Variational Convergence of Finite Networks

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60 KASUE uðxÞ ¼ 1 Z ud ðKÞ K þ EKðg ðx; Þ; uÞ; u 2 D½EKŠ; x 2 K: Applying this to the functions gzðx; Þ, we can verify the following identity: gzðx; yÞ ¼g ðz; zÞ g ðz; xÞ g ðz; yÞþg ðx; yÞ; x; y; z 2 K: ð2Þ Since RGðx; yÞ ¼gxðy; yÞ, direct computations show the following Theorem 2.3. The effective resistance on K is given by RGðx; yÞ ¼g ðx; xÞ 2g ðx; yÞþg ðy; yÞ; x; y 2 K: Now by taking a Radon measure such that supp ¼ G and using (2), we can derive the first identity of Theorem 2.1, from which the remaining one follows. Let g ðx; yÞ :¼ g ðx; yÞ C ; x; y 2 K; and Z G uðxÞ ¼ K g ðx; yÞuðyÞ d ðyÞ; u 2 L 2 ðK; Þ; where C is a positive constant given by C ¼ 1 Z 1 g ðx; yÞ d ðyÞ ¼ ðKÞ K 2 ðKÞ2 Z Z RGðy; zÞ d ðyÞd ðzÞ: K K Then G turns out to be the Green operator of L , that is, it satisfies I ¼ H þ L G on L2ðK; Þ; I ¼ H þ G L on D½L Š; H G ¼ G H ¼ 0; where we put H u ¼ 1 Z ud : ðKÞ K Let N ¼ #K þ1and i ði ¼ 0; 1; ...; N 1Þ the i-th eigenvalue of L , where 0 ¼ 0. We take a complete orthonormal system of eigenfunctions i with eigenvalue i in L2ðK; Þ. Then the Green kernel g is given by g ðx; yÞ ¼ XN 1 i¼1 1 i iðxÞ iðyÞ: Since RGðx; yÞ ¼g ðx; xÞ 2g ðx; yÞþg ðy; yÞ, as a result of Theorem 2.3, we obtain Corollary 2.4. It holds: In particular, one has RGðx; yÞ ¼ Z K Z K XN 1 i¼1 1 ð iðxÞ iðyÞÞ 2 ; x; y 2 K: i XN 1 1 RGðx; yÞd ðxÞd ðyÞ ¼2 ðKÞ : i¼1 i From the last identity of the corollary, it is easy to see that i where DðK; RGÞ ¼supfRGðx; yÞ jx; y 2 Kg. 2i DðK; RGÞ ; ðKÞ i ¼ 1; 2; ...; ð3Þ 2.3 Now we consider the case where a closed subset K of a compact metric graph G consists of a finite number of points fx1; ...; xNg ðN < þ1Þ. We define N functions ui ði ¼ 1; ...; NÞ on K by uiðxjÞ ¼ ij. Then ui 2 D½EKŠ. Let cij ¼ EKðui; ujÞ. For every fixed k, 1 k N, the definition of the Green functions reads
Variational Convergence of Finite Networks 61 X N ‘¼1 gxk ðxi; x‘Þc‘j ¼ ij; i 6¼ k; j 6¼ k; ð4Þ that is, defining ðN 1Þ ðN 1Þ matrices Ck and Gk respectively by Ck ¼ðcijÞ and Gk ¼ðgxkðxi; i 6¼ k; j 6¼ k, we have xjÞÞ, where GkCk ¼ IN 1; k ¼ 1; ...; N; where IN 1 stands for the unit matrix. By (4), we get XN XN k¼1 i;j¼1 gxk ðxi; xjÞcij ¼ NðN 1Þ: ð5Þ Since gxkðxi; xjÞ ¼ 1 2 ðRGðxi; xkÞþRGðxk; xjÞ RGðxi; xjÞÞ and PN j¼1 cij ¼ 0, we thus obtain the following: 1 X 2 N RGðxi; xjÞEK ðui; ujÞ ¼N 1 ð6Þ When K is the set of all vertecies V, cij ¼ 1=rði; jÞ ði 6¼ jÞ, and hence the identity reads 1 2 i;j¼1 X N i;j¼1;i 6¼ j RGðxi; xjÞ rði; jÞ ¼ N 1: This is a classical result due to R. M. Foster (cf. [7, 27]). P c N Now taking a canonical measure on K defined by ¼ i¼1 the following xi and using the identities of Corollary 2.4, we have Corollary 2.5. Under the conditions as above, one has and max 1 k n 1 2k ðn 1Þ c kð Þ 2 c n 1ð Þ max xi;x j2K RGðxi; xjÞ RGðxi; xjÞ; xi; xj 2 K; xi 6¼ xj: 2 c 1ð Þ Remark The second inequality in this corollary is proved in [2]. 2.4 Let V be a finite set, V ¼fx1; ...; xNg, and consider a nonnegative quadratic form E on the space ‘ðVÞ of functions on V such that Eðu; uÞ ¼0 if and only if u is constant. Let REðx; yÞ ¼maxfjuðxÞ uðyÞj2 =Eðu; uÞ ju 2 ‘ðVÞ; Eðu; uÞ > 0g ðx; y 2 VÞ. Then the arguments and the results as in the previous subsections 2.1 through 2.3 are valid, except the nonnegativity of the Green functions gzðx; yÞ of E; this relies on the maximum principle under the Markov property. In fact, if we assume that the form satisfies the Markov property, then we see that cij ¼ Eðui; ujÞ 0 for all i; j ¼ 1; ...; N with i 6¼ j, where fu1; ...; uNg is a canonical basis of ‘ðVÞ described in subsection 2.3. In this case, by letting E ¼ ffx; yg V j cij < 0g and rði; jÞ ¼ 1=cij, we obtain a network ðV; E; rÞ. In particular, Theorem 2.1 holds true without the Markov property, and hence we have the following Theorem 2.6. Let V be a finite set. Let E ð ¼ 1; 2Þ be nonnegative quadratic forms on ‘ðVÞ such that E ðu; uÞ ¼0 if and only if u is constant. Then E1 ¼ E2 if and only if RE1 ¼ RE2 . Given a subset K of V, we denote as before by EK a form on ‘ðKÞ defined by EKðu; uÞ ¼inffEðv; vÞ jv 2 ‘ðVÞ; vjK ¼ ug. Then we have RE ðx; yÞ ¼REðx; yÞ for all x; y 2 K. As a result of Theorem 2.6, we have the following K Corollary 2.7. Let K be a subset of a finite set V. Let E and F be respectively quadratic forms on ‘ðVÞ and ‘ðKÞ as above. Then F ¼ EK if and only if RF ðx; yÞ ¼REðx; yÞ for all x; y 2 K. Theorem 2.6 and Corollary 2.7 are respectively proved in [20], Theorem 2.1.12 and Corollary 2.1.13, under the condition of the Markov property, and they affermatively answer an question raised there. 2.5 In this part, we recall some definitions and results on resistance forms and resistance metrics from [20], Chap. 2. Let X be a set and ðE; D½EŠÞ a pair of a linear subspace D½EŠ of the space ‘ðXÞ of all functions on X and a nonnegative quadratic form E on D½EŠ. We call such a pair ðE; D½EŠÞ a resistance form on X if it satisfies the following conditions (RF-i) through (RF-v): (RF-i) Eðu; uÞ ¼0 if and only if u is constant on X. (RF-ii) Let be an equivalence relation on D½EŠ defined by u v if and only if u v is constant on X. Then ðD½EŠ= ; EÞ is a Hilbert space.
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