Variational Convergence of Finite Networks

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58 KASUE L2ð½0; rðeÞŠÞ. Then a canonical energy form EG on D½EGŠ is given by Z Z rðeÞ Let EGðu; vÞ ¼ G u 0 v 0 dsG ¼ X e2E 0 u 0 ðsÞv 0 ðsÞds; u; v 2 D½EGŠ: juðxÞ uðyÞj2 RGðx; yÞ ¼sup j u 2 D½EGŠ; EGðu; uÞ 6¼ 0 ; x; y 2 G: EGðu; uÞ The number RGðx; yÞ is called the effective resistance between the points x and y of G. The form EG has the following properties. (i) EG is symmetric and positive semi-definite, and EGðu; uÞ ¼0 if and only if u is constant. (ii) The form EGðu; vÞþuðoÞvðoÞ on D½EGŠ, where o is a fixed point of G, is closed in the sense that it provides a complete inner product on D½EGŠ. (iii) For a sequence of functions un in D½EGŠ whose energies EGðun; unÞ are uniformly bounded, if un uniformly converges to a function u in CðGÞ as n !1, then we have EGðu; vÞ ¼ lim n!1 EGðun; vÞ for all v 2 D½EGŠ. As a result, the energy form EG is lower semi-continuous on CðGÞ, that is, for a sequence of functions un 2 CðGÞ which uniformly converges to a function u 2 CðGÞ, we have EGðu; uÞ lim inf n!1 EGðun; unÞ ð þ1Þ: (iv) A function u 2 D½EGŠ satisfies juðxÞ uðyÞj 2 EGðu; uÞd R Gðx; yÞ; x; y 2 G; and hence by the definition of RG, this estimate is restated as follows: juðxÞ uðyÞj 2 EGðu; uÞ RGðx; yÞ; u 2 D½EGŠ; x; y 2 G; RGðx; yÞ d R Gðx; yÞ; x; y 2 G: In particular, we have juðxÞj EGðu; uÞ 1=2 RGðx; yÞ 1=2 þjuðyÞj; x; y 2 G: ð1Þ (v) EG satisfies the Markov property, that is, for u 2 D½EGŠ, u :¼ minfmaxf0; ug; 1g 2D½EGŠ and EGðu; uÞ EGðu; uÞ: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (vi) The embedding of ðD½EGŠ; EG þ 2 p oÞ into CðXÞ is compact, that is, for a sequence of functions in D½EGŠ which are uniformly bounded and whose energies are also uniformly bounded, there exists a subsequence that converges uniformly to a continuous function. (vii) The form EG is strong local, i.e., for u; v 2 D½EGŠ, EGðu; vÞ ¼0 if u is constant on the support of v. Now we are given a closed subset K of G. For any u 2 CðKÞ, let Au ¼fv2 D½EGŠ jv ¼ u on Kg and also F K ¼fu 2 CðKÞ jAu 6¼;g. Then in view of the properties (iii) and (iv), we see that for any u 2 F K, there exists uniquely a minimizer HK;u on Au, which is characterized as a function h 2 Au that satisfies EGðh; vÞ ¼0 for all v 2 D½EGŠ vanishing on K. Define a form EK on CðKÞ by D½EKŠ¼F K; EKðu; uÞ ¼EGðHK;u; HK;uÞ; u 2 D½EKŠ: Then E K satisfies the same properties of (i) through (vi) as EG; however (vii) does not hold true in general. We note that given a closed subset L of K and a function u 2 D½E K Š, E K ðu; vÞ ¼0 for all v 2 D½E K Š with supp v K n L if and only if EGðHK;u; vÞ ¼0 for all v 2 D½EGŠ with supp v G n L. In addition, let juðxÞ uðyÞj2 RKðx; yÞ ¼sup j u 2 D½EKŠ; EKðu; uÞ 6¼ 0 ; x; y 2 K: EKðu; uÞ Then RK is nothing but the restriction of the effective resistance RG of G to the set K, i.e., R K ðx; yÞ ¼RGðx; yÞ; x; y 2 K: Now the Markov property (v) implies that the maximum principle holds true for HK;u, that is, min K u HK;u max K u: Here we recall another important consequence from the Markov property that for u; v 2 D½EKŠ, uv 2 D½EKŠ and we have
