1^*(B) \}, \ E^{\prime \prime }= \{ B \in E ~|~ |df(B)| \le a_1^*(B)\}, \end{aligned}$$

we get

$$\begin{aligned} W_{r_1}^*(df)= & {} \sum _{B \in E} M_{1:B}^*(|df(B)|) \\= & {} \sum _{B \in E^{\prime }} M_{1:B}^*(|df(B)|) + \sum _{B \in E^{\prime \prime }} M_{1:B}^*(|df(B)|)\\= & {} \sum _{B \in E^{\prime }} M_{1:B}^*(|df(B)|) + \sum _{B \in E^{\prime \prime }} M_{3:B}^*(|df(B)|), \end{aligned}$$

so that $W_{r_1}^*(df)$ is finite, since $E^{\prime }$ is a finite subset of E. Thus the open mapping theorem shows that $L(G,r_3)=L(G,r_1)$, and there is a constant $\varLambda$ such that $\Vert df \Vert _{1:*} \le \varLambda \Vert df\Vert _{3:*}$ for all $f \in L(G,r_3)$. Since $r_{2:B}^{-1}(t) \ge r_{3:B}^{-1}(t), \ t \in [0,+\,\infty )$, we have $L(G,r_2) \subset L(G,r_3) =L(G,r_1)$ and $\Vert df \Vert _{3:*} \le \varLambda \Vert df\Vert _{2:*}$ for all $f \in L(G,r_2)$, and hence $\Vert df \Vert _{1:*} \le \varLambda \Vert df\Vert _{2:*}$ for all $f \in L(G,r_2)$. $\square$

Remark

In Theorem A.10, if $r_{1;B}^{-1}(t) \le r_{2:B}^{-1}(t)$ for $B \in E$ and all $t \ge 0$, then we can take $\gamma = 1$ (see [15], Theorem 5.8).

Corollary A.11

Let $r_i^{-1}$$(i=1,2)$ be conductance functions of $G=(V,E)$ as above. Let $\mathcal{R}(G,r_i)$$(i=1,2)$ be the Royden compactifications of G relative to the conductance functions $r_{i}^{-1}$. Then the identity map $\iota$ of V extends to a continuous map $\overline{\iota }$ from $\mathcal{R}(G,r_1)$ onto $\mathcal{R}(G,r_2)$ in such a way that $\overline{\iota } (\partial \mathcal{R}(G,r_1)) =\partial \mathcal{R}(G,r_1)$ and $\overline{\iota }(\varDelta (G,r_1)) \subset \varDelta (G,r_2)$. In particular, if $\partial \mathcal{R}(G,r_1)=\varDelta (G,r_1)$, then $\partial \mathcal{R}(G,r_2)=\varDelta (G,r)$ and $\overline{\iota }(\varDelta (G,r_1))= \varDelta (G,r_2)$.

In what follows, we introduce a capacity of subsets of the boundary and quasicontinuous functions on the compactification.

For a subset U of $\mathcal{R}(G,r)$ (resp. $\mathcal{K}_{Q}$), we write $\overline{U}^R$ (resp. $\overline{U}$) for the closure of U in $\mathcal{R}(G,r)$ (resp. $\mathcal{K}_{Q}$). We denote as before by $\pi _{Q}:\mathcal{R}(G,r)\rightarrow \mathcal{K}_{Q}$ and $\varDelta _{Q}$, respectively, the canonical projection of $\mathcal{R}(G,r)$ onto $\mathcal{K}_{Q}$ and the image of the harmonic boundary $\varDelta (G,r)$ by the projection, $\varDelta _{Q}=\pi _{Q}(\varDelta (G,r))$.

Definition

For a point $x \in V$ and a subset J of $\mathcal{K}_{Q} {\setminus } \{ x \}$, we let

$$\begin{aligned} \mathcal{A}_{x;J}= & {} \{ f \in L(G,r) ~|~ f(x)=0, f(y) \ge 1 \ \text{ for } \ y \in J \cap V, \\&\qquad \qquad \qquad \liminf _{y \in V \rightarrow \xi } f(y) \ge 1 \ \text{ for } \ \xi \in J \cap \partial \mathcal{K}_{Q} \}, \end{aligned}$$

and we assign to J a nonnegative number $\mathcal{C}(x,J)$ defined by

$$\begin{aligned} \mathcal{C}(x,J) = \inf \{ \Vert df\Vert _* ~|~ f \in \mathcal{A}_{x,J} \}. \end{aligned}$$

If x belongs to J, then $\mathcal{A}_{x;J}$ is empty; in this case, we assume $\mathcal{C}(x,J)=+\,\infty$.

We remark that

$$\begin{aligned} \mathcal{A}_{x,J}=\{ f \in L(G,r) ~|~ f(x)=0, \bar{f}\ge 1 \ \text{ on } \ \pi _{Q}^{-1}(J) \}. \end{aligned}$$

Proposition A.12

Fix a point $x \in V$ and let $J, J_1, J_2,\ldots$ be arbitrary subsets of $\mathcal{K}_{Q}$.

• (i) $\mathcal{C}(x,\emptyset )=0$.

• (ii) For $J_1 \subset J_2$, then $\mathcal{C}(x,J_1) \le \mathcal{C}(x,J_2)$.

