Approximation of partial differential equations on compact resistance spaces
aa r X i v : . [ m a t h . F A ] S e p APPROXIMATION OF PARTIAL DIFFERENTIAL EQUATIONS ON COMPACTRESISTANCE SPACES
MICHAEL HINZ , MELISSA MEINERT Abstract.
We consider linear partial differential equations on resistance spaces that are uniformly ellipticand parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Ourmain interest is to provide graph and metric graph approximations for their unique solutions. For familiesof equations with different coefficients on a single compact resistance space we prove that solutions haveaccumulation points with respect to the uniform convergence in space, provided that the coefficients remainbounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformlyalong a subsequence. For the special case of local resistance forms on finitely ramified sets we also considersequences of resistance spaces approximating the finitely ramified set from within. Under suitable assump-tions on the coefficients (extensions of) linearizations of the solutions of equations on the approximatingspaces accumulate or even converge uniformly along a subsequence to the solution of the target equation onthe finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.
Contents
1. Introduction 1Acknowledgements 32. Resistance forms and first order calculus 33. Linear equations of elliptic and parabolic type 74. Convergence of solutions on a single space 105. Convergence of solutions on varying spaces 146. Discrete and metric graph approximations 267. Restrictions of vector fields 31Appendix A. Generalized strong resolvent convergence 33References 34 Introduction
For several classes of fractal spaces, such as for instance p.c.f. self-similar sets, [12, 61–64, 75, 76], classicalSierpinski carpets, [9, 11], certain Julia sets, [90], Laaksø spaces, [91], diamond lattice type fractals, [1, 3, 35],and certain random fractals, [33,34], the existence of resistance forms in the sense of [65,67] has been proved.This allows to establish a Dirichlet form based analysis, [15, 27, 79], with respect to a given volume measure,and in particular, studies of partial differential equations on fractals, [8,64,93]. These results and many laterdevelopments based on them are motivated by a considerable body of modern research in physics suggestingthat in specific situations fractal models may be much more adequate than classical ones. The difficulty inthis type of analysis comes from the fact that on fractals many tools from traditional calculus (and evenmany tools used in the modern theory of metric measure spaces, see e.g., [29, 30]) are not available.For fractal counterparts of equations of linear second order equations, [22, 31], that do not involve firstorder terms - such as Poisson or heat equations for Laplacians - many results are known, [8,56,64,66,93], andthere are also studies of related semilinear equations without first order terms, [24,25]. More recently fractalcounterparts for equations involving first order terms have been suggested, [47,50–52], and a few more specificresults have been obtained, see for instance [48, 78]. The discussion of first order terms is of rather abstractnature, because on most fractals there is no obvious candidate for a gradient operator; instead, it has to be
Mathematics Subject Classification.
Key words and phrases.
Closed forms, semigroups, (metric) graphs, fractals, vector fields, varying spaces, Mosco convergence. , Research supported in part by the DFG IRTG 2235: ’Searching for the regular in the irregular: Analysis of singularand random systems’. onstructed from a given bilinear form in a subsequent step, [17, 18, 50, 55]. (An intuitive argument why thisconstruction cannot be trivial is the fact that for self-similar fractals, endowed with natural Hausdorff typevolume measures, volume and energy are typically singular, [14,40,41,43].) For a study of, say, counterpartsof second order equations, [31, Section 8], involving abstract gradient and divergence terms, it thereforeseems desirable to establish results which indicate that the equations have the correct physical meaning.In this article we consider analogs of linear elliptic and parabolic equations with first order terms onlocally compact separable resistance spaces, [65, 67]. We wish to point out that we use the word ’elliptic’ ina very broad (quadratic form) sense - the principal parts of our operators should rather be seen as fractalgeneralizations of hypoelliptic operators. Under suitable assumptions the equations admit unique weakrespectively semigroup solutions, Corollaries 3.2 and 3.3. We prove that if the resistance space is compactand we are given bounded sequences of coefficients, the corresponding solutions have uniform accumulationpoints, Corollary 4.3. If the sequences of coefficients converge, then the corresponding solutions convergein the L -sense and uniformly along subsequences, Theorem 4.1. For certain local resistance forms onfinitely ramified sets, [55, 97], we introduce an approximation scheme along varying spaces, general enoughto accommodate both discrete and metric graph approximations. If the coefficients are bounded in a suitablemanner, extensions of linearizations of solutions to the equations on the approximating spaces have uniformaccumulation points on the target space, Corollary 5.5. If the coefficients are carefully chosen, the solutionsconverge in an L -sense and the mentioned extensions converge uniformly along subsequences, Theorem 5.1.Combining these results, we obtain an approximation for more general coefficients, Theorem 5.2.For resistance forms on discrete and metric graphs the abstract gradient operators admit more familiarexpressions, Examples 2.1 and 2.2, and the bilinear forms associated with linear equations can be understoodin terms of the well-known analysis on graphs and metric graphs, Examples 3.1 and 3.2, see for instance [32,60]and [82,86]. The approximation scheme itself is of first order in the sense that it relies on the use of piecewiselinear respectively piecewise harmonic functions, and it resembles familiar finite element methods. Onemotivation to use this approach is that pointwise restrictions of piecewise harmonic functions on, say, theSierpinski gasket, are of finite energy on approximating metric graphs, [48], but for general energy finitefunctions on the Sierpinski gasket this is not true - the corresponding trace spaces on the metric graphs arefractional Sobolev spaces of order less than one, see for instance [94] and the reference cited there (and [42]for related results). Of course first order approximations have a certain scope and certain limitations.But keeping these in mind, we can certainly view our results as a strong first indication that the abstractlyformulated equations on the target space have the desired physical meaning, because their solutions appear asnatural limits of solutions to similar equations on more familiar geometries, where they are better understood.The established approximation scheme also provides a computational tool which could be used for numericalsimulations. Our results hold under rather minimal assumptions on the volume measure on the target space.For instance, in the situation of p.c.f. self-similar structures it is not necessary to specialize to self-similarHausdorff measures, [8, 64], or to energy dominant Kusuoka type measures, [40, 41, 75, 76].In [95, Section 6] a finite element method for a Poisson type equation on p.c.f. self-similar fractals wasdiscussed, and the use of an equivalent scalar product and a related orthogonal projection made it possibleto regard the approximation itself as the solution of a closely related equation. For equations involvingdivergence and gradient terms one cannot hope for a similarly simple mechanism. On the other hand,the construction of resistance forms itself is based on discrete approximations, [61–64], and in symmetricrespectively self-adjoint situations this can be used to obtain approximation results on the level of resistanceforms, [19], or Dirichlet forms, [87, 88]. In the latter case the dynamics of a partial differential equation ofelliptic or parabolic type for self-adjoint operators comes into play, and it can be captured using spectralconvergence results, [81, 89], possibly along varying Hilbert spaces, [77, 86]. The equations we have in mindare governed by operators that are not necessarily symmetric, but under some conditions on the coefficientsthey are still sectorial, [59,79]. This leads to the question of how to implement similar types on convergencesfor sectorial operators, and one arrives to a situation similar to those in [83] or [96]. The main difficulty ishow to correctly implement the convergence of drift and divergence terms. With [48, 87, 88] in mind a firstreflex might be to try to verify a type of generalized norm resolvent convergence as in [83], and to do so thefirst order terms would have to fit the estimate in their Definition 2.3, where particular (2.7e) is critical. Forconvergent sequences of drift and divergence coefficients, [50], on a single resistance space one can establishthis estimate with trivial identification operators (as adressed in their Example 2.5), but in the case of varyingspaces the interaction of identification operators with the first order calculus seems too difficult to handle. he convergence results in [96, Section 4] use the variational convergence studied in [98,99], which generalizesthe Mosco type convergence, [81], for generalized forms, [38], to the setup of varying Hilbert spaces, [77],and encodes a generalization of strong resolvent convergence. Also in the present article this variationalconvergence is used as a key tool: We verify the adequate Mosco type convergence along varying spaces ofthe bilinear forms associated with the equations, and by [38] and [98, 99] we can then conclude the L -typeconvergence of the solutions, see Theorem A.1. A significant difference between [96, Section 4] and ourresults is the way the first order terms are handled. There the approach from [5] is used, which relies heavilyon having a carr´e du champ operator, [15]. But this is an assumption which we wish to avoid, because - asmentioned above - interesting standard examples do not satisfy it. The target spaces for the approximationresult along varying spaces that we implement are assumed to be finitely ramified sets, [55,97], endowed withlocal regular resistance forms, [65, 67], satisfying certain assumptions. This class of fractals contains manyinteresting examples, [33, 34, 36, 80, 90], and in particular, p.c.f. self-similar fractals with regular harmonicstructures, [64], but it does not contain Sierpinski carpets, [9,11]. The cell structure of a finitely ramified setallows a transparent use of identification operators based on piecewise harmonic functions. The key propertyof resistance spaces that energy finite functions are continuous compensates the possible energy singularityof a given volume measure to a certain extent, in particular, we can use an inequality originally shown in [49]when handling the first order terms in the presence of an energy singular measure. Uniform energy boundsand the compactness of the space then allow to use Arzela-Ascoli type arguments to obtain subsequentiallimits in the sense of uniform convergence. Together with the L -type limit statements produced by thevariational convergence these limit points are then identified to be the solutions on the target space.The use of variational convergence schemes to study dynamical phenomena on certain geometries is awell-established idea, see for instance [57, 58, 77]. It was already a guiding theme in [81], and related resultsin different setups have been studied in a number of recent articles, see for instance [4, 23, 44, 54, 68, 69, 83,85–88, 96]. For fractal spaces variational schemes can provide a certain counterpart to homogenization: Inthe latter the effect of a complicated microstructure can be encoded in an equation for an effective materialif the problem is viewed at a certain mesoscopic scale. In analysis on fractals it may not be possible to findsuch a scale, and it is desirable to have a more direct understanding of how the microstructure determinesanalysis. This typically leads to non-classical rescalings when passing from discrete to continuous or fromsmooth to fractal. Although the present study is written specifically for resistance spaces, some aspects ofthe approximation scheme in Section 5 might also provide some guidance for schemes along varying spacesfor non-symmetric local or non-local operators on non-resistance spaces.In Section 2 we recall basics from the theory of resistance forms and explain items of the related firstorder calculus. We discuss bilinear forms including drift and divergence terms in Section 3, and followstandard methods, [22, 26, 31], to state existence, uniqueness and energy estimates for weak solutions toelliptic equations and (semigroup) solutions to parabolic equations. In Section 4 we prove convergence resultsfor equations on a single compact resistance space. We first discuss suitable conditions on the coefficients,then accumulation points and then strong resolvent convergence. Section 5 contains the approximationscheme along varying spaces for finitely ramified sets. We first state the basic assumptions and recordsome immediate consequences, then survey conditions on the coefficients and finally state the accumulationand convergence results. Section 6 discusses discrete approximations (Subsection 6.1), including classes ofexamples, metric graph approximations (Subsection 6.2), and short remarks on possible generalizations.Section 7 contains an auxiliary result on the restriction of vector fields for finitely ramified sets.We follow the habit to write E ( u ) for E ( u, u ) if E is a bilinear quantity depending on two arguments andboth arguments are the same. Acknowledgements
We thank Olaf Post, Jan Simmer, Alexander Teplyaev and Jonas T¨olle for helpful and inspiring conver-sations. 2.
Resistance forms and first order calculus
We recall the definition of resistance form, due to Kigami, see [64, Definition 2.3.1] or [65, Definition 2.8].By ℓ ( X ) we denote the space of real valued functions on a set X . Definition 2.1.
A resistance form ( E , F ) on a set X is a pair such that i) F ⊂ ℓ ( X ) is a linear subspace of ℓ ( X ) containing the constants and E is a non-negative definitesymmetric bilinear form on F with E ( u ) = 0 if and only if u is constant. (ii) Let ∼ be the equivalence relation on F defined by u ∼ v if and only if u − v is constant on X . Then ( F / ∼ , E ) is a Hilbert space. (iii) If V ⊂ X is finite and v ∈ ℓ ( V ) then there is a function u ∈ F so that u (cid:12)(cid:12) V = v . (iv) For x, y ∈ X R ( x, y ) := sup n ( u ( x ) − u ( y )) E ( u ) : u ∈ F , E ( u ) > o < ∞ . (v) If u ∈ F then ¯ u := max(0 , min(1 , u ( x ))) ∈ F and E (¯ u ) ≤ E ( u ) . To R one refers as the resistance metric associated with ( E , F ), [65, Definition 2.11], and to the pair( X, R ), which forms a metric space, [65, Proposition 2.10], we refer as resistance space . All functions u ∈ F are continuous on X with respect to the resistance metric, more precisely, we have(1) | u ( x ) − u ( y ) | ≤ R ( x, y ) E ( u ) , u ∈ F , x, y ∈ X. For any finite subset V ⊂ X the restriction of ( E , F ) to V is the resistance form ( E V , ℓ ( V )) defined by(2) E V ( v ) = inf n E ( u ) : u ∈ F , u (cid:12)(cid:12) V = v o , where a unique infimum is achieved. If V ⊂ V and both are finite, then ( E V ) V = E V .We assume X is a nonempty set and ( E , F ) is a resistance form on X so that ( X, R ) is separable . Thenthere exists a sequence ( V m ) m of finite subsets V m ⊂ X with V m ⊂ V m +1 , m ≥
1, and S m ≥ V m dense in( X, R ). For any such sequence ( V m ) m we have(3) E ( u ) = lim m E V m ( u ) , u ∈ F , as proved in [65, Proposition 2.10 and Theorem 2.14]. Note that for any u ∈ F the sequence ( E V m ( u )) m isnon-decreasing. Each E V m is of the form(4) E V m ( u ) = 12 X p ∈ V m X q ∈ V m c ( m ; p, q )( u ( p ) − u ( q )) , u ∈ F , with constants c ( m ; p, q ) ≥ p and q .We further assume that ( X, R ) is locally compact and that ( E , F ) is regular , i.e., that the space F ∩ C c ( X ) isuniformly dense in the space C c ( X ) of continuous compactly supported functions on ( X, R ), see [67, Definition6.2]. Definition 2.1 (v) implies that
F ∩ C c ( X ) is an algebra under pointwise multiplication and(5) E ( f g ) / ≤ k f k sup E ( g ) / + k g k sup E ( f ) / , f, g ∈ F ∩ C c ( X ) , see [67, Lemma 6.5].To introduce the first order calculus associated with ( E , F ), let ℓ a ( X × X ) denote the space of all realvalued antisymmetric functions on X × X and write(6) ( g · v )( x, y ) := g ( x, y ) v ( x, y ) , x, y ∈ X, for any v ∈ ℓ a ( X × X ) and g ∈ C c ( X ), where g ( x, y ) := 12 ( g ( x ) + g ( y )) . Obviously g · v ∈ ℓ a ( X × X ), and (6) defines an action of C c ( X ) on ℓ a ( X × X ), turning it into a module.By d u : F ∩ C c ( X ) → ℓ a ( X × X ) we denote the universal derivation,(7) d u f ( x, y ) := f ( x ) − f ( y ) , x, y ∈ X, and by(8) Ω a ( X ) := (X i g i · d u f i : g i ∈ C c ( X ) , f i ∈ F ∩ C c ( X ) ) , deviating slightly from the notation used in [48], the submodule of ℓ a ( X × X ) of finite linear combinations offunctions of form g · d u f . A quick calculation shows that for f, g ∈ F ∩ C c ( X ) we have d u ( f g ) = f · d u g + g · d u f . n Ω a ( X ) we can introduce a symmetric nonnegative definite bilinear form h· , ·i H by extending(9) h g · d u f , g · d u f i H := lim m →∞ X p ∈ V m X q ∈ V m c ( m ; p, q ) g ( p, q ) g ( p, q ) d u f ( p, q ) d u f ( p, q )linearly in both arguments, respectively, and we write k·k H = p h· , ·i H for the associated Hilbert seminorm.In Proposition 2.1 below we will verify that the definition of h· , ·i H does not depend on the choice of thesequence ( V m ) m .We factor Ω a ( X ) by the elements of zero seminorm and obtain the space Ω a ( X ) / ker k·k H . Givenan element P i g i · d u f i of Ω a ( X ) we write (cid:2) P i g i · d u f i (cid:3) H to denote its equivalence class. CompletingΩ a ( X ) / ker k·k H with respect to k·k H we obtain a Hilbert space H , we refer to it as the space of generalized L -vector fields associated with ( E , F ). This is a version of a construction introduced in [17, 18] and stud-ied in [13, 49–53, 55, 78], see also the related sources [20, 29, 30, 100]. The basic idea is much older, see forinstance [15, Exercise 5.9], it dates back to ideas of Mokobodzki and LeJan.The action (6) induces an action of C c ( X ) on H : Given v ∈ H and g ∈ C c ( X ), let ( v n ) n ⊂ Ω a ( X ) be suchthat lim n [ v n ] H = v in H and define g · v ∈ H by g · v := lim n [ g · v n ] H . Since (6) and (9) imply(10) k g · v k H ≤ k g k sup k v k H , it follows that the definition of g · v is correct. Given f ∈ F ∩ C c ( X ), we denote the H -equivalence class ofthe universal derivation d u f as in (7) by ∂f . By the preceding discussion we observe [ g · d u f ] H = g · ∂f forall f ∈ F ∩ C c ( X ) and g ∈ C c ( X ). It also follows that the map f ∂f defines a derivation operator ∂ : F ∩ C c ( X ) → H which satisfies the identity k ∂f k H = E ( f ) for any f ∈ F ∩ C c ( X ) and the Leibniz rule ∂ ( f g ) = f · ∂g + g · ∂f for any f, g ∈ F ∩ C c ( X ).To show the independence of h· , ·i H of the choice of the sequence ( V m ) m in (9) and to formulate laterstatements, we consider energy measures. For f ∈ F ∩ C c ( X ) there is a unique finite Radon measure ν f on X satisfying(11) Z X g dν f = E ( f g, f ) − E ( f , g ) , g ∈ F ∩ C c ( X ) , the energy measure of f , see for instance [56,66,76,97] and or [27,39–41,43,46]. It is not difficult to see thatfor any f ∈ F ∩ C c ( X ) and g ∈ C c ( X ) we have(12) Z X g dν f = 12 lim m →∞ X p ∈ V m X q ∈ V m c ( m ; p, q ) g ( p )( d u f ( p, q )) . Mutual energy measures ν f ,f for f , f ∈ F ∩ C c ( X ) are defined using (11) and polarization.According to the Beurling-Deny decomposition of ( E , F ), see [2, Th´eor`eme 1] (or [27, Section 3.2] for adifferent context), there exist a uniquely determined symmetric bilinear form E c on F ∩ C c ( X ) satisfying E c ( f, g ) = 0 whenever f ∈ F ∩ C c ( X ) is constant on an open neighborhood of the support of g ∈ F ∩ C c ( X )and a uniquely determined symmetric nonnegative Radon measure J on X × X \ diag such(13) E ( f ) = E c ( f ) + Z X Z X ( d u f ( x, y )) J ( dxdy ) , f ∈ F ∩ C c ( X ) . The form E c is called the local part of E , and by ν cf we denote the local part of the energy measure of afunction f ∈ F ∩ C c ( X ), i.e. the finite Radon measure (uniquely) defined as in (11) with E c in place of E .From (11) and (13) it is immediate that(14) Z X gdν f = Z gdν cf + Z X Z X g ( x )( d u f ( x, y )) J ( dxdy ) , f, g ∈ F ∩ C c ( X ) . Proposition 2.1.