Variational Convergence of Finite Networks 59 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EKðuv; uvÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kukC0 EKðv; vÞ þkvkC0 EKðu; uÞ (cf. [9], Theorem 1.4.2). In particular, D½EKŠ is a subalgebra of CðKÞ containing the unit element 1 and separating points of K. As a result, D½EKŠ is dense in CðKÞ, since K is compact. Now we consider the case where K consists of N points x1; ...; xN of G. We denote by ui ði ¼ 1; ...; NÞ the function on K defined by uiðxjÞ ¼ ij. Let cij ¼ EKðui; ujÞ ði; j ¼ 1; ...; NÞ. Since EKðui; 1Þ ¼0, we get PN j¼1 cij ¼ 0 ði ¼ 1; ...; NÞ, and hence we can deduce that EKðu; vÞ ¼ 1 X 2 N cijðuðxiÞ uðxjÞÞðvðxiÞ vðxjÞÞ; u; v 2 D½EKŠ: i;j¼1 Note that cij 0 if i 6¼ j, which follows from the Markov property of E K (cf. [20], p. 43). 2. Green Functions, Effective Resistance, and Laplace Operators 2.1 Let G be a compact connected metric graph corresponding to a finite network ðV; E; rÞ. For points a; x 2 G, in view of (1), we have uniquely a function gaðx; Þ 2 D½EGŠ satisfying EGðgaðx; Þ; vÞ ¼vðxÞ vðaÞ for all v 2 D½EGŠ, and gaðx; aÞ ¼0. Observe that gaðx; yÞ ¼gaðy; xÞ, since gaðx; yÞ ¼EGðgaðx; Þ; gaðy; ÞÞ ¼ EGðgaðy; Þ; gaðx; ÞÞ ¼ gaðy; xÞ. The functions ga are called the Green functions of the form EG. Given two points x; y 2 G, let HxyðzÞ ¼gyðz; xÞ=gyðx; xÞ ðz 2 GÞ. Then the effective resistance RG is expressed as RGðx; yÞ ¼EGðHxy; HxyÞ 1 ¼ gyðx; xÞð¼ gxðy; yÞÞ; x; y 2 G: In fact, we can show the following result, which will be verified after Theorem 2.2. Theorem 2.1. The following mutually equivalent identities hold: gzðx; yÞ ¼ 1 2 fRGðx; zÞþRGðz; yÞ RGðx; yÞg; RGðx; yÞ ¼gxðy; yÞ ¼gyðx; xÞ ¼gxðy; zÞþgyðx; zÞ for all x; y; z 2 G. Since gzðx; yÞ 0 by the maximum principle, the first identity implies that RG : G G ! R satisfies the triangle inequality and hence we have the following Corollary 2.2. The effective resistance RG is a metric on G, called the resistance metric of G, which induces the same topology as the Riemannian distance. Remarks (i) Let G and ðV; E; rÞ be as above. Then the resistance metric of G coincides with its Riemannian distance if and only if G is simply connected. Let ðV0 ; E0Þ be a subnetwork of ðV; EÞ, i.e., a pair of sets V0 V and E0 E such that x; y 2 V0 if fx; yg 2E0 . Let G0 be the metric graph associated with the subnetwork ðV0 ; E0 ; rjE0Þ and assume that G0 is connected. Then we have RGðx; yÞ RG 0ðx; yÞ; x; y 2 G0 : This is known as Rayleigh’s monotonicity principle. (ii) The identities in Theorem 2.1 are presented for finite networks in [27]. 2.2 Let be a Radon measure on G with support K and consider the Laplace operator L associated with the closed form EK on L2ðK; Þ. The domain D½L Š of L consists of functions u 2 D½EKŠ such that the functional on D½EKŠ defined by v ! EKðu; vÞ is continuous with respect to the norm of L2ðK; Þ, and for any u 2 D½L Š, L u is the unique function (in the closure of D½EKŠ in L2ðK; Þ) satisfying Z EKðv; uÞ ¼ v L ud ; v 2 D½EKŠ: Let g ðx; yÞ ¼ 1 Z gzðx; yÞ d ðzÞ; ðKÞ K x; y 2 K: Then by the definition of the Green functions, we have uðxÞ ¼ 1 Z ud ðKÞ K Z þ g ðx; yÞL uðyÞ d ðyÞ; K u 2 D½L Š; x 2 K; and hence K
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