Variational Convergence of Finite Networks

66 KASUE
I ðxÞ ¼
iðxÞ
p ffiffiffiffi ; x 2 K:
i
ffiffiffiffiffi
RE
Then I
p
gives rise to an isometric embedding of ðK;
2 Þ into ‘ . In fact, it satisfies the following properties:
RGðx; yÞ ¼kI ðxÞ I ðyÞk 2 ‘ 2;
g ðx; yÞ ¼ðI ðxÞ; I ðyÞÞ ‘ 2; x; y 2 K:
Now we introduce another embedding of K into ‘ 2 as follows:
J ðxÞ ¼ðe i=2
iðxÞÞ; x 2 K:
This embedding has the following properties:
kJ ðxÞ J ðyÞk 2 XN 1
‘ 2 ¼
i¼1
e i ð iðxÞ iðyÞÞ 2
¼ p ð1; x; xÞ 2p ð1; x; yÞþp ð1; y; yÞ; x; y 2 K;
where p ðt; x; yÞ denotes the kernel function of the semigroup expð tL Þ generated by L . Let
S ðx; yÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
p ð1; x; xÞ 2p ð1; x; yÞþp ð1; y; yÞ;
x; y 2 K:
Then S : K K !½0; þ1Þ is a distance on K and J is an isometric embedding of the metric space ðK; S Þ into ‘ 2 .It
is easy to see that
0 S ðx; yÞ REðx; yÞ 1=2 ; x; y 2 K:
Given a function u 2 D½EKŠ, we simply denote by eu the minimizer HK;u in the set fv 2 D½EXŠ jvjK ¼ ug (see Section
1). Correspondingly we denote by eI , eJ , ep and eS respectively the continuous extentions of the embeddings I , J ,
the kernel function p and the distance S . We note that the extended maps eI and eJ may not be injective on X and
hence eS may be degenerate somewhere on X.
4.2 Now consider the same situation as in Theorem 3.4 and let fðXn; RnÞg, ðY; RY; EÞ, fn : Xn ! Y and hn : Y ! Xn be as
before. Let us take a sequence of Radon measures n on Xn with supp n ¼ Xn such that the push-forward measure
fn n weakly converges to a Radon measure on Y as n !1. Let L be the Laplacian in L2ðK; Þ, where
K ¼ supp , associated with the induced form EK on CðKÞ and ep the continuous extension of the kernel function p of
the semigroup generated by L .
To state a result on spectral convergence, we need some definitions. We define a linear map Tn : D½EKŠ!L2ðXn; nÞ
by TnðuÞ ¼ fn ~uð¼ fn HK;uÞ; u 2 D½EKŠ. A sequence of functions un 2 L2ðXn; nÞ is said to L2 strongly (resp. L2 weakly) converge to a function u 2 L2ðK; Þ if there exists a sequence of functions vi in D½EKŠ such that
limi!1 lim supn!1 kTnðviÞ unkL2ðXn; nÞ ¼ 0 (resp. if, for every v 2 L2ðK; Þ and any sequence of vn 2 L2ðXn; nÞ
which L2-strongly converges to v, limn!1ðun; vnÞL2ðXn; nÞ ¼ðu; vÞL2ðK; Þ) (cf. [21]).
Theorem 4.1. Under the conditions above, the following assertions hold:
(i) Let fung be a sequence of functions un 2 L2ðXn; nÞ with supn kunkL2ðXn; nÞ < þ1 which L2-weakly converges to a
function u 2 L2ðK; Þ. Then for any >0, wn ¼ R n; un uniformly converges to w ¼ðR ; uÞ and Enðwn; wnÞ tends
to EKðw; wÞ as n !1, where R n; and R ; are respectively the resolvents of the Laplacians L and L .
n
(ii) The maps fn and hn approximate the kernel functions p and p in such a way that
n
lim
sup
n!1 t>0;x;y2Xn
e ðtþ1=tÞ jp n ðt; x; yÞ ep ðt; fnðxÞ; fnðyÞÞj ¼ 0
lim
n!1 sup e
t>0;a;b2Y
ðtþ1=tÞ jp ðt; hnðaÞ; hnðbÞÞ n ep ðt; a; bÞj ¼ 0
lim eS ða; fn hnðaÞÞ ¼ 0:
n!1 sup
a2Y
(iii) For each i, the i-th eigenvalue of L converges to that of L n
then the i-th eigenvalue diverges to infinity as n !1.
in the sense that if N ¼ #K < þ1 and i N,
Example Let ðV; E; rÞ be a finite network and E the associated form on ‘ðVÞ. Given a sequence of positive functions
on V, we have a sequence of Morkovian forms E on ‘ðVÞ defined by
n
n
E ðu; vÞ ¼Eðu= n n; v= nÞ; u; v 2 ‘ðVÞ:
Suppose that n converges to a nonnegative function 1 on V. Let n (resp. 1) be measures on V given by
nðuÞ ¼ P
x2V uðxÞ nðxÞ2 (resp. 1ðuÞ ¼ P
x2K uðxÞ 1ðxÞ2Þ, where K stands for the support of 1. Then as n !1,
the Laplacian L of E in L n 2ðV; nÞ converges to the Laplacian L of the induced form E 1 K in L2ðK; 1Þ in the sense