Suppose that closed balls in ( X, R ) are compact. Then for any f , f ∈ F ∩ C c ( X ) and g , g ∈ C c ( X ) we have h g · ∂f , g · ∂f i H = Z X g g dν ( c ) f ,f + Z X Z X g ( x, y ) g ( x, y ) d u f ( x, y ) d u f ( x, y ) J ( dxdy ) . In particular, the definition of the bilinear form h· , ·i H is independent of the choice of the sets V m . roof. Standard arguments show that for all v ∈ C c ( X × X \ diag) we have(15) 12 lim ε → lim m →∞ X x ∈ V m X y ∈ V m ,R ( x,y ) >ε c ( m ; x, y ) v ( x, y ) = Z X Z X v ( x, y ) J ( dxdy ) , see for instance [27, Section 3.2]. The particular case v = d u f , together with (13), then implies that(16) E c ( f ) = 12 lim ε → lim m →∞ X x ∈ V m X y ∈ V m ,R ( x,y ) ≤ ε c ( m ; x, y )( d u f ( x, y )) for any f ∈ F ∩ C c ( X ). We claim that given such f and g ∈ C c ( X ),(17) Z X g dν cf = 12 lim ε → lim m →∞ X x ∈ V m X y ∈ V m ,R ( x,y ) ≤ ε c ( m ; x, y ) g ( x, y ) ( d u f ( x, y )) . This follows from (11) and (16) and the fact thatlim ε → lim m →∞ X x ∈ V m X y ∈ V m ,R ( x,y ) ≤ ε c ( m ; x, y )( d u g ( x, y )) ( d u f ( x, y )) = 0 , which can be seen following the arguments in the proof of [48, Lemma 3.1]. Combining (15), applied to v = g · d u f , and (17), we obtain the desired result by polarization. (cid:3) As a consequence of Proposition 2.1 and dominated convergence we can define g · v for all v ∈ H and g ∈ C b ( X ) and (10) remains true for such v and g . Note also that if v , v ∈ H and g ∈ C b ( X ) then h g · v , v i H = h v , g · v i H . In the special cases of finite graphs, [32, 60], and compact metric graphs, [28, 70–73, 82, 86], the space H and the operator ∂ appear in a more familiar form. Examples . If (
V, ω ) is a finite simple weighted (unoriented) graph, [32], then E ( u ) = 12 X p ∈ V X q ∈ V ω ( p, q )( u ( p ) − u ( q )) , u ∈ ℓ ( V ) , is a resistance form on the finite set V , and it makes it a compact resistance space. In this case H isisometrically isomorphic to the space ℓ a ( V × V \ diag , ω ) of real-valued antisymmetric functions on V × V \ diag,endowed with the usual ℓ -scalar product, and for any f ∈ ℓ ( V ) the gradient ∂f ∈ H of f is the image of d u f ∈ ℓ a ( V × V \ diag , ω ) under this isometric isomorphism, see for instance [45, Section 3]. Examples . Let (
V, E ) be a finite simple (unoriented) graph and ( l e ) e ∈ E a finite sequence of positivenumbers. Consider the metric graph Γ obtained by identifying each edge e ∈ E with an oriented copy ofthe interval (0 , l e ) and considering different copies to be joined at the vertices the respective edges have incommon. Then the set X Γ = V ∪ S e ∈ E e , endowed with a natural topology, becomes a compact metricspace. For each u ∈ C ( X Γ ) let E Γ ( u ) = X e ∈ E E e ( u e ) , where E e ( u e ) = Z l e ( u e ( s )) ds, e ∈ E, and u e is the restriction of u to e ∈ E . If ˙ W , ( X Γ ) denotes the space of all u ∈ C ( X Γ ) such that E ( u ) < + ∞ then ( E Γ , ˙ W , ( X Γ )) is a resistance form making X Γ a compact resistance space. The space H is isometrically isomorphic to L e ∈ E L (0 , l e ), and for any f ∈ ˙ W , ( X Γ ) the gradient ∂f ∈ H is the imageunder this isometric isomorphism of ( f ′ e ) e ∈ E , where f ′ e ∈ L (0 , l e ) denotes the usual first derivative of f e ,seen as a function on (0 , l e ). For more precise descriptions and further details see [13, 55]. In Subsection 6.2we consider a scaled variant of this construction as in [48]. Remark . For convenience the above construction of the space H and the operator ∂ is formulated forresistance spaces. However, we wish to point out that the original construction does not need the specificproperties of a resistance space, it can be formulated for Dirichlet forms in very high generality, [17]. . Linear equations of elliptic and parabolic type
The considerations in this section are straightforward from standard theory for partial differential equa-tions, [31, Chapter 8], and Dirichlet forms, [27], see for instance [26].Let ( E , F ) be a resistance form on a nonempty set X so that ( X, R ) is separable and locally compact andassume that ( E , F ) is regular. In addition, assume that closed balls are compact. Let µ be a Borel measureon ( X, R ) such that for any x ∈ X and R > < µ ( B ( x, R )) < + ∞ . Then by [67, Theorem 9.4]the form ( E , F ∩ C c ( X )) is closable on L ( X, µ ) and its closure, which we denote by ( E , D ( E )), is a regularDirichlet form. In general we have D ( E ) ⊂ F ∩ L ( X, µ ), and in the special case that (
X, R ) is compact, D ( E ) = F , [67, Section 9]. Given α > E α ( f, g ) := E ( f, g ) + α h f, g i L ( X,µ ) , f, g ∈ D ( E ) , and we use an analogous notation for other bilinear forms. Recall that we also write E ( f ) to denote E ( f, f )and similarly for other bilinear quantities.By the closedness of ( E , D ( E )) the derivation ∂ , defined as in the preceding section, extends to a closedunbounded linear operator ∂ : L ( X, µ ) → H with domain D ( E ), we write Im ∂ for the image of D ( E ) under ∂ . The adjoint operator ( ∂ ∗ , D ( ∂ ∗ )) of ( ∂, D ( E )) can be interpreted as minus the divergence operator, and forthe generator ( L , D ( L )) of ( E , D ( E )) we have ∂f ∈ D ( ∂ ∗ ) whenever f ∈ D ( L ), and in this case, L f = − ∂ ∗ ∂f .3.1. Closed forms.
We call a symmetric bounded linear operator a : H → H a uniformly elliptic (in thesense of quadratic forms) if there are universal constants 0 < λ < Λ such that(19) λ k v k H ≤ h a v, v i H ≤ Λ k v k H , v ∈ H . As mentioned in the introduction, the phrase ’uniformly elliptic’ is interpreted in a wide sense, and (19)rather corresponds to a sort of energy equivalence, see for instance [10, Definition 2.17]. We follow [26] andsay that an element b ∈ H is in the Hardy class if there are constants δ ( b ) ∈ (0 , ∞ ) and γ ( b ) ∈ [0 , ∞ ) suchthat(20) k g · b k H ≤ δ ( b ) E ( g ) + γ ( b ) k g k L ( X,µ ) , g ∈ F ∩ C c ( X ) . Given uniformly elliptic a as in (19), b , ˆ b ∈ H in the Hardy class and c ∈ L ∞ ( X, µ ) we consider thebilinear form on
F ∩ C c ( X ) defined by(21) Q ( f, g ) = h a · ∂f, ∂g i H − h g · b, ∂f i H − (cid:10) f · ˆ b, ∂g (cid:11) H − h cf, g i L ( X,µ ) , f, g ∈ F ∩ C c ( X ) . We say that a densely defined bilinear form ( Q, D ( Q )) on L ( X, µ ) is semibounded if there exists some C ≥ Q ( f ) ≥ − C k f k L ( X,µ ) , f ∈ D ( Q ). If in addition ( D ( Q ) , e Q C +1 ) is a Hilbert space, where e Q denotes the symmetric part of Q , defined by(22) e Q ( f, g ) = 12 ( Q ( f, g ) + Q ( f, g )) , f, g ∈ D ( Q ) , then we call ( Q, D ( Q )) a closed form . In other words, we call ( Q, D ( Q )) a closed form if ( e Q, D ( Q )) is a closedquadratic form in the sense of [89, Section VIII.6]. We say that a closed form ( Q, D ( Q )) is sectorial if thereis a constant K > | Q C +1 ( f, g ) | ≤ K Q C +1 ( f ) / Q C +1 ( g ) / , f, g ∈ D ( Q ) , where C is as above. In other words, we call a closed form ( Q, D ( Q )) sectorial if ( Q C , D ( Q )) is a coerciveclosed form in the sense of [79, Definition 2.4].The following proposition follows from standard estimates and (20), we omit its proof. Proposition 3.1. (i)
Assume that a : H → H is symmetric and satisfies (19), c ∈ L ∞ ( X, µ ) and b, ˆ b ∈ H are in the Hardyclass and such that (23) λ := 12 (cid:18) λ − p δ ( b ) − q δ (ˆ b ) (cid:19) > . Then ( Q , F ∩ C c ( X )) is closable on L ( X, µ ) , and its closure ( Q , D ( E )) is a sectorial closed form. ii) If in addition c is such that (24) c := ess inf x ∈ X ( − c ( x )) − γ ( b ) + γ (ˆ b )2 λ > then ( Q , D ( E )) satisfies the bounds (25) λ E ( f ) + c k f k L ( X,µ ) ≤ Q ( f ) ≤ Λ ∞ E ( f ) + c ∞ k f k L ( X,µ ) , f ∈ D ( E ) , where (26) Λ ∞ := Λ + p δ ( b ) + q δ (ˆ b ) + 1 and c ∞ := γ ( b ) + γ (ˆ b )2 + k c k L ∞ ( X,µ ) . Remark . These conditions are chosen for convenience, we do not claim their optimality. Standardestimates using integrability conditions for vector fields, as for instance used in [96], do not apply unless oneassumes that energy measures are absolutely continuous with respect to µ , an assumption we wish to avoid.Suppose that the hypotheses of Proposition 3.1 (i) are satisfied. Let ( L ( Q ) , D ( L ( Q ) )) denote the infinites-imal generator of ( Q , D ( E )), that is, the unique closed operator on L ( X, µ ) associated with ( Q , D ( E )) bythe identity Q ( f, g ) = − (cid:10) L Q f, g (cid:11) L ( X,µ ) , f ∈ D ( L Q ) , g ∈ D ( E ) . A direct calculation shows the following.
Corollary 3.1.
Let the hypotheses of Proposition 3.1 (i) and (ii) be satisfied, let notation be as there andset (27) K = 1 λ (cid:18) Λ + p δ ( b ) + q δ (ˆ b ) + p γ ( b ) + q γ (ˆ b ) (cid:19) + 2 k c k L ∞ ( X,µ ) c + 1 . Then the generator ( L ( Q ) , D ( L ( Q ) )) satisfies the sector condition (28) | (cid:10) ( −L Q − ε ) f, g (cid:11) L ( X,µ ) | ≤ K (cid:10) ( −L Q − ε ) f, f (cid:11) / L ( X,µ ) (cid:10) ( −L Q − ε ) g, g (cid:11) / L ( X,µ ) ,f, g ∈ D ( L Q ) , for all ≤ ε ≤ c / . Linear elliptic and parabolic problems.
Suppose throughout this subsection that a , b , ˆ b and c satisfy the hypotheses of Proposition 3.1 (i) and (ii). It is straightforward to formulate equations of elliptictype. Given f ∈ L ( X, µ ), we say that u ∈ L ( X, µ ) is a weak solution to(29) L Q u = f if u ∈ D ( E ) and Q ( u, g ) = − h f, g i L ( X,µ ) for all g ∈ D ( E ). Remark . Formally, the generator ( L Q , D ( L Q )) of ( Q , D ( E )) has the structure L Q u = − ∂ ∗ ( a ∂u ) + b · ∂u + ∂ ∗ ( u · ˆ b ) + cu, so that equation (29) is seen to be an abstract version of the elliptic equationdiv( a ∇ u ) + b · ∇ u − div( ub ) + cu = f. It follows from the lower estimate in (25) that the Green operator G Q = ( −L Q ) − of L Q exists as abounded linear operator G Q : L ( X, µ ) → L ( X, µ ) and satisfies(30) Q ( G Q f, g ) = h f, g i L ( X,µ ) , f ∈ L ( X, µ ) , g ∈ D ( E ) . Corollary 3.2.