• (iii) $\mathcal{C}(x,*)$ is countably subadditive, i.e.,

  $$\begin{aligned} \mathcal{C}(x,\cup _{n=1}^{\infty } J_n ) \le \sum _{n=1}^{\infty } \mathcal{C}(x,J_n). \end{aligned}$$

• (iv) $\mathcal{C}(x,*)$ is an outer capacity, i.e.,

  $$\begin{aligned} \mathcal{C}(x,J)= \inf \{ \mathcal{C}(x,U) ~|~ U \ \text{ is } \text{ an } \text{ open } \text{ subset } \text{ of } \ \mathcal{K}(G,r) \ \text{ including } \ J \}. \end{aligned}$$

• (v) For a point y of V distinct from x, $\mathcal{C}(x,J)=0$ if and only if $\mathcal{C}(y,J)=0$.

Proof

(i) and (ii) are obvious. We prove the assertion (iii). We may assume that $\sum _{n=1}^{\infty } \mathcal{C}(x,J_n) < +\,\infty$. Let $\varepsilon$ be a positive number. For each n, in view of (2.1), we take a nonnegative function $f_n$ in $\mathcal{A}_{x;J_n}$ in such a way that $\Vert df_n \Vert _* \le \mathcal{C}(x,J_n) + 2^{-n}\varepsilon$. Let $g_n =\sum _{k=1}^n f_k$. Then $\{ g_n\}$ is an increasing sequence of nonnegative functions in L(G, r) with $g_n(x)=0$ satisfying $\Vert dg_m -dg_n \Vert _* \le \sum _{k=m+1}^n \Vert df_k\Vert _* \le 2^{-m}\varepsilon$ for $m<n$. Thus $\{ g_n \}$ is a Cauchy sequence in L(G, r). Let $g=\lim _{n \rightarrow \infty } g_n$. Then $g(x)=0$, $g \ge g_n$ for all n, and $\Vert dg\Vert _* \le \sum _{n=1}^{\infty } \Vert df_n \Vert _* \le \sum _{n=1}^{\infty } \mathcal{C}(x,J_n) +\varepsilon$. This implies that $g \in \mathcal{A}_{x;\cup J_n}$ and

$$\begin{aligned} \mathcal{C}(x,\cup _{n=1}^{\infty } J_n) \le \Vert dg \Vert _* \le \sum _{n=1}^{\infty } \mathcal{C}(x,J_n) +\varepsilon . \end{aligned}$$

Letting $\varepsilon \rightarrow 0$, we obtain

$$\begin{aligned} \mathcal{C}(x,\cup _{n=1}^{\infty } J_n) \le \sum _{n=1}^{\infty } \mathcal{C}(x,J_n). \end{aligned}$$

Now we prove the assertion (iv). Given a positive number $\varepsilon$, we take a nonnegative function f in $\mathcal{A}_{x;J}$ such that $\Vert df\Vert _* \le \mathcal{C}(x,J) + \varepsilon$. For any $\xi \in E \cap \partial \mathcal{K}_{Q}$, there exists a neighborhood $W_{\xi }$ of $\xi$ in $\mathcal{K}_{Q}$ such that $f(y) \ge 1-\varepsilon$ for all $y \in W_{\xi } \cap V$, since $\liminf _{y \in V \rightarrow \xi } f(y) \ge 1$. We define an open subset U in $\mathcal{K}_{Q}$ as follows:

$$\begin{aligned} U= (E \cap V) \cup ( \cup \{ W_{\xi } ~|~ \xi \in E \cap \partial \mathcal{K}_{Q} \}). \end{aligned}$$

Then for any $y \in U$, $f(y) \ge 1$ if $y \in J \cap V$, $f(y) \ge 1-\varepsilon$ if $y \in W_{\xi } \cap V$ for some $\xi \in E \cap \partial \mathcal{K}_{Q}$, and

$$\begin{aligned} \liminf _{z \in V \rightarrow y} f(z) = \liminf _{z \in V \cap W_{\xi } \rightarrow y} f(z) \ge 1-\varepsilon \end{aligned}$$

if $y \in W_{\xi } \cap \partial \mathcal{K}_{Q}$ for some $\xi \in E\cap \partial \mathcal{K}_{Q}$. Hence it follows that $f/(1-\varepsilon ) \in \mathcal{A}_{x;U}$, so that

$$\begin{aligned} \mathcal{C}(x,J) \le \mathcal{C}(x,U) \le \frac{1}{1-\varepsilon } \Vert df \Vert _* \le \frac{1}{1-\varepsilon } ( \mathcal{C}(x,J) + \varepsilon ). \end{aligned}$$

Letting $\varepsilon$ tend to zero, we get

$$\begin{aligned} \mathcal{C}(x,J)= \inf \{ \mathcal{C}(x,U) ~|~U \ \text{ is } \text{ an } \text{ open } \text{ subset } \text{ of } \ \mathcal{K}(G,r) \ \text{ including } \ J \}. \end{aligned}$$

To prove the assertion (v), we take a decreasing sequence of open subsets $\{ U_i\}$ of $\mathcal{K}_{Q}$ such that $E \subset U_i$ and $\mathcal{C}(x,J)=\lim _{i \rightarrow \infty } \mathcal{C}(x,U_i)$. We may assume that each $U_i$ satisfies the following property: $U_i$ includes any $\xi \in \overline{U_i} \cap \partial \mathcal{K}(G,r)$ which admits a neighborhood W of $\xi$ such that $W \cap V \subset U_i \cap V$. Then it follows that $\mathcal{A}_{x;U_i}$ is a closed convex subset of L(G, r). Hence there exists a unique $W^*$-minimizer $f_{x;U_i}$ in $\mathcal{A}_{x;U_i}$ such that $\Vert df_{x;U_i}\Vert _*=\mathcal{C}(x,U_i)$ (see Lemma 2.3); $f_{x;U_i}$ satisfies the following properties: $f_{x;U_i}(x)=0$, $f_{x;U_i}=1$ on $U_i\cap V$, $0< f_{x;U_i} < 1$ on $V {\setminus } (\{ x \}\cup U_i)$, and $\mathcal{L}_r f_{x;U_i} = 0$ in $V {\setminus } (\{ x \}\cup U_i)$. For a point $y \in V {\setminus } U_i$, we define a function

$$\begin{aligned} u_y(z)= \frac{ f_{x;U_i}(z)-f_{x;U_i}(y)}{1-f_{x;U_i}(y)}, \quad z \in V. \end{aligned}$$

