Variational Convergence of Finite Networks

62 KASUE
(RF-iii) For any finite subset V of X and for any u 2 ‘ðVÞ, there exists v 2 D½EŠ such that vjV ¼ u.
(RF-iv) For any x; y 2 X, supfjuðxÞ uðyÞj2 =Eðu; uÞ ju 2 D½EŠ; Eðu; uÞ > 0g is finite. The supremum is denoted by
REðx; yÞ, which will be called the effective resistance between the points x and y.
(RF-v) If u 2 D½EŠ, then u 2 D½EŠ and Eðu; uÞ Eðu; uÞ, where u is defined by uðxÞ ¼1 if uðxÞ 1, uðxÞ ¼uðxÞ if
0 < uðxÞ < 1, and uðxÞ ¼0 if uðxÞ 0.
We use RF ðXÞ to denote the collection of resistance forms on X. Given ðE; D½EŠÞ 2 RF ðXÞ and a finite subset V of
X, let
EVðu; uÞ ¼inffEðv; vÞ jv 2 D½EŠ; vjV ¼ ug; u 2 ‘ðVÞ:
Then ðEV ;‘ðVÞÞ 2 RF ðVÞ and RE ¼ RE on V. This implies that if ðE; D½EŠÞ 2 RF ðXÞ, then RE induces a metric on X,
V pffiffiffiffiffi called the resistance metric associated with the resistance form E on X. Obviously RE is also a distance on X; indeed,
so is it without the Markov property (RF-v), if we assume instead of property (RF-iii) that REðx; yÞ > 0 for all x; y 2 X
with x 6¼ y.
A function R : X X !½0; þ1Þ is by definition a resistance metric on X if, for any finite subset V X, there exists
a resistance form EV on ‘ðVÞ such that Rðx; yÞ ¼REV ðx; yÞ for all x; y 2 V, where REV is the effective resistance with
respect to the form EV. The collection of resistance metrics on X is denoted by RMðXÞ. Each ðE; D½EŠÞ 2 RF ðXÞ is
associated with RE 2 RMðXÞ.
pffiffiffiffiffi This correspondence is in fact bijective.
We note that, when ðX; REÞ
is separable, E can be thought to be a limit of finite dimensional forms; in fact, given an
increasing sequence fVmg of finite subsets of X such that [mVm is dense in X, a function u on X belongs to D½EŠ if and
only if limm!1 EVmðujVm ; ujVmÞ < þ1 and in this case, Eðu; uÞ ¼limm!1 EVmðujVm ; ujVmÞ, where E is the induced
Vm
form on ‘ðVmÞ as above.
Given ðE; D½EŠÞ 2 RF ðXÞ, ifX is the completion of X with respect to the resistance metric, then any u 2 D½EŠ is
naturally extended to a continuous function on X. Using this extention, E is regarded as the collection of functions on X.
Then ðE; D½EŠÞ is a resistance form on X and the resistance metric associated with ðE; D½EŠÞ on X is the natural
extension of the resistance metric associated with ðE; D½EŠÞ on X.
2.6 Before proceeding to the next section, we consider a wider class of quadratic forms for our purpose. Let ðE; D½EŠÞ
be a nonnegative quadratic form on a set X and
juðxÞ uðyÞj2
REðx; yÞ ¼sup j u 2 D½EŠ; Eðu; uÞ > 0 ; x; y 2 X:
Eðu; uÞ
pffiffiffiffiffi We suppose that 0 < REðx; yÞ < þ1 for all x; y 2 X with x 6¼ y. Then RE
pffiffiffiffiffi induces a distance on X, and the domain
D½EŠ is a subspace of CðX; REÞ.
Moreover we suppose that for a point o 2 X, a quadratic form defined by Eðu; vÞþ
uðoÞvðoÞ provides a complete inner product on D½EŠ, that is, ðD½EŠ; E þ 2 oÞ is a Hilbert space (this holds for any o 2 X
because of the assumption above).
pffiffiffiffiffi Now we suppose that the metric space ðX; REÞ
is separable. Then we are able to verify the following assertions:
pffiffiffiffiffi (i) When the form E is regarded as a functional on the space of continuous functions CðXÞ on X ¼ðX; REÞ
by
letting EðuÞ ¼Eðu; uÞ if u 2 D½EŠ and EðuÞ ¼þ1otherwise, pffiffiffiffiffi the functional E : CðXÞ !½0; þ1Š is lower semicontinuous
with respect to the uniform topology of CðX; REÞ.
(ii) Given a closed subset K of X, let
D½EKŠ¼fu 2 CðKÞ ju ¼ vjK 9v 2 D½EŠg;
EKðu; uÞ ¼inffEðv; vÞ jv 2 D½EŠ; vjK ¼ ug:
Then ðEK ; D½EKŠÞ is a quadratic form on CðKÞ with the same properties as E on X. In fact, there exists a unique
minimizer HK;u and the correspondence u 2 D½EKŠ!HK;u 2 D½EŠ is linear and injective. The minimizer is
characterized as a function h 2 D½EŠ which coincides with u on K and satisfies Eðh; vÞ ¼0 for all v 2 D½EŠ vanishing on
K. In addition, we see that supfjuðxÞ uðyÞj2 =EKðu; uÞ ju 2 D½EKŠ; EKðu; uÞ 6¼ 0g ¼Rðx; yÞ for all x; y 2 K.
(iii) If 1 2 D½EŠ and Eð1; 1Þ ¼0, then Theorem 2.1 is valid for this situation. If, in addition, E satisfies the Markov
property, then so does EK , and hence D½EKŠ\L1 is a subalgebra of CðKÞ containing the unit elememt 1 and separating
points of K;thus in this case, ðE; D½EŠÞ is a resistance form on X.
(iv) Let be a -finite Borel measure on X and K the support of . Let H be the L2-closure of D½EKŠ\L2ðK; Þ and
L the self-adjoint operator associated with the closed form EK in H with domain D½EKŠ\L2ðK; Þ. Given a positive
number , we use R ; to denote the resolvent operator ðL þ IÞ 1 of L . Suppose that
Z
ðKÞ < þ1; REðo; xÞ d ðxÞ < þ1
K
for some o 2 X (and hence any o). Then D½E K Š L 2 ðK; Þ and the embedding is compact with respect to the norm
ðE K ðu; uÞþ R u 2 d Þ 1=2 of D½E K Š. If, in addition, 1 2 D½EŠ and Eð1; 1Þ ¼0, then Theorem 2.3, Corollary 2.4 and
Corollary 2.5 hold true.