For any f ∈ L ( X, µ ) the function u = − G Q f ∈ D ( L Q ) is the unique weak solution to (29).It satisfies (31) Q ( u ) ≤ (cid:18) c + 4 c (cid:19) k f k L ( X,µ ) . Remark . The constant in (31) is chosen just for convenience. The only fact that matters is that it maybe chosen in a way that depends monotonically on c . roof. The first part is clear, the second follows from (30), Cauchy-Schwarz and because for any 0 < ε ≤ c / c as in (24) the operator L Q + ε generates a strongly continuous contraction semigroup, so that (cid:13)(cid:13) G Q f (cid:13)(cid:13) L ( X,µ ) = (cid:13)(cid:13)(cid:13)(cid:0) ε + ( − ε − L Q ) (cid:1) − f (cid:13)(cid:13)(cid:13) L ( X,µ ) ≤ ε k f k L ( X,µ ) . (cid:3) Remark . If c ∈ L ∞ ( X, µ ) does not satisfy (24), one can at least solve equations(32) L Q u − c u = f, where c > c defined as in (24) one has c + c >
0. The sectorial closed form(33) Q c ( f, g ) = Q ( f, g ) + c h f, g i L ( X,µ ) , f, g ∈ D ( E ) , satisfies (25), (26), (28) and (27) with c + c and k c k L ∞ ( X,µ ) + c in place of c and k c k L ∞ ( X,µ ) .Related parabolic problems can be discussed in a similar manner. Given ˚ u ∈ L ( X, µ ) we say that afunction u : (0 , + ∞ ) → L ( X, µ ) is a solution to the Cauchy problem(34) ∂ t u ( t ) = L Q u ( t ) , t > , u (0) = ˚ u, if u is an element of C ((0 , + ∞ ) , L ( X, µ )) ∩ C ([0 , + ∞ ) , L ( X, µ )), we have u ( t ) ∈ D ( L Q ) for any t > Remark . Problem (34) is an abstract version of the parabolic problem ∂ t u ( t ) = div( a ∇ u ( t )) + b · ∇ u ( t ) − div( u ( t )ˆ b ) + cu ( t ) , t > , u (0) = ˚ u. Let ( T Q t ) t> denote the strongly continuous contraction semigroup on L ( X, µ ) generated by L Q . Corollary 3.3.
For any ˚ u ∈ L ( X, µ ) the Cauchy problem (34) has the unique solution u ( t ) = T Q t ˚ u , t > .For any t > it satisfies u ( t ) ∈ D ( L Q ) and (35) Q ( u ( t )) ≤ (cid:18) C K t + 1 (cid:19) k ˚ u k L ( X,µ ) , where C K > is a constant depending only on the sector constant K in (28).Proof. Again the first part of the Corollary is standard. To see (35) recall that the operator ( L Q , D ( L Q ))satisfies the sector condition (28). Consequently the semigroup ( T Q t ) t> generated by ( L Q + ε, D ( L Q ))extends to a holomorphic contraction semigroup on the sector { z ∈ C : | Im z | ≤ K − Re z } , see for instance[59, Chapter XI, Theorem 1.24], or [79, Theorem 2.20 and Corollary 2.21]. By (25) zero is contained in theresolvent set of L Q . This implies that for any t > kL Q T Q t f k L ( X,µ ) ≤ C K t k f k L ( X,µ ) , f ∈ L ( X, µ ) , for some C K ∈ (0 , ∞ ) depending only on the sector constant K , as an inspection of the classical proofs of (36)shows, see for instance [21, Theorem 4.6], [84, Section 2.5, Theorem 5.2] or the explanations in [83, Section2]. Now (35) follows using (36), Cauchy-Schwarz and contractivity. (cid:3) Remark . Since weak solutions to (29) and solutions to (34) at fixed positive times are elements of D ( E ) ⊃ D ( L Q ), they are H¨older continuous of order 1 / X, R ) by (1).It is a trivial observation that if a ∈ C ( X ) satisfies(37) λ < a ( x ) < Λ , x ∈ X, then a , interpreted as a bounded linear map v a · v from H into itself, satisfies (19). Our main interest isto understand the drift terms and therefore we restrict attention to coefficients a of form (37) in the followingsections. Note that under condition (37) the function a may also be seen as a conformal factor, [7]. Remark . A discussion of more general diffusion coefficients a should involve suitable coordinates, see[40,53,97]. In view of the fact that natural local energy forms on p.c.f. self-similar sets have pointwise indexone, [13, 41, 76], assumption (37) does not seem to be unreasonably restrictive for this class of fractal spaces. n finite graphs, [32, 60], and compact metric graphs, [28, 70–73, 82, 86], the forms in (21) admit ratherfamiliar expressions. Examples . In the setup of Examples 2.1 and with a given volume function µ : V → (0 , + ∞ ) we obtain,accepting a slight abuse of notation, Q ( f, g ) = 12 X p ∈ V X q ∈ V ω ( p, q ) a ( p, q )( f ( p ) − f ( q ))( g ( p ) − g ( q )) − X p ∈ V X q ∈ V ω ( p, q ) g ( p, q ) b ( p, q )( f ( p ) − f ( q )) − X p ∈ V X q ∈ V ω ( p, q ) f ( p, q )ˆ b ( p, q )( g ( p ) − g ( q )) − X p ∈ V c ( p ) f ( p ) g ( p ) µ ( p )for all f, g ∈ ℓ ( V ) and any given coefficients a, c ∈ ℓ ( V ) and b, ˆ b ∈ ℓ ( V × V \ diag , ω ). Examples . Suppose we are in the same situation as in Examples 2.1 and µ is a finite Borel measure on X Γ that has full support and is equivalent to the Lebesgue-measure on each individual edge. Then, abusingnotation slightly, Q ( f, g ) = X e ∈ E Z l e a e ( s ) f ′ e ( s ) g ′ e ( s ) ds − X e ∈ E Z l e g e ( s ) b e ( s ) f ′ e ( s ) ds − X e ∈ E Z l e f e ( s )ˆ b e ( s ) g ′ e ( s ) ds − X e ∈ E Z l e c e ( s ) f e ( s ) g e ( s ) µ ( ds )for all f, g ∈ ˙ W , ( X Γ ) and all a ∈ C ( X Γ ), c ∈ L ∞ ( X Γ , µ ) and b, ˆ b ∈ L e ∈ E L (0 , l e ), where u e denoted therestriction to e ∈ E in the a.e. sense of an integrable function on X Γ .4. Convergence of solutions on a single space
In this section we define bilinear forms Q ( m ) on L ( X, µ ) by replacing a , b and ˆ b in (21) by coefficients a m b m and ˆ b m that may vary with m . To keep the exposition more transparent and since it is rather trivial tovary it, we keep c fixed. We consider the unique weak solutions to elliptic problems (29) and unique solutionsat fixed positive times of parabolic problems (34) with these coefficients. For a sequence ( a m ) m satisfying(19) uniformly in m , bounded sequences ( b m ) m and (ˆ b m ) m and small enough c , we can find accumulationpoints with respect to the uniform convergence on X of these solutions, and these accumulation points areelements of F , Corollary 4.3. If coefficients a , b , ˆ b and c are given and the sequences ( a m ) m , ( b m ) m and(ˆ b m ) m converge to a , b and ˆ b , respectively, then we can conclude the uniform convergence of the solutionsto the respective solutions of the target problem, Theorem 4.1.4.1. Boundedness and convergence of vector fields.
As in the preceding section we assume that (
X, R )is separable and locally compact, that ( E , F ) is regular and that µ is a Borel measure on ( X, R ) such thatfor any x ∈ X and R > < µ ( B ( x, R )) < + ∞ .Under a mild geometric assumption on µ any vector field b ∈ H satisfies the Hardy condition. We saythat µ has a uniform lower bound V if V is an non-decreasing function V : (0 , + ∞ ) → (0 , + ∞ ) so that(38) µ ( B ( x, r )) ≥ V ( r ) , x ∈ X, r > . The following proposition is a partial refinement of [49, Lemma 4.2].
Proposition 4.1.
Suppose that µ has the uniform lower bound V . Then for any g ∈ F ∩ C c ( X ) , any b ∈ H and any M > we have (39) k g · b k H ≤ M E ( g ) + V ( M k b k H ) k b k H k g k L ( X,µ ) , where V is the non-decreasing function V ( s ) = 2 V (cid:0) s (cid:1) , s > . n particular, any b ∈ H is in the Hardy class, and for any M > it satisfies the estimate (20) with δ ( b ) = M and γ ( b ) = V ( M k b k H ) k b k H . Moreover, for any λ > condition (23) holds if we choose M > /λ for both b and ˆ b . A proof of an inequality of type (39) had already been given in [49, Lemma 4.2], but the function V hadnot been specified and an unnecessary metric doubling assumption had been made. We comment on thenecessary modifications. Proof.
We may assume k b k H >
0. Let { B j } j be a countable cover of X by open balls B j of radius r =(2 M k b k H ) − . As in [49] we can use (1) to see that for any j and any x ∈ B j we have | g ( x ) | ≤ | g ( x ) − ( g ) B j | + 2( g ) B j ≤ E ( g ) r + 2( g ) B j , where we use the shortcut notation ( f ) B = µ ( B ) R B f dµ . Setting B = ∅ and C j = B j \ S j − i =0 B i , j ≥
1, weobtain a countable Borel partition { C j } j of X with C j ⊂ B j , j ≥
1. Then for any x ∈ X we have | g ( x ) | ≤ E ( g ) r + 2 X j ( g ) B j C j ( x ) ≤ E ( g ) r + 2 V ( r ) k g k L ( X,µ ) , and using (10) we arrive at the claimed inequality. (cid:3) We record two consequences of Proposition 4.1. The first states that if the norms of vector fields in asequence are uniformly bounded then we may choose uniform constants in the Hardy condition (20).
Corollary 4.1.
Suppose that µ has a uniform lower bound. If ( b m ) m ⊂ H satisfies sup m k b m k H < + ∞ thenfor any M > there is a constant γ M > independent of m such that (20) holds for each b m with constants δ ( b m ) = M and γ ( b m ) = γ M .Proof. Let V be defined as in Proposition 4.1. Since V is increasing we can take(40) γ M := V ( M sup m k b m k H ) sup m k b m k H . (cid:3) The second consequence is a continuity statement.
Corollary 4.2.
Suppose that µ has a uniform lower bound. If b ∈ H and ( b m ) m ⊂ H is a sequence with lim m b m = b in H then for any g ∈ C c ( X ) we have lim m k g · b m − g · b k H = 0 . Proof.
This is immediate from the definition of the function V in Proposition 4.1 and the fact that theuniform lower bound V of µ is strictly positive and increasing. (cid:3) Accumulation points.
For the rest of this section we assume that ( E , F ) is a resistance form on anonempty set X so that ( X, R ) is compact, and that µ is a finite Borel measure on ( X, R ) with a uniformlower bound V . Note that by compactness ( E , F ) is regular.For each m let a m ∈ C ( X ) satisfy (37) with the same constants 0 < λ < Λ. Suppose
M > λ := λ/ − /M > b m , ˆ b m ∈ H satisfy(41) sup m k b m k H < + ∞ and sup m (cid:13)(cid:13) ˆ b m (cid:13)(cid:13) H < + ∞ . Let γ M be as in (40), let ˆ γ M defined in the same way with the ˆ b m replacing the b m and suppose that c ∈ L ∞ ( X, µ ) is such that(42) c := ess inf x ∈ X ( − c ( x )) − γ M + ˆ γ M λ − /M > . Then by Proposition 3.1 and Corollary 4.1 the forms(43) Q ( m ) ( f, g ) := h a m · ∂f, ∂g i H − h g · b m , ∂f i H − (cid:10) f · ˆ b m , ∂g (cid:11) H − h cf, g i L ( X,µ ) , f, g ∈ F , are sectorial closed forms on L ( X, µ ). They satisfy (25) with δ ( b m ) = δ (ˆ b m ) = 1 /M and γ M , ˆ γ M replacing γ ( b ), γ (ˆ b ) in (26). Their generators ( L Q ( m ) , D ( L Q ( m ) )) satisfy the sector conditions (28) with the same sector onstant K . As a consequence we observe uniform energy bounds for the solutions of (29) and (34). Wewrite Q ( m ) ,α for the form defined as E α in (18) but with Q ( m ) in place of E . Proposition 4.2.
Let a m , b m , ˆ b m and c be as above such that (41) and (42) hold. (i) If f ∈ L ( X, µ ) and u m is the unique weak solution to (29) with L Q ( m ) in place of L , then we have sup m Q ( m ) , ( u m ) < + ∞ . (ii) If ˚ u ∈ L ( X, µ ) and u m is the unique solution to (34) with L Q ( m ) in place of L , then for any t > we have sup m Q ( m ) , ( u m ( t )) < + ∞ .Proof. Since (42) and (28) hold with the same constants c and K for all m , the statements follow fromCorollaries 3.2 and 3.3. (cid:3) The compactness of X implies the existence of accumulation points in C ( X ). Corollary 4.3.
Let a m , b m , ˆ b m and c be as above such that (41) and (42) are satisfied. (i) If f ∈ L ( X, µ ) and u m is the unique weak solution to (29) with L Q ( m ) in place of L Q , then eachsubsequence of ( u m ) m has a subsequence converging to a limit e u ∈ F uniformly on X . (ii) If ˚ u ∈ L ( X, µ ) and u m is the unique solution to (34) with L Q ( m ) in place of L Q , then for each t > each subsequence of ( u m ( t )) m has a further subsequence converging to a limit e u t ∈ F uniformly on X . At this point we can of course not claim that the C ( X )-valued function t e u t has any good propertiesor significance. Proof.
Since all Q ( m ) satisfy (25) with the same constants, Proposition 4.2 implies that sup m E ( u m ) < + ∞ .By [67, Lemma 9.7] the embedding F ⊂ C ( X ) is compact, hence ( u m ) m has a subsequence that convergesuniformly on X to a limit e u . To see that e u ∈ F , note that also this subsequence is bounded in F andtherefore has a further subsequence that converges to a limit e w ∈ F weakly in L ( X, µ ), as follows from aBanach-Saks type argument. This forces e w = e u . Statement (ii) is proved in the same manner. (cid:3) Strong resolvent convergence.
Let ( E , F ) and µ be as in the preceding subsection. Let a ∈ F besuch that (37) holds with constants 0 < λ < Λ and let ( a m ) m ⊂ C ( X ) be such that(44) lim m k a m − a k sup = 0 . Without loss of generality we may then assume that also the functions a m satisfy (37) with the very sameconstants 0 < λ < Λ. Suppose
M > λ := λ/ − /M >
0. Let b, ˆ b ∈ H and let( b m ) m ⊂ H and (ˆ b m ) m ⊂ H be sequences such that(45) lim m k b m − b k H = 0 and lim m (cid:13)(cid:13) ˆ b m − ˆ b (cid:13)(cid:13) H = 0 . Note that this implies (41). Let γ M be as in (40) and ˆ γ M similarly but with the ˆ b m , and suppose that c ∈ L ∞ ( X, µ ) satisfies (42). Let Q be as in (21) and Q ( m ) as in (43).The next theorem states that the solutions to (29) and (34) depend continuously on the coefficients a , b and ˆ b . It is based on [38, Theorem 3.1]. Theorem 4.1.