Then $u_y$ belongs to $\mathcal{A}_{y;U_i}$, so that we have

$$\begin{aligned} \mathcal{C}(y,U_i) \le \Vert du_y \Vert _* = \frac{\Vert df_{x,U_i}\Vert _*}{1-f_{x;U_i}(y)} = \frac{\mathcal{C}(x,U_i)}{1-f_{x,U_i}(y)}. \end{aligned}$$

Taking a subsequence if necessarily, we may assume that $f_{x;U_i}$ converges to a function $f_x$ in L(G, r) as $i \rightarrow \infty$. Then $f_{x;U_i}(y)$ tends to $f_x(y)$ as $i \rightarrow \infty$, so that we get

$$\begin{aligned} \mathcal{C}(y,J) = \lim _{i \rightarrow \infty } \mathcal{C}(y,U_i) \le \lim _{i \rightarrow \infty } \frac{\mathcal{C}(x,U_i)}{1-f_{x,U_i}(y)}= \frac{\mathcal{C}(x,J)}{1-f_x(y)}. \end{aligned}$$

Similarly we obtain

$$\begin{aligned} \mathcal{C}(x,J) \le \frac{\mathcal{C}(y,J)}{1-f_y(x)}. \end{aligned}$$

This completes the proof of Proposition A.12. $\square$

Lemma A.13

For a point $x \in V$ and a function $f \in L(G,r)$ with $f(x)=0$, and for a positive number t, we have

$$\begin{aligned} \mathcal{C}(x, \{ y \in V ~|~ f(y)\ge t \} \cup \{ \xi \in \partial \mathcal{K}_{Q} ~|~ \liminf _{y \in V \rightarrow \xi } f(y) \ge t \}) \le \frac{1}{t} \Vert df\Vert _*. \end{aligned}$$

Proof

  Let $E_t=\{ y \in V ~|~ f(y)\ge t \} \cup \{ \xi \in \partial \mathcal{K}_{Q} ~|~ \liminf _{y \in V \rightarrow \xi } f(y) \ge t \}$. Then $(1/t)f \in \mathcal{A}_{x,E_t}$, so that $\mathcal{C}(x, E_t) \le \Vert (1/t)df\Vert _*=(1/t) \Vert df\Vert _*$. $\square$

Proposition A.14

For any $x \in V$, we have

$$\begin{aligned} \mathcal{C}(x,\partial \mathcal{K}_{Q}) =\frac{1}{\lambda (\mathcal{P}_{x,\infty })}. \end{aligned}$$

Proof

For $f \in \mathcal{A}_{x,\partial \mathcal{K}_{Q}}$, $k=\max \{1-f,0\}$. Then $\bar{k}$ vanishes on the harmonic boundary $\varDelta (G,r)$, so that k belongs to $L_0(G,r)$ and satisfies $k(x)=1$. This shows that $\mathcal{C}(x,\varDelta _{Q}) \ge \inf \{ \Vert dg \Vert _* ~|~ g(x)=1, \ g \in L_0(G,r) \}$. On the other hand, for $g \in \ell _0(V)$ with $g(x)=1$, $f=1-g$ belongs to $\mathcal{A}_{x,\partial \mathcal{K}_{Q}}$. Since the set of functions g in $\ell _0(V)$ with $g(x)=1$ is dense in $\{ u~|~ u\in L_0(G,r), u(x)=1\}$, it follows that $\inf \{ \Vert df \Vert _* ~|~ f \in \mathcal{A}_{x,\partial \mathcal{K}_{Q}} \} \le \inf \{ \Vert dg \Vert _* ~|~ g(x)=1, g \in L_0(G,r) \}$. Moreover it follows from (2.2) that $\inf \{ \Vert dg \Vert _* ~|~ g(x)=1, g \in L_0(G,r) \}= \inf \{ \Vert dg \Vert _* ~|~ 0\le g \le 1, \ g(x)=1, g \in L_0(G,r)\}$. These prove that $\mathcal{C}(x,\varDelta _{Q})=\mathcal{C}(x,\partial \mathcal{K}_{Q}) = \inf \{ \Vert dg \Vert _* ~|~ g(x)=1, \ g \in L_0(G,r) \}=\lambda (\mathcal{P}_{x,\infty })^{-1}$. $\square$

Proposition A.15

Let (G, r) be a nonparabolic network. For a nonempty relatively open subset J of $\varDelta _{Q}$, $\mathcal{C}(x,J)$ is positive. In particular, an isolated point of $\varDelta _{Q}$ (if it exists) has positive capacity.