Let a , a m , b , b m , ˆ b and ˆ b m be as above such that (44) and (45) hold. Then lim m L Q ( m ) = L Q in the strong resolvent sense, and the following hold. (i) If f ∈ L ( X, µ ) , u and u m are the unique weak solutions to (29) and to (29) with L Q ( m ) in place of L Q , respectively, then lim m u m = u in L ( X, µ ) . Moreover, there is a sequence ( m k ) k with m k ↑ + ∞ such that lim k u m k = u uniformly on X . (ii) If ˚ u ∈ L ( X, µ ) , and u and u m are the unique solutions to (34) and to (34) with L Q ( m ) in place of L , then for any t > we have lim m u m ( t ) = u ( t ) in L ( X, µ ) . Moreover, for any t > there is asequence ( m k ) k with m k ↑ + ∞ such that lim k u m k ( t ) = u ( t ) uniformly on X .Proof. By [38, Theorem 3.1], the claimed strong resolvent convergence and the stated convergences in L ( X, µ ) follow once we have verified the conditions in Definition A.2, see Theorem A.1 and Remark A.3 inAppendix A. The statements on uniform convergence then also follow using Corollary 4.3. ithout loss of generality we may assume that the function c ∈ L ∞ ( X, µ ) satisfies condition (42). Ifnot, proceed similarly as in Remark 3.4 and replace c by c − c , where c > c as defined as in (42) we have c + c >
0, and consider the forms ( Q ( m ) ,c , F ) with generators( L Q ( m ) − c , D ( L Q ( m ) )). If lim m →∞ L Q ( m ) − c = L Q − c in the KS-generalized strong resolvent sensethen also lim m →∞ L Q ( m ) = L Q in the KS-generalized strong resolvent sense. The statements on uniformconvergence then follow using Corollary 4.3 and Remark 3.4, note that for all m and u ∈ F we have Q ( m ) ( u ) ≤ Q ( m ) ,c ( u ).Thanks to (23), (24), (25) and (26) together with Proposition 4.1 and Corollaries 4.1 and 4.2 we can finda constant C > m we have(46) C E ( f ) ≤ Q ( m ) , ( f ) ≤ C − E ( f ) , f ∈ F . Suppose that lim m u m = u weakly in L ( X, µ ) with lim m Q ( m ) , ( u m , u m ) < + ∞ . Then there is a sub-sequence ( u m k ) k such that sup k Q ( m k ) , ( u m k ) < + ∞ , and by (46) we have sup k E ( u m k , u m k ) < + ∞ . Asubsequence of ( u m k ) k converges to a limit u E ∈ F weakly in ( F , E ) and standard Banach-Saks type argumentshows that u E = u , what proves condition (i) in Definition A.2.To verify condition (ii) in Definition A.2 suppose that ( m k ) k be a sequence of natural numbers with m k ↑ ∞ , lim k u k = u weakly in L ( X, µ ) with sup k Q ( m k ) , ( u k , u k ) < ∞ and u ∈ F . By (46) we havesup k E ( u k ) < ∞ , what implies that ( u k ) k has a subsequence ( u k j ) j converging to u ∈ F weakly in F anduniformly on X , and such that its averages N − P Nj =1 u k j converge to u in F . Here the statement on uniformconvergence is again a consequence of the compact embedding F ⊂ C ( X ), [67, Lemma 9.7]. Combined withthe weak convergence in L ( X, µ ) it follows that ( u k j ) j converges weakly to u in ( F / ∼ , E ). Moreover, using(13), the convergence of averages and the linearity of d u we may assume that ( d u u k j ) j converges to d u u weakly in L ( X × X \ diag , J ). As a consequence, we also havelim j E c ( u k j , v ) = lim j E ( u k j , v ) − lim j Z X Z X d u u k j ( x, y ) d u v ( x, y ) J ( dxdy )= E ( u, v ) − Z X Z X d u u ( x, y ) d u v ( x, y ) J ( dxdy )= E c ( u, v )for all v ∈ F .Now let w ∈ F . Then we have | Q ( m kj ) ( w, u k j ) − Q ( w, u ) | ≤ | Q ( m kj ) ( w, u k j ) − Q ( w, u k j ) | + | Q ( w, u k j − u ) | . (47)Since c is kept fixed, the first summand on the right hand side of the inequality (47) is bounded by (cid:12)(cid:12)(cid:12)D ( a m kj − a ) · ∂w, ∂u k j E H (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:10) u k j · ( b m kj − b ) , ∂w (cid:11) H (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)D w · (ˆ b m kj − ˆ b ) , ∂u k j E H (cid:12)(cid:12)(cid:12) ≤ k a m kj − a k sup E ( w ) / E ( u k j ) / + k u k j k sup k b m kj − b k H E ( w ) / + k w k sup k ˆ b m kj − ˆ b k H E ( u k j ) / , where we have used Cauchy-Schwarz and (10). By the hypotheses on the coefficients and the boundednessof ( u k j ) j in energy and in uniform norm this converges to zero. The second summand on the right hand sideof (47) is bounded by | (cid:10) ∂w, a · ∂ ( u k j − u ) (cid:11) H | + |h ( u k j − u ) · b, ∂w i H | + |h w · ˆ b, ∂ ( u k j − u ) i H | + |h cw, u k j − u i L ( X,µ ) | . The last summand in this line obviously converges to zero, and also the second does, note that |h ( u k j − u ) · b, ∂w i H | ≤ k u k j − u k sup k b k H E ( w ) / by Cauchy-Schwarz and (10). By Proposition 2.1 we have (cid:10) ∂w, a · ∂ ( u k j − u ) (cid:11) H = Z X a dν cw,u kj − u + Z X Z X a ( x, y ) d u w ( x, y ) d u ( u k j − u )( x, y ) J ( dxdy ) . Since k ad u w k L ( X × X \ diag ,J ) ≤ k a k sup E ( w ) / , the double integral converges to zero by the weak convergenceof ( d u u k j ) j to d u u in L ( X × X \ diag , J ). By (5) we have sup j E ( au k j ) / < + ∞ and E ( wu k j ) / < + ∞ . Thinning out the sequence ( u k j ) j once more we may, using the arguments above, assume that im j E c ( au k j , v ) = E c ( au, v ) and lim j E c ( wu k j , v ) = E c ( wu, v ) for all v ∈ F . Then also Z X a dν cw,u kj − u = 12 (cid:8) E c ( aw, u k j − u ) + E c ( a ( u k j − u ) , w ) − E c ( w ( u k j − u ) , a ) (cid:9) converges to zero. Together this implies that lim j (cid:10) ∂w, a · ∂ ( u k j − u ) (cid:11) H = 0. Finally, note that by theLeibniz rule for ∂ , (cid:10) ˆ b, w · ∂ ( u k j − u ) (cid:11) H = (cid:10) ˆ b, ∂ ( w ( u k j − u )) (cid:11) H − (cid:10) ˆ b, ( u k j − u ) · ∂w (cid:11) H . As before we see easily that the second summand on the right hand side goes to zero. For the first, letˆ b = ∂f + η be the unique decomposition of ˆ b ∈ H into a gradient ∂f of a function f ∈ F and a ’divergencefree’ vector field η ∈ ker ∂ ∗ . Then (cid:10) ˆ b, ∂ ( w ( u k j − u )) (cid:11) H = (cid:10) ∂f, ∂ ( w ( u k j − u )) (cid:11) H = E ( f, w ( u k j − u )) , which converges to zero by the preceding arguments. Combining, we see thatlim j Q ( n kj ) ( w, u k j ) = Q ( w, u ) , and since w ∈ F was arbitrary, this implies condition (ii) in Definition A.2. (cid:3) Examples . The basic requirements for Theorem 4.1 are that the resistance form ( E , F ) is regular, thespace ( X, R ) is compact, and that the Borel measure µ on ( X, R ) has a uniform lower bound. In particular, µ does not have to satisfy a volume doubling property. Possible examples can for instance be found amongstfinite graphs, [32, 60], compact metric graphs, [28, 70–73, 82, 86], p.c.f. self-similar sets, [12, 61–64, 75, 76, 80],classical Sierpinski carpets, [9, 11], certain Julia sets, [90], and certain random fractals, [33, 34].5. Convergence of solutions on varying spaces
In this section we basically repeat the approximation program from Section 4, but now on varying resis-tance spaces. More precisely, we study the convergence of suitable linearizations of solutions to (29) and(34) on approximating spaces X ( m ) to solutions to these equations on X . We establish these results for thecase that X is a finitely ramified set, [55, 97], endowed with a local resistance form. Possible generalizationsare commented on in Section 6.3.5.1. Setup and basic assumptions.
We describe the setup we consider and the assumptions under whichthe results of this section are formulated. They are standing assumptions for all results in this section andwill not be repeated in the particular statements.We recall the notion of finitely ramified cell structures as introduced in [97, Definition 2.1].
Definition 5.1. A finitely ramified set X is a compact metric space which has a cell structure { X α } α ∈A and a boundary (vertex) structure { V α } α ∈A such that the following hold: (i) A is a countable index set; (ii) each X α is a distinct compact connected subset of X ; (iii) each V α is a finite subset of X α ; (iv) if X α = S kj =1 X α j then V α ⊂ S kj =1 X α j ; (v) there is a filtration {A n } n such that (v.a) each A n is a finite subset of A , A = { } , and X = X ; (v.b) A n ∩ A m = ∅ if n = m ; (v.c) for any α ∈ A n there are α , ..., α k ∈ A n +1 such that X α = S kj =1 X α j ; (vi) X α ′ ∩ X α = V α ′ ∩ V α for any two distinct α, α ′ ∈ A n ; (vii) for any strictly decreasing infinite sequence X α ) X α ) ... there exists x ∈ X such that T n ≥ X α n = { x } .Under these conditions the triple ( X, { X α } α ∈A , { V α } α ∈A ) is called a finitely ramified cell structure . We write V n = S α ∈A n V α and V ∗ = S n ≥ V n , note that V n ⊂ V n +1 for all n . Suppose that ( E , e F ) is aresistance form on V ∗ . It can be written in the form (3) with e F in place of F , where the forms E V m are therestrictions of E to V m as in (2) and (4). Any function in e F is continuous in (Ω , R ), where Ω denotes the R -completion of V ∗ , and therefore has a unique extension to a continuous function on Ω. Writing F for the pace of all such extensions, we obtain a resistance form ( E , F ) on Ω. To avoid topological difficulties in thispaper, we make the following assumption. Assumption . We have Ω = X and the resistance metric R is compatible with the original topology.In view of [36, Section 4], [80, Section 7] and the well-established theory in [64, Section 3.3] one couldrephrase this by saying we consider a regular harmonic structure. As a consequence, ( X, R ) is a compactand connected metric space and ( E , F ) is a regular resistance form on X , local in the sense that if f ∈ F isconstant on an open neighborhood of the support of g ∈ F , then E ( f, g ) = 0, see for instance [97, Theorem3 and its proof].Given m ≥ v ∈ ℓ ( V m ) there exists a unique function h m ( v ) ∈ F such that h m ( v ) | V m = v in ℓ ( V m ) and E ( h m ( v )) = E V m ( v ) = min {E ( u ) : u ∈ F , u | V m = v } , see [65, Proposition 2.15]. This function h m ( v ) is called the harmonic extension of v , and as usual we saythat a function u ∈ F is m -harmonic if u = h m ( u | V m ). We write H m ( X ) to denote the space of m -harmonicfunctions on X and write H m u := h m ( u | V m ), u ∈ F , for the projection from F onto H m ( X ). It is well knownand can be seen as in [93, Theorem 1.4.4] that(48) lim m E ( u − H m u ) = 0 , u ∈ F , and using (1) it follows that also lim m k u − H m u k sup = 0, where k·k sup denotes the supremum norm. Con-sequently the space H ∗ ( X ) = [ m ≥ H m ( X )is dense in F w.r.t. the seminorm E / and in C ( X ) w.r.t. the supremum norm. We write H m ( X ) / ∼ for the space of m -harmonic functions on X modulo constants. For each m the space H m ( X ) / ∼ is afinite dimensional, hence closed subspace of ( F / ∼ , E ), and since H m = , the operator H m is easily seento induce an orthogonal projection in ( F / ∼ , E ) onto H m ( X ) / ∼ , which we denote by the same symbol.Clearly H ∗ ( X ) / ∼ is dense in ( F / ∼ , E ).We now state the main assumptions under which we implement the approximation scheme. They areformulated in a way that simultaneously covers approximations schemes by discrete graphs and by metricgraphs as discussed in Sections 6.1 and 6.2, respectively.Let diam R ( A ) denote the diameter of a set A in ( X, R ). The following assumption requires E to becompatible with the cell structure in the following ’uniform’ metric sense. Assumption . We have lim n →∞ max α ∈A n diam R ( X α ) = 0.We now assume that ( X ( m ) ) m is a sequence of subsets X ( m ) ⊂ X such that for each m ≥ X ( m ) ⊂ X ( m +1) and X ( m ) = S α ∈A m X ( m ) α where for any α ∈ A m the set X ( m ) α satisfies V α ⊂ X ( m ) α ⊂ X α . For any m ≥ E ( m ) , F ( m ) ) be a resistance form on X ( m ) so that ( X ( m ) , R ( m ) ) is topologicallyembedded in ( X, R ). We also assume that the spaces ( X ( m ) , R ( m ) ) are compact, this implies that theresistance forms ( E ( m ) , F ( m ) ) are regular. By ν ( m ) f we denote the energy measure of a function f ∈ F ( m ) ,defined as in (11) with ( E ( m ) , F ( m ) ) in place of ( E , F ). The energy measures ν ( m ) f may be interpreted asBorel measures on X . Remark . For spaces, forms, operators, coefficients and measures indexed by m and connected to X andthe form ( E , F ) we will use a subscript index m , similar objects corresponding to the spaces X ( m ) and theforms ( E ( m ) , F ( m ) ) will be indexed by a superscript ( m ), unless stated otherwise. For functions we willgenerally use a subscript index.We make some further assumptions. The first expresses a connection between the resistance forms interms of m -harmonic functions. Assumption . i) For each m the pointwise restriction u u | X ( m ) defines a linear map from H m ( X ) into F ( m ) whichis injective and satisfies(49) E ( m ) ( u | X ( m ) ) = E ( u ) , u ∈ H m ( X ) . (ii) We have(50) ν u = lim m ν ( m ) H m ( u ) | X ( m ) , u ∈ F , in the sense of weak convergence of measures on X .As a trivial consequence of (50) we have(51) E ( u ) = lim m →∞ E ( m ) ( H m ( u ) | X ( m ) ) , u ∈ F . Remark . For approximations by discrete graphs (50) follows from (51) and (12). For metric graphapproximations (50) is verified in Subsection 6.2 below, the use of products in (11) hinders a direct conclusionof (50) from (51).Now let H m ( X ( m ) ) := { u | X ( m ) : u ∈ H m ( X ) } denote the image of H m ( X ) under the pointwise restriction u u | X ( m ) , which by (49) induces an isometryfrom ( H m ( X ) / ∼ , E ) onto the Hilbert space ( H m ( X ( m ) ) / ∼ , E ( m ) ). The space H m ( X ( m ) ) is a closed linearsubspace of F ( m ) and the space H m ( X ( m ) ) / ∼ is a closed linear subspace of F ( m ) / ∼ . Let H ( m ) m denote theprojection from F ( m ) onto H m ( X ( m ) ). It satisfies H ( m ) m = and induces an orthogonal projection from( F ( m ) / ∼ , E ( m ) ) onto H m ( X ( m ) ) / ∼ so that in particular,(52) E ( m ) ( H ( m ) m v ) ≤ E ( m ) ( v ) , v ∈ F ( m ) . Let id F ( m ) denote the identity operator in F ( m ) . We need an assumption on the decay of the operators id F ( m ) − H ( m ) m as m goes to infinity. By k·k sup ,X ( m ) we denote the supremum norm on X ( m ) . Assumption . (i) For any sequence ( u m ) m with u m ∈ F ( m ) such that sup m E ( m ) ( u m ) < + ∞ we have(53) lim m (cid:13)(cid:13) u m − H ( m ) m u m (cid:13)(cid:13) sup ,X ( m ) = 0 . (ii) For u, w ∈ H n ( X ) we have(54) lim m E ( m ) ( u | X ( m ) w | X ( m ) − H ( m ) m ( u | X ( m ) w | X ( m ) )) = 0 . Remark . For discrete graph approximations as in Subsection 6.1 we have H ( m ) m v = v , v ∈ F ( m ) , so thatAssumption 5.4 is trivial.Now let µ and µ ( m ) be a finite Borel measures on X and X ( m ) , respectively, which assign positive massto each nonempty open subset of the respective space. Then by [67, Theorem 9.4] and [97, Theorem 3] theforms ( E , F ) and ( E ( m ) , F ( m ) ) are regular Dirichlet forms on L ( X, µ ) and L ( X ( m ) , µ ( m ) ), and the Dirichletform ( E , F ) is strongly local in the sense of [27].We make an assumption on the connection between the spaces L ( X, µ ) and L ( X ( m ) , µ ( m ) ) and itsconsistency with the projections and pointwise restrictions. By Ext m : H m ( X ( m ) ) → H m ( X ) we denote theinverse of the bijection u u | X ( m ) from H m ( X ) onto H m ( X ( m ) ). Assumption . (i) The measures µ and µ ( m ) admit a uniform lower bound in the following sense: There is a non-increasing function V : N → (0 , + ∞ ) such that for any m we have µ ( X α ) ≥ V ( m ), α ∈ A m , andmoreover, µ ( m ) ( X ( m ) α ) ≥ V ( m ), α ∈ A m .(ii) There are linear operators Φ m : L ( X, µ ) → L ( X ( m ) , µ ( m ) ) such that(55) sup m k Φ m k L ( X,µ ) → L ( X ( m ) ,µ ( m ) ) < + ∞ , (56) lim m →∞ (cid:13)(cid:13) Φ m u (cid:13)(cid:13) L ( X ( m ) ,µ ( m ) ) = k u k L ( X,µ ) , u ∈ L ( X, µ ) , nd for any n and u ∈ H n ( X ) we have(57) lim m k Φ ∗ m Φ m u − u k L ( X,µ ) = 0 , where for any m the symbol Φ ∗ m denotes the adjoint of Φ m .(iii) For any sequence ( u m ) m ⊂ F with sup m E ( u m ) < + ∞ we have(58) lim m →∞ (cid:13)(cid:13) Φ m u m − u m | X ( m ) (cid:13)(cid:13) L ( X ( m ) ,µ ( m ) ) = 0 . (iv) For any sequence ( u m ) m with u m ∈ F ( m ) such that sup m E ( m )1 ( u m ) < + ∞ we have(59) sup m (cid:13)(cid:13)(cid:13) Ext m H ( m ) m u m (cid:13)(cid:13)(cid:13) L ( X,µ ) < + ∞ . Let H and H ( m ) denote the spaces of generalized L -vector fields associated with ( E , F ) and ( E ( m ) , F ( m ) ),respectively. The corresponding gradient operators we denote by ∂ and ∂ ( m ) . If a , b , ˆ b and c satisfy thehypotheses of Proposition 3.1 (i) then Q ( f, g ) := h a ∂f, ∂g i H − h g · b, ∂f i H − (cid:10) f · ˆ b, ∂g (cid:11) H − h cf, g i L ( X,µ ) , f, g ∈ F , defines a sectorial closed form ( Q , F ) on L ( X, µ ). If a and c are suitable continuous functions on X and b ,ˆ b , b ( m ) and ˆ b ( m ) are vector fields of a suitable form, then we can define sectorial closed forms ( Q ( m ) , F ( m ) )on the spaces L ( X ( m ) , µ ( m ) ), respectively, by(60) Q ( m ) ( f, g ) := h a | X ( m ) · ∂f, ∂g i H ( m ) − D g · b ( m ) , ∂f E H ( m ) − (cid:10) f · ˆ b ( m ) , ∂g (cid:11) H ( m ) − h c | X ( m ) f, g i L ( X ( m ) ,µ ( m ) ) , f, g ∈ F ( m ) . In Subsection 5.4 below we observe that under simple boundedness assumptions the solutions of (29) and(34) (for fixed t >
0) associated with the forms Q ( m ) on the spaces X ( m ) accumulate in a suitable sense,see Proposition 5.2. In Theorem 5.1 in Subsection 5.5 we then conclude that they actually converge to thesolutions to the respective equation associated with the form Q , as announced in the introduction. In thepreparatory Subsections 5.2 and 5.3 we record some consequences of the assumptions and discuss possiblechoices for b , ˆ b , b ( m ) and ˆ b ( m ) .5.2. Some consequences of the assumptions.
We begin with some well-known conclusions.
Lemma 5.1. (i)
For any p, q ∈ V m we have R ( m ) ( p, q ) = R ( p, q ) . In particular, diam R ( V α ) = diam R ( m ) ( V α ) for any m ≥ n and α ∈ A n . (ii) We have diam R ( X α ) = diam R ( V α ) for any n and α ∈ A n , and diam R ( m ) ( X ( n ) α ) = diam R ( m ) ( V α ) if m ≥ n .Proof. To see (i) note that for any p, q ∈ V m we have, by a standard conclusion and using (49) and (52), R ( p, q ) − = min {E ( u ) : u ∈ H m ( X ) : u ( p ) = 0 , u ( q ) = 1 } = min n E ( m ) ( u | X ( m ) ) : u ∈ H m ( X ) , u ( p ) = 0 , u ( q ) = 1 o = R ( m ) ( p, q ) − . If the first statement (ii) were not true we could find p ∈ X α ∩ V ∗ and q ∈ ( X α ∩ V ∗ ) \ V α such that R ( p, q ) > R ( p, q ′ ) for all q ′ ∈ V α . This would imply that there exists some u ∈ H n ( X ) with u ( p ) = 0 and E ( u ) = 1 such that u ( q ) < u ( q ′ ) for all q ′ ∈ V α . However, this contradicts the maximum principle forharmonic functions on the cell X α . The second statement follows similarly. (cid:3) Also the following is due to Assumption 5.3.
Corollary 5.1.
For any f , f ∈ H n ( X ) and g , g ∈ C ( X ) we have lim m D g | X ( m ) · ∂ ( m ) ( f | X ( m ) ) , g | X ( m ) · ∂ ( m ) ( f | X ( m ) ) E H ( m ) = h g · ∂f , g · ∂f i H . roof. If all E ( m ) ’s are local then by Proposition 2.1 we have (cid:13)(cid:13)(cid:13) g | X ( m ) · ∂ ( m ) ( f | X ( m ) ) (cid:13)(cid:13)(cid:13) H ( m ) = Z X ( m ) ( g | X ( m ) ) dν ( m ) ,cf | X ( m ) = Z X g dν ( m ) f | ( m ) X for all f ∈ H n ( X ) and g ∈ C ( X ), where ν ( m ) ,cf denotes the local part of the energy measure of f with respectto ( E ( m ) , F ( m ) ), and by (50) this converges to Z X g dν f = k g · ∂f k H . Suppose now that the E ( m ) ’s have nontrivial jump measures J ( m ) . If f ∈ H n ( X ) and g ∈ H n ′ ( X ) havedisjoint supports then by Proposition 2.1, the locality of E ( m ) ,c , (49) and the locality of E we have − m Z X Z X f ( x ) g ( y ) J ( m ) ( dxdy ) = lim m Z X ( m ) Z X ( m ) ( f ( x ) − f ( y ))( g ( x ) − g ( y )) J ( m ) ( dxdy )= lim m E ( m ) ( f, g )= E ( f, g )= 0 . (61)Given f, g ∈ C ( X ) with disjoint supports, we can, by the proof of [97, Theorem 3], find sequences offunctions ( f j ) j and ( g j ) j from H ∗ ( X ) approximating f and g uniformly and disjoint compact sets K ( f ) ⊂ X and K ( g ) ⊂ X such that all f j and g j are supported in K ( f ) and K ( g ), respectively. Therefore (61) andthe arguments used in the proof of [27, Theorem 3.2.1] imply that lim m J ( m ) = 0 vaguely on X × X \ diag.For functions f ∈ H n ( X ) and g ∈ C ( X ) we therefore havelim m Z X ( m ) Z X ( m ) ( d u g ( x, y )) ( d u f ( x, y )) J ( m ) ( dxdy ) = 0 , as can be seen using the arguments in the proof of [48, Lemma 3.1]. On the other hand we have k g · ∂f k H = lim m (cid:26)Z X ( m ) g dν ( m ) ,cf + 12 Z X ( m ) Z X ( m ) ( g ( x ) + g ( y ))( d u f ( x, y )) J ( m ) ( dxdy ) (cid:27) for such f and g by (50) and (14). Combining and taking into account Proposition 2.1 we can conclude that k g · ∂f k H = lim m (cid:26)Z X ( m ) g dν ( m ) ,cf + 12 Z X ( m ) Z X ( m ) ( g ( x, y )) ( d u f ( x, y )) J ( m ) ( dxdy ) (cid:27) = lim m (cid:13)(cid:13)(cid:13) g | X ( m ) · ∂ ( m ) ( f | X ( m ) ) (cid:13)(cid:13)(cid:13) H ( m ) , from which the stated result follows by polarization. (cid:3) Another consequence, in particular of Assumption 5.5, is the convergence of the L -spaces and the energydomains in the sense of Definition A.1. Corollary 5.2. (i)
We have (62) lim m →∞ L ( X ( m ) , µ ( m ) ) = L ( X, µ ) in the KS-sense with identification operators Φ m as above. (ii) We have (63) lim m →∞ F ( m ) = F in the KS-sense with identification operators u ( H m u ) | X ( m ) mapping from F into F ( m ) respec-tively. (iii) If f ∈ F and ( f m ) m is a sequence of functions f m ∈ F ( m ) such that lim m f m = f KS-strongly w.r.t.(63) then we also have lim m f m = f KS-strongly w.r.t. (62). roof. Statement (i) is immediate from (56). To see statement (ii) let u ∈ F . If x ∈ V is fixed, we have H m u ( x ) = u ( x ) for any m and therefore, by (1) and (48),lim m k u − H m u k L ( X,µ ) ≤ µ ( X ) lim m k u − H m u k ≤ µ ( X ) diam R ( X ) lim m E ( u − H m u ) = 0 . Using (55), we obtain lim m k Φ m H m u − Φ m u k L ( X,µ ) = 0, and combining with (58) and (56),lim m k ( H m u ) | X ( m ) k L ( X ( m ) ,µ ( m ) ) = lim m k ( H m u ) | X ( m ) − Φ m H m u k L ( X ( m ) ,µ ( m ) ) + lim m k Φ m H m u k L ( X ( m ) ,µ ( m ) ) = lim m k Φ m u k L ( X ( m ) ,µ ( m ) ) = k u k L ( X,µ ) . Together with (51) this shows that lim m E ( m )1 (( H m u ) | X ( m ) ) = E ( u ) for all u ∈ F . To see (iii) note thataccording to the hypothesis, there exist ϕ n ∈ F such thatlim n E ( ϕ n − f ) = 0 and lim n lim m E ( m )1 (( H m ϕ n ) | X ( m ) − f m ) = 0 . This implies lim n k ϕ n − f k L ( X,µ ) = 0 and lim n lim m k ( H m ϕ n ) | X ( m ) − f m k L ( X ( m ) ,µ ( m ) ) = 0. Conditions(56) and (58), applied to the constant function , yield lim m µ ( m ) ( X ( m ) ) = µ ( X ), and in particular,(64) sup m µ ( X ( m ) ) < + ∞ . We may therefore use (58) to conclude lim m k ( H m ϕ n ) | X ( m ) − Φ m H m ϕ n k L ( X ( m ) ,µ ( m ) ) = 0 for any n , so that(65) lim n lim m k Φ m H m ϕ n − f m k L ( X ( m ) ,µ ( m ) ) = 0 . Let x ∈ V . Then, since H m ϕ n ( x ) = ϕ ( x ) for all m and n , the resistance estimate (1) implieslim m k H m ϕ n − ϕ n k L ( X,µ ) = 0for all n . Together with (55) it follows thatlim n lim m k Φ m H m ϕ n − Φ m ϕ n k L ( X ( m ) ,µ ( m ) ) ≤ (cid:18) sup m k Φ m k L ( X,µ ) → L ( X ( m ) ,µ ( m ) ) (cid:19) lim n lim m k H m ϕ n − ϕ n k L ( X,µ ) = 0 , and combining with (65) we obtain lim n lim m k Φ m ϕ n − f m k L ( X ( m ) ,µ ( m ) ) = 0. (cid:3) In the sequel we will say ’KS-weakly’ resp. ’KS-strongly’ if we refer to the convergence (62) and say’KS-weakly w.r.t. (63)’ resp.’KS-strongly w.r.t. (63)’ if we refer to the convergence (63). We finally recorda property of KS-weak convergence that will be useful later on.
Lemma 5.2. If lim m f m = f KS-weakly and w ∈ F then there is a sequence ( m k ) k with m k ↑ + ∞ such that lim k w | X ( mk ) f m k = wf KS-weakly.Proof.
For any w ∈ F we have lim m w | X ( m ) = w KS-strongly by (58). Fix w ∈ F . Clearlysup m k w | X ( m ) f m k L ( X ( m ) ,µ ( m ) ) < + ∞ by the boundedness of w , hence lim k w | X ( mk ) f m k = e w KS-weakly for some e w ∈ L ( X, µ ) and some sequence( m k ) k . For any v ∈ F we have vw ∈ F and trivially ( vw ) | X ( m ) = v | X ( m ) w | X ( m ) , hence h e w, v i L ( X,µ ) = lim k h w | X ( mk ) f m k , v | X ( mk ) i L ( X ( mk ) ,µ ( mk ) ) = lim k h f m k , ( vw ) | X ( mk ) i L ( X ( mk ) ,µ ( mk ) ) = h f, wv i L ( X,µ ) = h wf, v i L ( X,µ ) , what by the density of F in L ( X, µ ) implies e w = wf and therefore the lemma. (cid:3) .3. Boundedness and convergence of vector fields.
We provide a version of Proposition 4.1 for finitelyramified sets. Recall the notation from Assumption 5.5.
Proposition 5.1. (i)
Given b ∈ H and M > let n be such that max α ∈A n diam R ( X α ) ≤ M k b k H . Then for all g ∈ F we have k g · b k H ≤ M E ( g ) + 2 V ( n ) k b k H k g k L ( X,µ ) . (ii) Suppose b ( m ) ∈ H ( m ) , M > and n ≤ m are such that max α ∈A n diam R ( m ) ( X ( n ) α ) ≤ M (cid:13)(cid:13) b ( m ) (cid:13)(cid:13) H ( m ) . Then for all g ∈ F ( m ) we have (cid:13)(cid:13) g · b ( m ) (cid:13)(cid:13) H ( m ) ≤ M E ( m ) ( g ) + 2 V ( n ) (cid:13)(cid:13) b ( m ) (cid:13)(cid:13) H ( m ) (cid:13)(cid:13) g (cid:13)(cid:13) L ( X ( m ) ,µ ( m ) ) . Proof.
We use the shortcut ( g ) X α = µ ( X α ) R X α g dµ . For any α ∈ A n and x ∈ X α have, by (1), | g ( x ) − ( g ) X α | ≤ E ( g ) / diam R ( X α ) / and therefore | g ( x ) | ≤ E ( g ) diam R ( X α ) + 2( g ) X α ≤ M k b k H E ( g ) + 2 V ( n ) k g k L ( X,µ ) . Creating a finite partition of X from the cells X α , α ∈ A n , we see that the preceding estimate holds for all x ∈ X , and using (10) we obtain (i). Statement (ii) is similar. (cid:3) Similarly as in Corollary 4.1, uniform norm bounds on the vector fields allow to choose uniform constantsin the Hardy condition (20).
Corollary 5.3.
Suppose b ( m ) ∈ H ( m ) are such that sup m (cid:13)(cid:13) b ( m ) (cid:13)(cid:13) H ( m ) < + ∞ . Then for any M > thereexist γ M > and n such that for each m ≥ n the coefficient b ( m ) satisfies (20) with δ ( b ( m ) ) = M and γ ( b ( m ) ) = γ M .Proof. By Lemma 5.1 and Assumption 5.2 we can find n such thatsup m ≥ n max α ∈A n diam R ( m ) ( X ( m ) α ) ≤ M sup m (cid:13)(cid:13) b ( m ) (cid:13)(cid:13) H ( m ) . Setting(66) γ M := 2 V ( n ) sup m ≥ n (cid:13)(cid:13) b ( m ) (cid:13)(cid:13) H ( m ) we obtain the desired result by Proposition 5.1. (cid:3) To formulate an analog of Corollary 4.2 for varying spaces we need a certain compatibility of the vectorfields involved. One rather easy way to ensure the latter is to focus on suitable elements of the moduleΩ a ( X ) and their equivalence classes in H and H ( m ) which then define vector fields b on X and b ( m ) on X ( m ) suitable to allow an approximation procedure. Given an element of Ω a ( X ) of the special form P i g i · d u f i with g i ∈ C ( X ) and f i ∈ H n ( X ), let b defined as its H -equivalence class (cid:2) P i g i · d u f i (cid:3) H as in Section 2, thatis,(67) b := X i g i · ∂f i . By Assumption 5.3 we have f i | X ( m ) ∈ F ( m ) for all i and m , so that P i g i | X ( m ) · d u ( f i | X ( m ) ) is an element ofΩ a ( X ( m ) ). We define b ( m ) to be its H ( m ) -equivalence class (cid:2) P i g i | X ( m ) · d u ( f i | X ( m ) ) (cid:3) H ( m ) , that is,(68) b ( m ) := X i g i | X ( m ) · ∂ ( m ) ( f i | X ( m ) ) . he following convergence result may be seen as a partial generalization of (50). It is immediate fromCorollary 5.1 and bilinear extension. Corollary 5.4.