Proof

Let U be an open subset of $\mathcal{K}_{Q}$ such that $J=U \cap \varDelta _{Q}$. We take an open subset W of $\mathcal{K}_{Q}$ in such a way that $\overline{W} \subset U$ and $F:= W \cap \varDelta _{Q}$ is not empty and included in J. Let $\rho$ be a continuous function on $\mathcal{K}_{Q}$ such that $\rho (x)=0$, $0 \le \rho \le 1$, $\mathrm{supp}~\rho \subset U$ and $\rho =1$ on F. Let h be the Perron solution to the equation: $\mathcal{L}_r u=0$ on $V {\setminus } \{ x \}$ which coincides with $\rho \circ \pi _{Q}$ on the harmonic boundary $\varDelta (G,r)$ and vanishes at x. We note that $h>0$ on $V{\setminus } \{ x \}$.

We take a sequence $\{ f_n\}$ of nonnegative functions in $\mathcal{A}_{x,J}$ in such a way that $\lim _{n \rightarrow \infty } \Vert df_n\Vert _*=\mathcal{C}(x,J)$. We further decompose the functions $f_n$ as $f_n=h_n+g_n$ as above. Then $\overline{h_n}=\overline{f_n} \ge \rho \circ \pi _{Q}$ on the harmonic boundary $\varDelta (G,r)$, and hence it follows from Theorem 36 that $h_n \ge h$ on V. This allows us to conclude that J has positive capacity; otherwise, $W^*(df_n)$ and hence $W^*(dh_n)$ tend to zero as $n \rightarrow \infty$, and as a result, $h_n$ pointwise converges to zero as $n \rightarrow \infty$. This is absurd. This completes the proof of the proposition. $\square$

Now we prove the following

Proposition A.16

Let (G, r) be a nonparabolic network and Q a family of bounded functions in L(G, r). For a point $x \in V$ and a nonempty subset J of $\partial \mathcal{K}_{Q}$, it holds that

$$\begin{aligned} \frac{1}{\mathcal{C}(x,J)} \le \lambda (\mathcal{Q}_{x,J}) \le \lambda (\mathcal{P}_{x,J}). \end{aligned}$$

Proof

For any $f \in \mathcal{A}_{x;J}$ and $c \in \mathcal{Q}_{x,J}$, we have

$$\begin{aligned} L_{ \Vert df \Vert _* }c = \sum _{i=0}^{\infty } |f(c(i))-f(c(i+1))| \ge |f(c(n))| \end{aligned}$$

for all n, so that

$$\begin{aligned} L_{ \Vert df \Vert _* } c \ge 1, \end{aligned}$$

since $\liminf _{n \rightarrow \infty } f(c(n)) \ge 1$. Then it follows that

$$\begin{aligned} \frac{1}{ \Vert df \Vert _*} \le \frac{ \inf \{ L_{|df|} c ~|~ c \in \mathcal{Q}_{x,E} \}}{\Vert df \Vert _*} \le \lambda (\mathcal{Q}_{x,J}). \end{aligned}$$

This holds for all $f \in \mathcal{A}_{x;J}$, so that we obtain

$$\begin{aligned} \mathcal{C}(x,J)^{-1} \le \lambda (\mathcal{Q}_{x,J})=\lambda (\mathcal{P}_{x,J}). \end{aligned}$$

This completes the proof of Proposition A.16. $\square$

Corollary A.17

Let (G, r) be a nonparabolic network and Q a family of bounded functions in L(G, r) such that there exists a countable and dense subfamily relative to the uniform topology. Suppose that $\mathcal{C}(x,\{ \xi \})=0$ for any $\xi \in \varDelta _{Q}$. Then $\varDelta _{Q}$ is a perfect subspace of $\mathcal{K}_{Q}$.

Proof

It follows from the assumption and Proposition A.16 that $\lambda (\mathcal{P}_{x,\{ \xi \}})=+\,\infty$ for all $\xi \in \varDelta _{\varGamma }$. so that we can apply Corollary 52 in [15]. $\square$

Definition

We say that a property regarding points in $\mathcal{K}_{Q}$ holds quasi-everywhere if the set J of points for which it fails has capacity zero, that is, $\mathcal{C}(x,J)=0$ for some $x \in V {\setminus } J$.

In what follows, we fix a vertex $x_0 \in V$.

Definition

Let $f :\mathcal{K}_{Q} \rightarrow \overline{ {R} } (={R} \cup \{ \pm \infty \})$ be an extended real valued function on $\mathcal{K}_{Q}$. We say that f is quasicontinuous on $\mathcal{K}_{Q}$ if for any $\varepsilon >0$, there exists an open subset U of $\mathcal{K}_{Q}$ such that $\mathcal{C}(x_0,U) < \varepsilon$ and the restriction $f_{| \mathcal{K}_{Q} {\setminus } U}$ of f to the closed subset $\mathcal{K}_{Q} {\setminus } U$ is continuous on $\mathcal{K}_{Q} {\setminus } U$, $f_{| \mathcal{K}_{Q} {\setminus } U} \in C^0(\mathcal{K}_{Q} {\setminus } U)$. For a quasicontinuous function f on $\mathcal{K}_{Q}$, we let

$$\begin{aligned} \mathop {\mathcal{C}\mathrm{-ess}\sup }_{\partial \mathcal{K}_{Q} } \phi = \inf \{ t\in {R} ~|~ \mathcal{C}(x_0,\{ \xi \in \partial \mathcal{K}_{Q} ~|~ \phi (\xi ) > t \}) =0 \}, \\ \mathop {\mathcal{C}\mathrm{-ess}\inf }_{\partial \mathcal{K}_{Q} } \phi = \sup \{ t \in {R} ~|~ \mathcal{C}(x_0,\{ \xi \in \partial \mathcal{K}_{Q} ~|~ f(\xi ) < t \}) =0 \}. \end{aligned}$$

Lemma A.18

Suppose that Q is dense in L(G, r). Then a function $f \in L(G,r)$ extends to a quasicontinuous function $\tilde{f}$ on $\mathcal{K}_{Q}$.

Proof

Let f be a function in L(G, r) such that $f(x_0)=0$. Then we have a sequence of functions $\rho _n$ in Q such that $\rho _n(x_0)=0$ and $\lim _{n \rightarrow \infty } \Vert d\rho _n-df\Vert _*=0$. We may assume that $\Vert d\rho _{n+1}-d\rho _n\Vert _* \le 2^{-2n}$. Let $W_n= \{ x \in \mathcal{K}_{Q} ~|~ |\rho _{n+1}(x)-\rho _n(x)| > 2^{-n} \}$ and $W_n'=\cup _{k \ge n+1} W_k$. Then it follows from Lemma A.13 that $\mathcal{C}(x_0,W_n) \le 2^n \Vert d\rho _{n+1}-d\rho _n\Vert _* \le 2^{-n}$, and hence we get by Proposition A.12 (iii)

$$\begin{aligned} \mathcal{C}(x_0,W_n') \le \sum _{k=n+1}^{\infty } \mathcal{C}(x_0,W_k) \le 2^{-n}. \end{aligned}$$

Consider an open subset $U_n$ of $\mathcal{K}_{Q}$ given by

$$\begin{aligned} U_n= W_n'\cup \{ x \in \mathcal{K}_{Q} ~|~ x \ \text{ has } \text{ a } \text{ neighborhood } \ W \ \text{ such } \text{ that } \ W \cap V \subset W_n' \}. \end{aligned}$$

Then we have

$$\begin{aligned} \mathcal{C}(x_0,U_n) = \mathcal{C}(x_0, W_n') \le 2^{-n}. \end{aligned}$$

Let

$$\begin{aligned} F_n = (\mathcal{K}_{Q} {\setminus } U_n)\cap V. \end{aligned}$$

Then it holds that $\overline{F_n} = \mathcal{K}_{Q} {\setminus } U_n$. For any $x \in \overline{F_n}$ and $i,j> N >n$, we have

$$\begin{aligned} |\rho _i(x)-\rho _j(x)| \le \sum _{k=N+1}^{\infty }|\rho _{k+1}(x)-\rho _{k}(x)| \le 2^{-N}, \end{aligned}$$

so that $\{ \rho _i\}_{\overline{F_n}}$ converges uniformly to a continuous function $f_n$ on $\overline{F_n}$. Since $\rho _i$ converges pointwise to f on V, we see that $f_n=f$ on $F_n$. Since $f_n=f_m$ on $\overline{F_m}$ if $m < n$, we can define a function $\tilde{f}$ on $\cup \overline{F_n}$ by $\tilde{f}(x)=f_n(x)$ for $x \in \overline{F_n}$. Letting $\tilde{f}=0$ on $\cap U_n$, we get a quasicontinuous function $\tilde{f}$ on $\mathcal{K}_{Q}$ which coincides with f on V. $\square$

Theorem A.19

Let (G, r) be a nonparabolic network and Q a dense family of bounded functions of L(G, r). Given a countable family of functions $f_{i}$ in L(G, r) which extend to quasicontinuous functions $\tilde{f}_i$ on $\mathcal{K}_{Q}$, there exists an increasing sequence $\{F_n\}$ of subsets of V satisfying the following properties:

• (i) $V=\cup _n F_n$, $\lim _{n \rightarrow \infty } \mathcal{C}(x_0,\mathcal{K}_{Q} {\setminus } \overline{F_n})=0$, and $f_{i|\overline{F_n}} \in C^0(\overline{F_n})$ for all i and n;

• (ii) $\cup \overline{F_n}^R \cap \varDelta (G,r)$ is dense in $\varDelta (G,r)$;

• (iii) $\tilde{f}_{i} \circ \pi _{Q}= \overline{f_{i}}$ on $\cup _n \overline{F_n}^R$, $\mathop {\mathcal{C}\mathrm{-ess} \sup }_{\partial \mathcal{K}_{Q}} \tilde{f}_i= \sup _{ \varDelta (G,r)} \overline{f_{i}}$, $\mathop {\mathcal{C}\mathrm{-ess}\inf }_{\partial \mathcal{K}(G,r)} \tilde{f}_i = \inf _{ \varDelta (G,r)} \overline{\phi _{i}}$ for each $f_i$.

Proof

For each $f_{i}$, we take open subsets $U_n^{(i)}$$(n=1,2,\ldots )$ in $\mathcal{K}_{Q}$ such that $\mathcal{C}(x_0,U_n^{(i)}) \le 2^{-n-i-1}$ and $f_{ i |\mathcal{K}_{Q} {\setminus } U_n^{(i)}} \in C^0(\mathcal{K}_{Q}{\setminus } U_n^{(i)})$. Let $U_n'= \cup _{k \ge n} (\cup _{i}U_k^{(i)})$. Then it follows from Proposition A.12 (iii) that $\mathcal{C}(x_0, U_n') \le 2^{-n-1}$. Define open subsets $U_n$ of $\mathcal{K}_{Q}$ by

$$\begin{aligned} U_n= U_n'\cup \{ \xi \in \partial \mathcal{K}_{Q} | \text{ there } \text{ exists } \text{ a } \text{ neighborhood } \, W \, \text{ of } \ \xi \, \text{ such } \text{ that } \ W \cap V \subset U_n' \}. \end{aligned}$$

Then it follows from the definition of the capacity that $\mathcal{C}(x_0,U_n)=\mathcal{C}(x_0,U_n') \le 2^{-n-1}$. Let

$$\begin{aligned} F_n = (\mathcal{K}_{Q} {\setminus } U_n) \cap V. \end{aligned}$$

Then $\overline{F_n}^K=\mathcal{K}_{Q} {\setminus } U_n$ and $\cup F_n=V$.