Suppose b and b ( m ) are as in (67) and (68) and g ∈ C ( X ) . Then we have (69) lim m (cid:13)(cid:13)(cid:13) g | X ( m ) · b ( m ) (cid:13)(cid:13)(cid:13) H ( m ) = k g · b k H . Remark . One might argue that an analog of Corollary 4.2 in terms of a simple restriction of vector fields b ∈ H to X ( m ) would be more convincing than Corollary 5.4. However, as H and H ( m ) are obtained bydifferent factorizations, it is not obvious how to correctly define a restriction operation on all of H . Usingthe finitely ramified cell structure one can introduce restrictions b | X ( m ) to X ( m ) of certain types of vectorfields b ∈ H and obtain an counterpart of (69) with these restrictions b | X ( m ) in place of the b ( m ) ’s. Thisauxiliary observation is discussed in Section 7, it is not needed for our main results.5.4. Accumulation points.
Let a ∈ C ( X ) satisfy (37) with 0 < λ < Λ, suppose
M > λ := λ/ − /M > b ( m ) , ˆ b ( m ) ∈ H ( m ) satisfy(70) sup m (cid:13)(cid:13)(cid:13) b ( m ) (cid:13)(cid:13)(cid:13) H ( m ) < + ∞ and sup m || ˆ b ( m ) || H ( m ) < + ∞ . Let γ M be as in (66) and ˆ γ M similarly but with the ˆ b ( m ) in place of b ( m ) and suppose that c ∈ C ( X ) satisfies(42). Then for each m the form ( Q ( m ) , F ( m ) ) as in (60) is a closed form on L ( X ( m ) , µ ( m ) ), and (25) holdswith δ ( b ( m ) ) = δ (ˆ b ( m ) ) = 1 /M and with γ M , ˆ γ M in place of γ ( b ), γ (ˆ b ) in (26). There is a constant K > m the generator ( L Q ( m ) , D ( L Q ( m ) )) of ( Q ( m ) , F ( m ) ) obeys the sector condition (28) withsector constant K . As a consequence, we can observe the following uniform energy bounds on solutions toelliptic and parabolic equations similar to Proposition 4.2. Proposition 5.2.
Let a , b ( m ) , ˆ b ( m ) and c be as above such that (70) and (42) hold. (i) If f ∈ L ( X, µ ) , and u m is the unique weak solution to (29) with L Q ( m ) in place of L and f m = Φ m f in place of f then we have sup m Q ( m )1 ( u m ) < + ∞ . (ii) If ˚ u ∈ L ( X, µ ) , and u m is the unique solution to (34) in L ( X ( m ) , µ ( m ) ) with L Q ( m ) in place of L and with initial condition ˚ u m = Φ m ˚ u then for any t > we have sup m Q ( m )1 ( u m ( t )) < + ∞ .Proof. Since (42) and (28) hold with the same constants c and K for all m , Corollaries 3.2 and 3.3 togetherwith (55) yield that sup m Q ( m )1 ( u m ) ≤ (cid:16) c + c (cid:17) k f k L ( X,µ ) and sup m Q ( m )1 ( u m ( t )) ≤ (cid:0) C K t + 1 (cid:1) k ˚ u k L ( X,µ ) and the results follow. (cid:3) Remark . Proposition 5.2 needs only Assumption 5.5 (i) and (ii). Assumption 5.3, Assumption 5.4 andAssumption 5.5 (iii) and (iv) are not needed.
Remark . The hypotheses of Proposition 5.2 imply that (( Q ( m ) , F ( m ) )) m is an equi-elliptic family in thesense of [83, Definition 2.1].By the compactness of X we can find accumulation points in C ( X ) for extensions to X of linearizationsof solutions. The next corollary may be seen as an analog of Corollary 4.3. Recall the definitions of theprojections H ( m ) m and the extension operators Ext m . Corollary 5.5.
Let a , b ( m ) , ˆ b ( m ) and c be as above such that (70) and (42) hold. (i) If f ∈ L ( X, µ ) , and u m is the unique weak solution to (29) with L Q ( m ) in place of L and f m = Φ m f in place of f then each subsequence ( u m k ) k of ( u m ) m has a further subsequence ( u m kj ) j such that (Ext m kj H ( m kj ) m kj u m kj ) j converges to a limit e u ∈ C ( X ) uniformly on X . (ii) If ˚ u ∈ L ( X, µ ) , and u m is the unique solution to (34) in L ( X ( m ) , µ ( m ) ) with L Q ( m ) in place of L and with initial condition ˚ u m = Φ m ˚ u then for any t > each subsequence ( u m k ( t )) k of ( u m ( t )) m hasa further subsequence ( u m kj ( t )) j such that (Ext m kj H ( m kj ) m kj u m kj ( t )) j converges to a limit e u t ∈ C ( X ) uniformly on X . .5. Generalized strong resolvent convergence.
The next result is an analog of Theorem 4.1 on varyingspaces, it uses notions of convergence along a sequence of varying Hilbert spaces, [77,98], see Appendix A. Thekey ingredient is Theorem A.1 - a special case of [99, Theorem 7.15, Corollary 7.16 and Remark 7.17], whichconstitute a natural generalization of [38, Theorem 3.1] to the framework of varying Hilbert spaces, [77].
Theorem 5.1.
Suppose that (71) b = X i g i · ∂f i and ˆ b = X i ˆ g i · ∂ ˆ f i are finite linear combinations with f i , ˆ f i , g i , ˆ g i ∈ H n ( X ) as in (67) and for any m let (72) b ( m ) := X i g i | X ( m ) · ∂ ( m ) ( f i | X ( m ) ) and ˆ b ( m ) := X i ˆ g i | X ( m ) · ∂ ( m ) ( ˆ f i | X ( m ) ) as in (68). Let a ∈ H n ( X ) be such that (19) holds and let c ∈ C ( X ) . Then lim m L Q ( m ) = L Q in theKS-generalized resolvent sense, and the following hold. (i) If f ∈ L ( X, µ ) , u is the unique weak solution to (29) on X and u m is the unique weak solution to(29) on X ( m ) with L Q ( m ) and Φ m f in place of L Q and f , then we have lim m u m = u KS-strongly.Moreover, there is a sequence ( m k ) k with m k ↑ + ∞ such that lim k Ext m k H ( m k ) m k u m k = u uniformlyon X . (ii) If ˚ u ∈ L ( X, µ ) , u is the unique solution to (34) on X and u m is the unique weak solution to(34) on X ( m ) with L Q ( m ) and Φ m ˚ u in place of L Q and ˚ u , then for any t > we have we have lim m u m = u KS-strongly. Moreover, for any t > there is a sequence ( m k ) k with m k ↑ + ∞ suchthat lim k Ext m k H ( m k ) m k u m k ( t ) = u ( t ) uniformly on X . A version of Theorem 5.1 for more general coefficients is stated below in Theorem 5.2. The proof ofTheorem 5.1 makes use of the following key fact.
Lemma 5.3.
Suppose ( n k ) k is a sequence with n k ↑ + ∞ and ( u k ) k is a sequence with u k ∈ L ( X ( n k ) , µ ( n k ) ) converging to u ∈ L ( X, µ ) KS-weakly and satisfying sup k E ( n k )1 ( u k ) < ∞ . Then we have u ∈ F , and thereis a sequence ( k j ) j with k j ↑ + ∞ such that (i) lim j u n kj = u KS-weakly w.r.t. (63), and moreover, for any f ∈ F and any sequence ( f j ) j such that f j ∈ F ( n kj ) and lim j f j = f KS-strongly w.r.t. (63) along ( n k j ) j we have (73) lim j E ( n kj ) ( f j , u n kj ) = E ( f, u ) . (ii) lim j Ext n kj H ( n kj ) n kj u n kj = u uniformly on X .Proof. Let v k := Ext n k H ( n k ) n k u k . By hypothesis and (49) we have(74) sup k E ( v k ) = sup k E ( n k ) ( H ( n k ) n k ( u k )) ≤ sup k E ( n k ) ( u n k ) < + ∞ . Since v k | X ( nk ) = H ( n k ) n k u k , (74), (64) and (53) allow to conclude that(75) lim k k v k | X ( nk ) − u k k L ( X ( nk ) ,µ ( nk ) ) = 0 , what implies that lim k v k | X ( nk ) = u KS-weakly.We now claim that for any n and any w ∈ H n ( X ) we have(76) lim k h w, v k i L ( X,µ ) = h w, u i L ( X,µ ) . We clearly have lim k Φ n k w = w KS-strongly. Therefore h w, u i L ( X,µ ) = lim k h Φ n k w, v k | X ( nk ) i L ( X nk ,µ ( nk ) ) , and using (58) and (74) this limit is seen to equallim k h Φ n k w, Φ n k v k i L ( X nk ,µ ( nk ) ) = lim k (cid:10) Φ ∗ n k Φ n k w, v k (cid:11) L ( X,µ ) . pplying (57) we arrive at (76). By (74), and since (59) implies sup k k v k k L ( X,µ ) < + ∞ , we can find asequence ( k j ) j with lim j k j = + ∞ such that ( u k j ) j converges KS-weakly w.r.t. (63) to a limit u E ∈ F and( v k j ) j converges weakly in L ( X, µ ) to a limit u E ∈ F . Since S n ≥ H n ( X ) is dense in L ( X, µ ) we have u E = u by (76), what shows that u ∈ F . We now verify that(77) u E = u E . For any w ∈ H n ( X ) the equalities E ( w, u E ) = lim j n E ( w, v k j ) + (cid:10) w, v k j (cid:11) L ( X,µ ) o = lim j n E ( w, v k j ) − (cid:10) Φ ∗ n kj Φ n kj w, v k j (cid:11) L ( X,µ ) o = lim j n E ( n kj ) ( w | X ( nkj ) , v k j | X ( nkj ) ) − (cid:10) Φ n kj w, Φ n kj v k j (cid:11) L ( X ( nkj ) ,µ ( nkj ) ) o hold, the second and third equality due to (57) and (49), respectively. Using (58) twice on the secondsummands in the last line, the above limit is seen to equallim j n E ( n kj ) ( w | X ( nkj ) , v k j | X ( nkj ) ) − (cid:10) w | X ( nkj ) , v k j | X ( nkj ) (cid:11) L ( X ( nkj ) ,µ ( nkj ) ) o . For j so large that n k j ≥ n the function w | X ( nkj ) is an element of H n kj ( X ( n kj ) ), so that by orthogonality in F ( n kj ) we can replace v k j | X ( nkj ) = H ( n kj ) n kj u k j in the first summand by u k j . In the second term we can makethe same replacement by (53) and (64), so that the above can be rewrittenlim j n E ( n kj ) ( w | X ( nkj ) , u k j ) − (cid:10) w | X ( nkj ) , u k j (cid:11) L ( X ( nkj ) ,µ ( nkj ) ) o = lim j E ( n kj )1 ( w | X ( nkj ) , u k j )= E ( w, u E ) , because lim j w | X ( nkj ) = w KS-strongly w.r.t. (63). Since S n ≥ H n ( X ) is dense in F , this implies (77) andtherefore the first statement of (i), so far for the sequence ( u k j ) j . The statement on the limit (73) in (i)follows by Corollary 5.2.To save notation in the proof of (ii) we now write ( u k ) k for the sequence ( u k j ) j extracted in (i). Let x ∈ V . Then (1) implies that ( v k − v k ( x )) k is an equicontinuous and equibounded sequence of functionson X , so that by Arzel`a-Ascoli we can find a subsequence ( v k j − v k j ( x )) j which converges uniformly on X to a function w x ∈ C ( X ). Since µ is finite, this implies lim j v k j − v k j ( x ) = w x in L ( X, µ ). By (58) and(74) we also have lim j (cid:13)(cid:13)(cid:13) v k j | X ( nkj ) − v k j ( x ) − Φ n kj ( v k j − v k j ( x )) (cid:13)(cid:13)(cid:13) L ( X ( nkj ) ,µ ( nkj ) ) = 0 , so that combining, we see that lim j ( v k j | X ( nkj ) − u k j | X ( nkj ) ( x )) = w x KS-strongly and therefore also KS-weakly. Since lim k v k | X ( nk ) = u KS-weakly by (75), we may conclude that lim k v k | X ( nk ) ( x ) = u − w x KS-weakly. In particular, by [77, Lemma 2.3],sup j | v k j | X ( nkj ) ( x ) | µ ( X ( n kj ) ) / = sup j (cid:13)(cid:13)(cid:13) v k j | X ( nkj ) ( x ) (cid:13)(cid:13)(cid:13) L ( X ( nkj ) ,µ ( nkj ) ) < + ∞ . Since lim m µ ( m ) ( X ( m ) ) = µ ( X ) > v k j | X ( nkj ) ( x ) is a bounded sequence of real numbersand therefore has a subsequence converging to some limit z ∈ R . Keeping the same notation for thissubsequence, we can use (58) and (64) to conclude that lim j (cid:13)(cid:13) v k j | X ( nkj ) ( x ) − Φ n kj z (cid:13)(cid:13) L ( X ( nkj ) ,µ ( nkj ) ) = 0,hence lim j v k j | X ( nkj ) ( x ) = z KS-weakly and therefore necessarily z = u − w x . This implies that lim j v k j =lim j ( v k j − v k j ( x )) + lim j v k j ( x ) = u uniformly on X as stated in (ii). Clearly the statements in (i) remaintrue for this subsequence. (cid:3) We prove Theorem 5.1.
Proof.