Now for each $U_n$, we take a function $u_n \in \mathcal{A}_{x_0;U_n}$ in such a way that $\Vert du_n \Vert _* \le 2^{-n}$. Then we decompose the function $u_n$ as follows:

$$\begin{aligned} u_n = h_n + g_n, \end{aligned}$$

where $h_n \in L(G,r)$ satisfies $h_n(x_0)=0$ and $\mathcal{L}_r h_n=0$ on $V {\setminus } \{ x_0 \}$, and $g_n \in L_0(G,r)$ satisfies $g_n(x_0)=0$ (see Theorem 3.1). We observe here that for a point $\xi$ of $\mathcal{R}(G,r)$, $\xi$ stays outside $\cup _n \overline{F_n}^R$ if and only if $\xi \in \partial \mathcal{R}(G,r)$ and for any n, there exists a sequence $\{y_i\}$ in $U_n \cap V$ such that $y_i$ converges to $\xi$ in $\mathcal{R}(G,r)$. This shows that for a point $\xi$ in $\mathcal{R}(G,r) {\setminus } \cup _n \overline{F_n}^R$, we have $\overline{u_n}(\xi )=\lim _{i \rightarrow \infty } u_n(y_i) \ge 1$, since $u_n(y_i) \ge 1$. Thus we see that for any n, $\overline{u_n} \ge 1$ on $\mathcal{R}(G,r) {\setminus } \cup _n \overline{F_n}^R$. This implies that $\overline{h_n} = \overline{u_n} \ge 1$ on $\varDelta (G,r) {\setminus } \cup _n \overline{F_n}^R$ if it is not empty.

To show that $\cup \overline{F_n}^R \cap \varDelta (G,r)$ is dense in $\varDelta (G,r)$, we suppose contrarily that there exists an open subset W in $\mathcal{R}(G,r)$ such that $\varDelta (G,r) \cap W \not = \emptyset$ and $\cup \overline{F_n}^R \cap \varDelta (G,r) \cap W =\emptyset$. Then for $\xi \in \varDelta (G,r) \cap W$, we take a continuous function $\rho$ on $\varDelta (G,r)$ in such a way that $|\rho | \le 1/2$, $\rho (\xi )=1/2$, $\rho =-1/2$ on $\varDelta (G,r) {\setminus } W$. Then we can find an r-harmonic function $f \in L(G,r)$ such that $|\overline{f}-\rho | < 1/4$ on $\varDelta (G,r)$. Let h be an r-harmonic function in L(G, r) such that $h-\max \{ f,0\} \in L_0(G,r)$. Then it holds that $0 \le \overline{h} \le 3/4$, $1/4 \le \overline{h}(\xi ) \le 3/4$, and $\overline{h}=0$ on $\varDelta (G,r) {\setminus } W$. Thus $\overline{h_n} \ge \overline{h}$ on $\varDelta (G,r)$. Then by Theorem 3.4, we see that $h_n \ge h$ on V. Since

$$\begin{aligned} \lim _{n \rightarrow \infty } W^*(dh_n) \le \lim _{n \rightarrow \infty } W^*(du_n) =0, \end{aligned}$$

we can conclude that $h_n$ tends to zero as $n \rightarrow \infty$. This shows that h vanishes everywhere on V. This is absurd. Thus $\cup \overline{F_n}^R \cap \varDelta (G,r)$ must be dense in $\varDelta (G,r)$.

Now we suppose that $f_{i|V}$ extends to a continuous function $\overline{f_{i}}$ on $\mathcal{R}(G,r)$. Then for any $\xi \in \overline{F_n}^R$, if a sequence $\{y_j \}$ in $F_n$ converges to $\xi$ in $\mathcal{R}(G,r)$, then it converges to $\pi _{\varGamma }(\xi )$ in $\mathcal{K}_{Q}$, so that we have

$$\begin{aligned} \overline{f_{i}}(\xi )= \lim _{y_j \rightarrow \xi \in F_n} f_{i}(y_j)=\tilde{f}_{i}(\pi _{\varGamma }(\xi )). \end{aligned}$$

This shows that $\tilde{f}_{i} \circ \pi _{Q}= \overline{f_{i}}$. Moreover we have

$$\begin{aligned} \mathcal{C}\mathrm{-ess~sup}\{ \tilde{f}_i(\xi ) ~|~ \xi \in \partial \mathcal{K}_{Q} \}= & {} \mathcal{C}\mathrm{-ess~sup}\{ \tilde{f}_i(\xi ) ~|~ \xi \in \varDelta _{Q} \} \\= & {} \mathcal{C}\mathrm{-ess~sup}\{ \tilde{f}_{i}(\xi ) ~|~ \xi \in \cup \overline{F_n} \cap \varDelta _{Q} \} \\= & {} \sup _n \max \{ f_{i}(\xi ) ~|~ \xi \in \overline{F_n} \cap \varDelta _{Q} \} \\= & {} \sup _n \max \{ \overline{f_{i}}(\eta ) ~|~ \eta \in \overline{F_n}^R\cap \varDelta (G,r) \} \\= & {} \sup \{ \overline{f_{i}}(\eta ) ~|~ \eta \in \varDelta (G,r) \}, \end{aligned}$$

where the last equality follows from the assertion (ii). This completes the proof of Theorem A.19. $\square$

As a result of Theorem A.19, we can prove a comparison principle for functions in L(G, r) on the r-Laplacian $\mathcal{L}_r$.