Since the operators L Q ( m ) obey the sector condition (28) with the same sector constant, Theorem A.1will imply the desired convergence, provided that the forms Q ( m ) and Q satisfy the conditions in DefinitionA.2. Corollary 5.5 then takes care of the claimed uniform convergences. ithout loss of generality we may (and do) assume that the function c ∈ C ( X ) satisfies condition (42).Otherwise we use the same shifting argument as in the proof of Theorem 4.1, the statements on uniformconvergence then follow using Corollary 5.5.By (23), (24), (25) and (26) together with Proposition 4.1 and Corollaries 5.3 and 5.4 we can find aconstant C > m we have(78) C E ( m )1 ( f ) ≤ Q ( m )1 ( f ) ≤ C − E ( m )1 ( f ) , f ∈ F ( m ) . To check condition (i) in Definition A.2, suppose that ( u m ) m is a sequence with u m ∈ L ( X ( m ) , µ ( m ) )converging KS-weakly to a function u ∈ L ( X, µ ) and such that lim m Q ( m )1 ( u m ) < + ∞ . It has a subsequence( u m k ) k which by (78) satisfies sup k E ( m k ) ( u m k ) < + ∞ , and by Lemma 5.3 we then know that u ∈ F , whatimplies the condition.To verify condition (ii), suppose that u ∈ F , ( m k ) k is a sequence with m k ↑ + ∞ and that u k ∈ L ( X ( m k ) , µ ( m k ) ) are such that lim k u k = u KS-weakly and sup k Q ( m k )1 ( u k ) < + ∞ . By (78) we havesup k E ( m k )1 ( u k ) < + ∞ . Now let w ∈ H n ( X ). Clearly lim m w | X ( m ) = w KS-strongly. By Lemma 5.2we may assume that along ( m k ) k we also have lim k a | X ( mk ) u k = au and lim k ( w ˆ g i ) | X ( mk ) u k = w ˆ g i u KS-weakly for all i , otherwise we pass to a suitable subsequence. By (5) also sup k E ( m k )1 ( a | X ( mk ) u k ) < + ∞ andsup k E ( m k )1 (( w ˆ g i ) | X ( mk ) u k ) < + ∞ . By Lemma 5.3 we can therefore find a sequence ( k j ) j as stated so that (i)and (ii) in Lemma 5.3 hold simultaneously for the sequences ( u k j ) j , ( a | X ( mkj ) u k j ) j and (( w ˆ g i ) | X ( mkj ) u k j ) j with limits u , au and w ˆ g i u , respectively. Our first claim is that(79) lim j D ∂ ( m kj ) ( w | X ( mkj ) ) , a | X ( mkj ) · ∂ ( m kj ) u k j E H ( mkj ) = h ∂w, a · ∂u i H . To see this note first that by the Leibniz rule for ∂ ( m kj ) each element of the sequence on the left hand sideequals D ∂ ( m kj ) ( w | X ( mkj ) ) , ∂ ( m kj ) ( a | X ( mkj ) u k j ) E H ( mkj ) − D ∂ ( m kj ) ( w | X ( mkj ) ) , u k j · ∂ ( m kj ) ( a | X ( mkj ) ) E H ( mkj ) . The first term converges to h ∂w, ∂ ( au ) i H by (73). In the second summand we can replace u k j by H ( m kj ) m kj u k j ,note that by (10) and (53) we havelim j (cid:13)(cid:13)(cid:13) ( u m kj − H ( m kj ) m kj u k j ) · ∂ ( m kj ) ( a | X ( mkj ) ) (cid:13)(cid:13)(cid:13) H ( mkj ) = 0 . By Lemma 5.3 (ii) we also havelim j (cid:13)(cid:13)(cid:13) ( H ( m kj ) m kj u k j − u | X ( mkj ) ) · ∂ ( m kj ) ( a | X ( mkj ) ) (cid:13)(cid:13)(cid:13) H ( mkj ) = 0 , so that lim j D ∂ ( m kj ) ( w | X ( mkj ) ) , u k j · ∂ ( m kj ) ( a | X ( mkj ) ) E H ( mkj ) = lim j D ∂ ( m kj ) ( w | X ( mkj ) ) , u · ∂ ( m kj ) ( a | X ( mkj ) ) E H ( mkj ) = h ∂w, u · ∂a i H by Corollary 5.4 and polarization. Using the Leibniz rule for ∂ we arrive at (79). We next claim that(80) lim j D w | X ( mkj ) · ˆ b ( m kj ) , ∂ ( m kj ) u m kj E H ( mkj ) = D w · ˆ b, ∂u E H . Each element of the sequence on the left hand side is a finite linear combination with summands D ∂ ( m kj ) ( ˆ f i | X ( mkj ) ) , ∂ ( m kj ) (( w ˆ g i ) | X ( mkj ) u k j ) E H ( mkj ) − D ∂ ( m kj ) ( ˆ f i | X ( mkj ) ) , u k j · ∂ ( m kj ) (( w ˆ g i ) | X ( mkj ) ) E H ( mkj ) . The first term converges to D ∂ ˆ f i , ∂ ( w ˆ g i u ) E H by (73). To see that(81) lim j D ∂ ( m kj ) ( ˆ f i | X ( mkj ) ) , u k j · ∂ ( m kj ) (( w ˆ g i ) | X ( mkj ) ) E H ( mkj ) = D ∂ ˆ f i , u · ∂ ( w ˆ g i ) E H et ε > n ′ so that by (48) we have(82) E ( H n ′ ( w ˆ g i ) − w ˆ g i ) / < ε k u k − E ( ˆ f i ) − / . For any j so that m k j ≥ n ′ we have H ( m kj ) m kj (( w ˆ g i ) | X ( mkj ) ) = H m kj ( w ˆ g i ) | X ( mkj ) = H n ′ ( w ˆ g i ) | X ( mkj ) and by (54) therefore(83) E ( m kj ) ( H n ′ ( w ˆ g i ) | X ( mkj ) − ( w ˆ g i ) | X ( mkj ) ) / < ε E ( f i ) − / E ( u ) − / for large enough j . Since as before we can replace u k j by u | X ( mkj ) , (83) shows thatlim j | D ∂ ( m kj ) ( ˆ f i | X ( mkj ) ) , u k j · ∂ ( m kj ) (( w ˆ g i ) | X ( mkj ) ) E H ( mkj ) − D ∂ ( m kj ) ( ˆ f i | X ( mkj ) ) , u · ∂ ( m kj ) ( H n ′ ( w ˆ g i ) | X ( mkj ) ) E H ( mkj ) | < ε . By Corollary 5.4 and (82) we havelim j | D ∂ ( m kj ) ( ˆ f i | X ( mkj ) ) , u · ∂ ( m kj ) ( H n ′ ( w ˆ g i ) | X ( mkj ) ) E H ( mkj ) − D ∂ ˆ f i , u · ∂ ( w ˆ g i ) E H | < ε . Since ε was arbitrary, we can combine these two estimates to conclude (81) and therefore (80). The identity(84) lim j D u k j · b ( m kj ) , ∂ ( m kj ) ( w | X ( mkj ) ) E H ( mkj ) = h u · b, ∂w i H follows by linearity from the fact that by Lemma 5.3 (ii) and Corollary 5.4 we havelim j D ( u k j g i | X ( mkj ) ) · ∂ ( m kj ) f i , ∂ ( m kj ) ( w | X ( mkj ) ) E H ( mkj ) = lim j D ( ug i | X ( mkj ) ) · ∂ ( m kj ) f i , ∂ ( m kj ) ( w | X ( mkj ) ) E H ( mkj ) = h ( ug i ) · ∂f i , ∂w i H . Together with the obvious identitylim j D ( cw ) | X ( mkj ) , u k j E L ( X ( mkj ) , µ ( mkj ) ) = h cw, u i L ( X,µ ) , formulas (79), (80) and (84) imply lim j Q ( m kj ) ( w | X ( mkj ) , u k j ) = Q ( w, u ) , what shows condition (ii) in Definition A.2. (cid:3) Theorems 4.1 and 5.1 together allow an approximation result involving more general coefficients.
Theorem 5.2.
Let a ∈ F be such that (19) holds with < λ < Λ . Let b, ˆ b ∈ H and let c ∈ C ( X ) . Then wecan find a ( m ) n ∈ F ( m ) and b ( m ) n , ˆ b ( m ) n ∈ H ( m ) such that for any n and m the forms Q ( n,m ) ( f, g ) = h a n | X ( m ) · ∂f, ∂g i H ( m ) − (cid:10) g · b ( m ) n , ∂f (cid:11) H ( m ) − (cid:10) f · ˆ b ( m ) n , ∂g (cid:11) H ( m ) − h c | X ( m ) f, g i L ( X ( m ) ,µ ( m ) ) , f, g ∈ F ( m ) (85) are sectorial closed forms on L ( X ( m ) , µ ( m ) ) , respectively. Moreover, writing ( L Q ( n,m ) , D ( L Q ( n,m ) )) for thegenerator of the form ( Q ( n,m ) , D ( Q ( n,m ) )) , we can observe the following. (i) If f ∈ L ( X, µ ) , u is the unique weak solution to (29) on X and u ( m ) n is the unique weak solutionto (29) on X ( m ) with L Q ( n,m ) and Φ m f in place of L Q and f , then there are sequences ( m k ) k and ( n l ) l with m k ↑ + ∞ and n l ↑ + ∞ so that lim l lim k (cid:13)(cid:13) Ext m k H ( m k ) m k u ( m k ) n l − u (cid:13)(cid:13) sup = 0 . ii) If ˚ u ∈ L ( X, µ ) , u is the unique solution to (34) on X and u ( m ) n is the unique weak solution to (34)on X ( m ) with L Q ( n,m ) and Φ m ˚ u in place of L Q and ˚ u , then for any t > there are sequences ( m k ) k and ( n l ) l with m k ↑ + ∞ and n l ↑ + ∞ so that lim l lim k (cid:13)(cid:13) Ext m k H ( m k ) m k u ( m k ) n l ( t ) − u ( t ) (cid:13)(cid:13) sup = 0 . Remark . By [6, Corollary 1.16] we can find a sequence ( l k ) k with l k ↑ + ∞ such thatlim k (cid:13)(cid:13) Ext m k H ( m k ) m k u ( m k ) n lk − u (cid:13)(cid:13) sup = 0in the situation of Theorem 5.2 (i) and similarly for (ii).The following is a straightforward consequence of the density of H ∗ ( X ) in F , we omit its short proof. Lemma 5.4.
The space of finite linear combinations P i g i ∂f i with g i , f i ∈ H ∗ ( X ) is dense in H . We prove Theorem 5.2.
Proof.
Given a ∈ F , let ( a n ) n ⊂ H ∗ ( X ) be a sequence approximating a uniformly on X and such thatall a n satisfy (19) with the same constants 0 < λ < Λ as a . Let M > λ := λ/ − /M >
0. By Lemma 5.4 there exist b n := X i g n,i ∂f n,i and ˆ b n := X i ˆ g n,i ∂ ˆ f n,i with f n,i , ˆ f n,i , g n,i , ˆ g n,i ∈ H ∗ ( X ) that approximate b and ˆ b in H , respectively. For each n we can proceed asin (68) and consider the elements b ( m ) n := X i g n,i | X ( m ) · ∂ ( m ) ( f n,i | X ( m ) ) and ˆ b ( m ) n := X i ˆ g n,i | X ( m ) · ∂ ( m ) ( ˆ f n,i | X ( m ) )of H ( m ) . With γ M and ˆ γ M as in (66) and assuming that, without loss of generality, c ∈ C ( X ) satisfies (42),we can conclude that for each n and each sufficiently large m the forms ( Q ( n,m ) , D ( Q ( n,m ) )) as in (85) with D ( Q ( n,m ) ) = F ( m ) are closed forms in L ( X ( m ) , µ ( m ) ).To prove (i), suppose that f ∈ L ( X, µ ) and u is the unique weak solution to (29) on X . Let u ( m ) n be theunique weak solution to (29) on X ( m ) with L Q ( n,m ) and Φ m ( f ) in place of L Q and f . By Theorem 5.1 we canfind a sequence ( m k ) k with m k ↑ + ∞ so that lim k →∞ Ext m k H ( m k ) m k u ( m k )1 = u uniformly on X . Repeatedapplications of Theorem 5.1 allow to thin out ( m k ) k further so that for any n we have (cid:13)(cid:13) Ext m k H ( m k ) m k u ( m k ) j − u j (cid:13)(cid:13) sup < − n , j ≤ n, provided that k is greater than some integer k n depending on n . On the other hand Theorem 4.1 allows tofind a sequence ( n l ) l with n l ↑ + ∞ such that lim l →∞ u n l = u uniformly on X , and combining these facts,we obtain (i). Statement (ii) is proved in the same manner. (cid:3) Discrete and metric graph approximations
Discrete approximations.
We describe approximations in terms of discrete Dirichlet forms, our no-tation follows that of Subsection 5.1. Let ( E , F ) be a local regular resistance form on the compact space( X, R ), obtained under Assumption 5.1 as in Section 5.1, and suppose that also Assumption 5.2 is satisfied.Let X ( m ) = V m , E ( m ) = E V m and F ( m ) = ℓ ( V m ) be the discrete energy forms on the finite subsets V m asin (3). Clearly Assumption 5.3 is satisfied, note that for every u ∈ H m ( X ) we have E V m ( u | V m ) = E ( u ) andthat (50) is immediate from (12). Since every element of ℓ ( V m ) is the pointwise restriction of a function in H m ( X ), the operator H ( m ) m is the identity operator id F ( m ) , so that Assumption 5.4 is trivially satisfied, aspointed out in Remark 5.3.Now let µ be a finite Borel measure on X such that for any m the value V ( m ) := inf α ∈A m µ ( X α ) is strictlypositive. Following [87] we define, for each m , a measure µ ( m ) on V m by µ ( m ) ( { p } ) := Z X ψ p,m ( x ) dµ ( x ) , p ∈ V m , here ψ p,m ∈ H m ( X ) is the (unique) harmonic extension to X of the function { p } on V m . Since X ( m ) α = V α and P p ∈ V α ψ p,m ( x ) = 1 for all m , α ∈ A m and x ∈ X α , we have µ ( m ) ( X ( m ) α ) = X p ∈ V α µ ( m ) ( { p } ) = Z X X p ∈ V α ψ p,m ( x ) µ ( dx ) ≥ µ ( X α ) ≥ V ( m )for all m and α ∈ A m , so that Assumption 5.5 (i) is seen to be satisfied.For each m let Φ m be a linear operator Φ m : L ( X, µ ) → ℓ ( V m , µ ( m ) ) defined byΦ m f ( p ) := 1 µ ( m ) ( { p } ) h f, ψ p,m i L ( X,µ ) , p ∈ V m , f ∈ L ( X, µ ) . In [87, proof of Theorem 1.1] it was shown that for each m the adjoint Φ ∗ m of Φ m equals the harmonicextension operator Ext m : ℓ ( V m , µ ( m ) ) → H m ( X ),Ext m v = X p ∈ V m v ( p ) ψ p,m , v ∈ ℓ ( V m , µ ( m ) )which satisfies k Ext m f k L ( X,µ ) ≤ k f k ℓ ( V m ,µ ( m ) ) for all f ∈ ℓ ( V m , µ ( m ) ). Consequently (55) is fulfilled, andalso (59) holds. The function ψ p,m is supported on the union of all X α , α ∈ A m , which contain the point p .By Assumption 5.2 we therefore have(86) lim m →∞ sup p ∈ V m diam R (supp ψ p,m ) = 0 . If a sequence ( u m ) m ⊂ F is such that sup m E ( u m ) < ∞ then by (1) it is equicontinuous, and combined with(86) it follows that given ε > p ∈ V m sup x ∈ ψ p,m | u m ( p ) − u m ( x ) | < ε whenever m is large enough, and consequently k Φ m u m − u m | V m k ℓ ( V m ,µ ( m ) ) ≤ X p ∈ V m µ ( m ) ( { p } ) (cid:18)Z X | u m ( x ) − u m ( p ) | ψ p,m ( x ) dµ ( x ) (cid:19) < ε for such m , note that P p ∈ V m ψ p,m ( x ) = 1 for all m and x ∈ X . This shows (58). For every u ∈ F it followsthatlim m →∞ k u | V m k ℓ ( V m ,µ ( m ) ) = lim m →∞ X p ∈ V m Z X (cid:2) ( u ( p ) − u ( x ))( u ( p ) + u ( x )) + u ( x ) (cid:3) ψ p,m ( x ) dµ ( x ) = k u k L ( X,µ ) , since u is bounded and lim m P p ∈ V m R X ( u ( p ) − u ( x )) ψ p,m ( x ) dµ ( x ) = 0 by (86) as above, proving (56). Toverify the remaining condition (57) note that for u ∈ H n ( X ) we have k Φ ∗ m Φ m u − u k L ( X,µ ) ≤ k Φ ∗ m k ℓ ( V m ,µ ( m ) ) → L ( X,µ ) k Φ m u − u | V m k ℓ ( V m ,µ ( m ) ) + k Φ ∗ m ( u | V m ) − u k L ( X,µ ) , and since Φ ∗ m ( u | V m ) = H m u the last summand is bounded by diam R ( X ) / E ( H m u − u ) / µ ( X ) / . Using(48), (55) and (58) condition (57) now follows. Examples . It is well known that p.c.f. self-similar structures form a subclass of finitely ramified sets.Because of its importance, and since we will discuss metric graph approximations for this subclass in thenext section, we provide some details. Let (
K, S, { F j } j ∈ S ) be a connected post-critically finite (p.c.f.) self-similar structure , see [64, Definitions 1.3.1, 1.3.4 and 1.3.13]. The set of finite words w = w w ...w m oflength | w | = m over the alphabet S is denoted by W m := S m , and we write W ∗ = S m ≥ W m . Given a word w ∈ W m we write F w = F w ◦ F w ◦ ... ◦ F w m and use the abbreviations K w := F w ( K ) and V w := F w ( V ). Then( K, { K w } w ∈ W ∗ , { V w } w ∈ W ∗ ) is a finitely ramified cell structure in the sense of Definition 5.1. We considerthe discrete sets V m := ∪ | w | = m V w , m ≥
0, and assume that (( E V m , ℓ ( V m ))) m is a sequence of Dirichlet formsassociated with a regular harmonic structure on K , [64, Definitions 3.1.1 and 3.1.2], that is, there existconstants r j ∈ (0 , j ∈ S , a Dirichlet form E V ( u ) = P p ∈ V P q ∈ V c (0; p, q )( u ( p ) − u ( q )) on ℓ ( V ), forall m ≥ E V m ( u, v ) = X w ∈ W m r − w E ( u ◦ F w , v ◦ F w ) , u, v ∈ ℓ ( V m ) , here r w := r w . . . r w m for w = w ...w m , and ( E V m +1 ) V m = E V m for all m ≥
0. The regularity of the harmonicstructure implies in particular that Ω = K , [64, Theorem 3.3.4], and the limit (3) defines a (self-similar)local regular resistance form ( E , F ) on K . Assumptions 5.1 and 5.2 are clear from general theory, [64]. Examples . Further examples which fit into the above scheme are for instance non-self-similar resistanceforms on Sierpinski gaskets associated with regular harmonic structures, [80], certain energy forms on randomSierpinski gaskets, [33,34], finitely ramified graph-directed sets with a regular harmonic structure, [36, Section4, in particular p. 18], or basilica Julia sets with a regular harmonic structure, [90, Theorem 3.9].6.2.