Corollary A.20

Let (G, r) and Q be as in Theorem A.19. For functions $u, v \in L(G,r)$, $u \ge v$ on V if $\varDelta _{r} u \ge \varDelta _{r}v$ on V and $\tilde{u} \ge \tilde{v}$ quasi-everywhere on $\varDelta _{Q}$.

Lemma A.21

Let $\{ U_n \}_{n=1}^{\infty }$ be a decreasing sequence of open subsets of $\mathcal{K}_{Q}$ such that $\mathcal{C}(x_0,U_n) \le 2^{-2n}$, where $x_0$ is a fixed point of V. Then there exists a decreasing sequence of nonnegative functions $\{ v_j \}$ in L(G, r) such that $\Vert dv_j \Vert _* \le 2^{-j}$ and $\liminf _{y \in V \rightarrow x} v_j(y) \ge n-j$ for all $x \in U_n$ whenever $n > j$.

Proof

Since $\mathcal{C}(x_0,U_n) \le 2^{-2n}$, there is a nonnegative function $f_n \in \mathcal{A}_{x_0;U_n}$ such that $\Vert df_n \Vert _* \le 2^{-n}$. Let $v_j=\sum _{k=j+1}^{\infty } f_k$, $j=1,2,\ldots$. For $n>j$, $v_j \ge f_{j+1}+\cdots +f_{n}$, so that $v_j(x) \ge n-j$ if $x \in U_n \cap V$ and $\liminf _{y \in V \rightarrow x} v_j(y) \ge n-j$ if $x \in U_n\cap \partial \mathcal{K}_{Q}$. This completes the proof of Lemma A.21. $\square$

When $q=0$, we can prove the following

Theorem A.22

Let (G, r) be a nonparabolic network and Q a dense family of bounded functions of L(G, r). A quasicontinuous function $\phi$ on $\partial \mathcal{K}_{Q}$ which extends to a function $f \in L(G,r)$ is resolutive for the equation : $\mathcal{L}_ru=0$ on V, and the Perron solution $\mathcal{H}_0\phi$ of $\phi$ coincides with Hf.

Proof

To show that $\mathcal{H}_0\phi \ge Hf$ on V, in view of Theorem A.19, we find an increasing sequence of subsets $F_n$ of V in such a way that $V=\cup F_n$, $\lim _{n \rightarrow \infty } \mathcal{C}(x_0,U_n)=0$, where we put $U_n=\mathcal{K}_{Q} {\setminus } \overline{F_n}$, $f_{|\overline{F_n}} \in C^0(\overline{F_n})$, $Hf_{|\overline{F_n}} \in C^0(\overline{F_n})$, and $f=Hf$ on $\cup \overline{F_n} \cap \mathcal{K}_{Q}$. By Lemma A.21, we have a nonincreasing sequence $\{ \psi _n \}$ of nonnegative quasicontinuous functions on $\mathcal{K}_{Q}$ such that $\psi _{n|V} \in L(G,r)$, $\Vert d\psi _n\Vert _* \le 2^{-n}$, and $\liminf _{y \in V \rightarrow x} \psi _{j+m}(y) \ge m$ for all $x \in U_{j+m}$.

We assume first that f is nonnegative, so that Hf is also nonnegative. Let $f_j=Hf + \psi _j$ and let $u_j =S^+f_j$ be as in Theorem 3.5$(q=0)$ which is the unique $W^*$-minimizer in $\{ u \in L(G,r) ~|~ u-f_j \in L_0(G,r), \ u \ge f_j\}$. We are given a point $\xi \in \partial \mathcal{K}_{Q}$. Let m and $\varepsilon$ be respectively a positive integer and a positive number. If $\xi \in U_{j+m}$, then $\liminf _{y \in V \rightarrow \xi } f_j(y) \ge \liminf _{y \in V \rightarrow \xi } \psi _{j+m}(y) \ge m$. If $\xi \in \overline{F_{j+m}}$, then we take a neighborhood W of $\xi$ such that $|Hf(y)-Hf(\xi )|=|Hf(y)-\phi (\xi )| < \varepsilon$ for all $y \in W \cap \overline{F_{j+m}}$, so that we have $f_j(y) \ge Hf(y) \ge Hf(\xi ) -\varepsilon = \phi (\xi )-\varepsilon$ for all $y \in W \cap \overline{F_{j+m}}$. For $y \in W \cap U_{j+m}$, then $f_j(y) \ge \psi _j(y) \ge m$. Thus we see that $\liminf _{y \in V \rightarrow \xi } f_j(y) \ge \min \{ f(x)-\varepsilon ,m \}$. Letting m tend to $+\,\infty$ and $\varepsilon$ go to 0, we get $\liminf _{y \in V \rightarrow \xi }f_j(y) \ge \phi (\xi )$. Since $u_j \ge f_j$, we see that $u_j \in \overline{\mathcal{F}}_{\phi }$ for any j, and hence $u_j \ge \overline{\mathcal{H}}_0 \phi$ on V.

We note here that $f_j$ converges to Hf in L(G, r) as $n \rightarrow \infty$ and hence $u_j$ converges to Hf in L(G, r) (see Theorem 3.9). Therefore it follows from Lemma 2.4 that $Hf \ge \overline{\mathcal{H}}_0 \phi$ on V.

Now we assume that f is bounded from below, that is, $f \ge - \alpha$ for some constant $\alpha$. Then we apply the above estimate to $f+\alpha$, and obtain $H(f+\alpha )\ge \overline{\mathcal{H}}_0 (\phi +\alpha )$. Since $\overline{\mathcal{H}}_0(\phi +\alpha ) = \overline{\mathcal{H}}_0(\phi )+\alpha$ and $H(f+\alpha )=H(f)+\alpha$, we see that $Hf \ge \overline{\mathcal{H}}_0 \phi$ on V.

Consider a function $\max \{ f, -m\}$, where m is a positive integer, and apply The above estimate so that we obtain $H(\max \{ f,-m\})\ge \overline{\mathcal{H}}_0 (\max \{ \phi , -m\})$. Letting m tend to infinity, $\max \{ f, -m\}$ converges to f in L(G, r), and hence $H(\max \{ f, -m\})$ converges to Hf in L(G, r) (see Theorem 3.1 (ii)), so that $H(\max \{ f, -m\})$ pointwise converges to Hf in V. We thus conclude that $Hf\ge \overline{\mathcal{H}}_0 \phi$ on V, since $\overline{\mathcal{H}}_0 \phi \le \overline{\mathcal{H}}_0 (\max \{ \phi , -m\})$. Since $\underline{\mathcal{H}}_0 \phi =-\overline{\mathcal{H}}_0 (-\phi )$ and $Hf=-H(-f)$, we see that $Hf \le \underline{\mathcal{H}}_0 \phi$ on V, thus we can conclude that $Hf=\overline{\mathcal{H}}_0 \phi =\underline{\mathcal{H}}_0 \phi$. This completes the proof of Theorem A.22. $\square$

Theorem A.23

Let (G, r) be a nonparabolic network and Q a family of bounded functions of L(G, r) such that Q is dense in L(G, r) and Q admits a countable subfamily which is dense there with respect to the uniform topology. For a function $f \in L(G,r)$ which extends to a quasicontinuous function $\tilde{f}$ on $\mathcal{K}_{Q}$, we have

$$\begin{aligned} \lim _{i \rightarrow \infty } f(c(i))= \tilde{f}(c(\infty )) \end{aligned}$$

for almost all paths $c \in \mathcal{P}_{\infty }$.

Proof

We take an increasing family of subsets $F_n$ of V in such a way that $\lim _{n \rightarrow \infty } \mathcal{C}(x_0,U_n)=0$, where $U_n=\mathcal{K}_{Q}{\setminus } \overline{F_n}$, and $f_{|\overline{F_n}} \in C^0(\overline{F_n})$. Moreover in view of Lemma 510, we find a nonincreasing sequence of quasicontinuous functions $\psi _j:\mathcal{K}_{Q} \rightarrow \overline{ {R} }$ such that $\psi _{j|V} \in L(G,r)$, $\psi _j(x_0)=0$, $\lim _{j \rightarrow \infty } \Vert d\psi _j\Vert _*=0$, and $\liminf _{y \in V \rightarrow x} \psi _j(y) \ge m$ for all $x \in U_{j+m}$, where $x_0$ is a fixed point of V. Let $\mathcal{Q}$ be the set of paths c in $\mathcal{P}_{x_0,\infty }$ such that for any m, infinitely many c(i) belongs to $U_m$. Given positive integers j, m, we find c(n) in $U_{j+m}$ for any $c \in \mathcal{Q}$ so that

$$\begin{aligned} L_{\Vert d \psi _j \Vert _*} c = \sum _{i=0}^{\infty } |\psi _j(c(i+1))-\psi _j(i)| \ge \psi _j(c(n))-\psi _j(x_0) \ge m. \end{aligned}$$

Thus it follows that

$$\begin{aligned} \lambda (\mathcal{Q}) \ge \frac{ \inf \{ L_{ \Vert d\psi _j\Vert _* } c ~|~ c \in \mathcal{Q} \}}{ \Vert d\psi _j\Vert _* } \ge \frac{m}{ \Vert d\psi _j \Vert _*}. \end{aligned}$$

The right-hand side tends to infinity as $j \rightarrow \infty$. Thus we see that $\lambda (\mathcal{Q})=\infty$. Now for $c \in \mathcal{P}_{x_0,\infty } {\setminus } \mathcal{Q}$, c(n) stays in $F_m$ for some m and all but finite n. This implies that $c(\infty ) \in \overline{F_m} \cap \partial \mathcal{K}_{Q}$. Hence we get

$$\begin{aligned} \lim _{n \rightarrow \infty } f(c(n))= \tilde{f}(c(\infty )), \quad c(\infty ) \in \cup \overline{F_m} \cap \partial \mathcal{K}_{\varGamma }. \end{aligned}$$

This completes the proof of Theorem A.23. $\square$

As a result of Theorems A.19 and A.23, we have the following

Corollary A.24

For $f \in L(G,r)$, the following are equivalent to each other:

• (i) $f \in L_0(G,r)$;

• (ii) $\lim _{i\rightarrow \infty } f(c(i))=0$ for almost all paths $c \in \mathcal{P}_{\infty }$;

• (iii) $\tilde{f}$ vanishes quasi-everywhere on $\varDelta _{Q}$,