Metric graph approximations.
We describe approximations in terms of local Dirichlet forms onmetric graphs (also called ’cable-systems’ in [10]). We follow the method in [48] and therefore specify to thecase where X is a post-critically finite self-similar set K . Let the setup and notation be as in Examples 6.1.For each m ≥ V m as the vertex set of a finite simple (unoriented) graph G m = ( V m , E m )with two vertices p, q ∈ V m being the endpoints of the same edge e ∈ E m if there is a word w of length | w | = m such that F − w p, F − w q ∈ V and c (0; F − w , F − w q ) >
0. For each m and e ∈ E m let l e be a positivenumber and identify the edge e with an oriented copy of the interval (0 , l e ) of length l e , we write i ( e ) and j ( e ) for the initial and the terminal vertex of e , respectively. This yields a sequence (Γ m ) m ≥ of metricgraphs Γ m , and for each m the set X Γ m = V m ∪ S e ∈ E m e , endowed with the natural length metric, becomesa compact metric space See [48] for details and further references. By construction we have X Γ m ⊂ X Γ m +1 and X Γ m ⊂ K for each m .On the space X Γ m we consider the bilinear form ( E Γ m , ˙ W , ( X Γ m )), where E Γ m ( f ) := X w ∈ W m r − w X e ∈ E m , e ⊂ K w l e E e ( f e ) and E e ( f e ) = Z l e ( f ′ e ( t )) dt and ˙ W , ( X Γ m ) = { f = ( f e ) e ∈ E m ∈ C ( X Γ m ) : f e ∈ ˙ W , ( e ) , E Γ m ( f ) < + ∞} . Here f e is the restriction of f to e ∈ E m and ˙ W , ( e ) is the homogeneous Sobolev space consisting of locallyLebesgue integrable functions g on the edge e such that E e ( g ) := Z l e ( g ′ ( s )) ds < + ∞ , where the derivative g ′ of g is understood in the distributional sense. Each form E e , e ∈ E m , satisfies(88) ( f e ( s ) − f e ( s ′ )) ≤ l e E e ( f e )for any f ∈ ˙ W , ( X Γ m ) and any s, s ′ ∈ e . See [48] for further details. We approximate K , endowed with( E , F ) as in Examples 6.1, by the spaces X ( m ) = X Γ m carrying the resistance forms E ( m ) = E Γ m withdomains F ( m ) = ˙ W , ( X Γ m ).To a function f ∈ ˙ W , ( X Γ m ) which is linear on each edge e ∈ E m we refer as edge-wise linear function,and we denote the closed linear subspace of ˙ W , ( X Γ m ) of such functions by EL m . If f ∈ EL m , then itsderivative on a fixed edge e is the constant function f ′ e = l − e ( f ( j ( e )) − f ( i ( e ))), so that(89) E e ( f e ) = Z l e ( f ′ e ( t )) dt = 1 l e ( f ( j ( e )) − f ( i ( e ))) on each e ∈ E m . For a general function f ∈ ˙ W , ( X Γ m ) formula (89) becomes an inequality in which theleft hand side dominates the right hand side. Given a function g ∈ ℓ ( V m ) it has a unique extension h to X Γ m which is edge-wise linear, h ∈ EL m . In particular, if f ∈ H m ( K ) is an m -piecewise harmonic functionon the p.c.f. self-similar set K then its pointwise restriction f | X Γ m to X Γ m is a member of EL m , and E Γ m ( f | X Γ m ) = E ( f ). Since any such f ∈ H m ( K ) is uniquely determined by its values on V m ⊂ X Γ m , thisrestriction map is injective, and Assumption 5.3 (i) is seen to be satisfied. Assumption 5.3 (ii) is verified inthe following lemma. By ν ( m ) f we denote the energy measures associated with the form ( E Γ m , ˙ W , ( X Γ m )). Lemma 6.1.
For any f ∈ F we have ν f = lim m →∞ ν ( m ) H m ( f ) | X Γ m weakly on K . roof. For f ∈ F and nonnegative g ∈ C ( K ) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)Z K gdν f (cid:19) − (cid:18)Z K gdν H m ( f ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k g k sup E ( f − H m ( f )) , see [27, Section 3.2]. This implies the relation R K g dν f = lim m R K g dν H m ( f ) , which by the standard decom-position g = g + − g − remains true for arbitrary g ∈ C ( K ). For any m we have Z K g dν H m ( f ) = X w ∈ W m r − w X e ∈ E m ,e ⊂ K w l e ( H m ( f ) ′ e ) g e ( i ( e ))by (4), here H m ( f ) ′ e ∈ R denotes the slope of the restriction H m ( f ) e of H m ( f ) to e . On the other hand, Z K g dν ( m ) H m ( f ) | X Γ m = X w ∈ W m r − w X e ∈ E m ,e ⊂ K w l e ( H m ( f ) ′ e ) Z l e g e ( t ) dt, and given ε > e ∈ E m sup s,t ∈ e | g ( s ) − g ( t ) | < ε whenever m is large enough, and in this case, (cid:12)(cid:12)(cid:12)(cid:12)Z K g dν H m ( f ) − Z K g dν ( m ) H m ( f ) | X Γ m (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε X w ∈ W m r − w X e ∈ E m ,e ⊂ K w l e ( H m ( f ) ′ e ) = ε E Γ m ( H m ( f ) | X Γ m ) ≤ ε E ( f ) . Combining, it follows that lim m R K g dν f = lim m R K g dν H m ( f ) | X Γ m . (cid:3) We verify condition (53) in Assumption 5.4 in the present setup. It states that the small oscillations onthe interior of individual edges in X Γ m subside uniformly for sequences of functions with a uniform energybound. Lemma 6.2.
Let ( f m ) m be a sequence of functions f m ∈ ˙ W , ( X Γ m ) such that sup m E Γ m ( f m ) < + ∞ and f m | V m = 0 for all m . Then lim m k f m k sup ,X Γ m = 0 .Proof. By (88) we have sup t ∈ e | ( f m ) e ( t ) | ≤ l e Z l e (( f ′ m ) e ( t )) dt on each e ∈ E m and consequently k f m k ,X Γ m ≤ X e ∈ E m sup t ∈ e | ( f m ) e ( t ) | ≤ (max j ∈ S r j ) m sup n E Γ n ( f n ) . (cid:3) By H Γ m we denote the orthogonal projection in ˙ W , ( X Γ m ) onto EL m . Given f m ∈ ˙ W , ( X Γ m ) it clearlyfollows that f m − H Γ m f m ∈ ˙ W , ( X Γ m ), we have ( f m − H Γ m f m ) | V m = 0 and E Γ m ( f m − H Γ m f m ) ≤ E Γ m ( f m ).We verify (54) in Assumption 5.4. Lemma 6.3.
Given f, g ∈ H n ( X ) , we have lim m →∞ E Γ m (cid:0) f | X Γ m g | X Γ m − H Γ m (cid:0) f | X Γ m g | X Γ m (cid:1)(cid:1) = 0 . Proof.
We first note that for any m ≥ n the functions f e and g e are linear on any fixed e ∈ E m , f e ( t ) = f e (0) + f ′ e · t and g e ( t ) = g e (0) + g ′ e · t, t ∈ [0 , l e ] , with slopes f ′ e ∈ R and g ′ e ∈ R , respectively. Therefore E e ( f e ) = l e ( f ′ e ) for each such e and l e ( f ′ e ) ≤ X | w | = m X e ∈ E m ,e ∈ K w l e ( f ′ e ) ≤ (max i r i ) m sup m ≥ n E Γ m (cid:0) f | X Γ m (cid:1) = (max i r i ) m E ( f ) , (90)similarly for the function g . Since( f g ) e ( t ) = f e ( t ) g e ( t ) = f e (0) g e (0) + g e (0) f ′ e · t + f e (0) g ′ e · t + f ′ e g ′ e · t nd therefore in particular H Γ m (( f g ) | e ) ( t ) = f e (0) g e (0) + tl e ( f e ( l e ) g e ( l e ) − f e (0) g e (0))= f e (0) g e (0) + tl e (cid:0) f ′ e g ′ e l e + ( f e (0) g ′ e + g e (0) f ′ e ) l e (cid:1) we obtain (( f g ) e − H Γ m (( f g ) | e )) ( t ) = f ′ e g ′ e t − f ′ e g ′ e l e t, t ∈ [0 , l e ] . This implies that for any edge e ∈ E m we have E e ((( f g ) e − H Γ m (( f g ) | e )) ( t )) = ( f ′ e g ′ e ) Z l e (2 t − l e ) dt = 13 ( f ′ e g ′ e ) l e , t ∈ [0 , l e ] . Summing up over e ∈ E m and using (90), we see that E Γ m ( f | X Γ m g | X Γ m − H Γ m (cid:0) f | X Γ m g | X Γ m (cid:1) )= X | w | = m r − w X e ∈ E m ,e ⊂ K w l e E e ( f | X Γ m g | X Γ m − H Γ m (cid:0) f | X Γ m g | X Γ m (cid:1) ) ≤ X | w | = m r − w X e ∈ E m ,e ⊂ K w l e ( f ′ e ) ( g ′ e ) ≤
13 (max i r i ) m E ( f ) X | w | = m r − w X e ∈ E m ,e ⊂ K w l e g ′ e = 13 (max i r i ) m E ( f ) E ( g ) . (cid:3) In what follows let µ be a finite Borel measure on K so that V ( m ) := inf | w | = m µ ( K w ) > m .Given an edge e ∈ E m we set(91) ψ e,m ( x ) := 1deg m ( i ( e )) ψ i ( e ) ,m ( x ) + 1deg m ( j ( e )) ψ j ( e ) ,m ( x ) , x ∈ K, to obtain a function ψ e,m which satisfies(92) X e ∈ E m h ψ e,m , i L ( K,µ ) = X p ∈ V m ψ p,m ( x ) = 1 , x ∈ K. We endow the space X Γ m with the measure µ ( m ) := µ Γ m which on each individual edge e ∈ E m equals1 l e (cid:18)Z K ψ e,m ( x ) µ ( dx ) (cid:19) λ | e , here λ denotes the one-dimensional Lebesgue measure. Writing X ( m ) w for X Γ m ∩ K w = V w = S e ∈ E m ,e ⊂ K w e ,we see that µ Γ m ( X ( m ) w ) = X e ∈ E m ,e ⊂ K w Z K ψ e,m ( x ) µ ( dx ) ≥ Z K w ψ e,m ( x ) µ ( dx ) = µ ( K w ) ≥ V ( m ) , so part (i) of Assumption 5.5 is satisfied. The remaining conditions in Assumption 5.5 (ii)-(iv) now followfrom results in [48]: If for each m we consider the linear operator Φ m : L ( X Γ m , µ Γ m ) → L ( K, µ ) definedby Φ m u ( t ) = X e ∈ E m e ( t ) h u, ψ e,m i L ( K,µ ) (cid:0)R K ψ e,m dµ (cid:1) , u ∈ L ( K, µ ) , then (55) and (56) are satisfied by [48, Prop. 4.1] (there the operator Φ m is denoted by J ∗ ,m ), and a proof of(57) is provided in [48, Lemma C.3]. Condition (58) follows from Lemma [48, Lemma C.2] (there the pointwiserestriction of m -harmonic functions to X Γ m is denoted by e J ,m ). For the operators Ext m H Γ m : ˙ W , ( X Γ m ) → m ( K ) (denoted by J ,m in Lemma [48, Lemma C.2]) we can use [48, Lemma C.2] and [48, Prop. 4.1] tosee that if ( f m ) m is a sequence of functions f m ∈ ˙ W , ( X Γ m ) with sup m E Γ m ( f m ) < ∞ then k Ext m H Γ m f m k L ( K,µ ) ≤ k f m k L ( X Γ m ,µ Γ m ) + C (max i r i ) m/ sup m E Γ m ( f m ) / with a positive constant C depending only on N . Consequently also (59) is satisfied.6.3. Short remarks on possible generalizations.
Although not covered by the above results, we con-jecture that under suitable additional conditions one can produce similar results for p.c.f. self-similar setswith non-regular harmonic structures, diamond lattice fractals, [1, 3, 35], Laaksø spaces, [91], and compactfractafolds, [92]. Well-known general results, [65, Proposition 2.10 and Theorem 2.14], motivate the ques-tion how to implement discrete or metric graph approximations for the Sierpinski carpet, endowed with itsstandard energy form. Another question is how to establish approximations by graph-like manifolds, [88],for non-symmetric forms of type (21), and a transparent discussion of drift and divergence terms should bequite interesting. A further open question is how to establish approximations in energy norm. This wouldmost likely have to involve second order splines as for instance discussed in [95] for the case of the Sierpinskigasket endowed with its standard energy form and the self-similar Hausdorff measure. Several tools used inthe present paper rely heavily on the use of linear and harmonic functions, and second order versions arenot so straightforward to see. A question in a different direction, particularly interesting in connection withprobability, [16], is how to approximate equations involving nonlinear first order terms. There are results onthe convergence of certain non-linear operators along varying spaces, [99], but they do not cover these cases.7.
Restrictions of vector fields
As mentioned in Remark 5.4, a finitely ramified cell structure also permits a restriction operation forspecific vector fields. As discussed in [48] the spaces Im ∂ and F / ∼ are isometric as Hilbert spaces, andsimilarly for Im ∂ ( m ) and F ( m ) / ∼ . Recall also that for each m the pointwise restriction u u | X ( m ) is anisometry from H m ( X ) / ∼ onto H m ( X ( m ) ) / ∼ . Therefore (67) and (68) give rise to a well defined restrictionof gradients of n -harmonic functions: Given f ∈ H n ( X ) and m ≥ n we can define the restriction of ∂f to X ( m ) by(93) ( ∂f ) | X ( m ) := ∂ ( m ) ( f | X ( m ) ) , and this operation is an isometry from ∂ ( H m ( X )) onto ∂ ( m ) ( H m ( X ( m ) )), see for instance [48, Subsection4.4]. In the sequel we assume, in addition to the assumptions made in Section 5, that for each m and each α ∈ A m the form E α ( u ) = P p ∈ V α P q ∈ V α c ( m ; p, q )( u ( p ) − u ( q )) , u ∈ F , is irreducible on V α . Following [55]we define subspaces H m of H by H m := (cid:26) X α ∈A m X α ∂h α : h α ∈ H m ( X ) for all α ∈ A m (cid:27) . Then H m ⊂ H m +1 for all m , [55, Lemma 5.3], and S m ≥ H m is dense in H , [55, Theorem 5.6]. To generalize(93) we now define a pointwise restriction of elements of H m to X ( m ) by(94) (cid:18) X α ∈A m X α ∂h α (cid:19) | X ( m ) := X α ∈A m X ( m ) α ∂ ( m ) ( h α | X ( m ) ) , and clearly this restriction operation maps H m into H ( m ) . Thanks to the finitely ramified cell structureof X it is straightforward to see that this definition is correct. The following auxiliary result is parallel toCorollary 5.4. Lemma 7.1.
For any b ∈ H n and any g ∈ C ( X ) we have (95) lim m k g | X ( m ) · b | X ( m ) k H ( m ) = k g · b k H . Proof.
Let ε >
0. Choose n g ≥ n sufficiently large such thatsup β ∈A ng sup x,y ∈ X β | g ( x ) − g ( y ) | < ε P α ∈A n E ( h α ) . or all β ∈ A n g choose x β ∈ X β \ V n g and define e g ( x ) := g ( x β ) if x ∈ X β \ V n g and e g ( x ) := 0 if x ∈ V n g .Then we we have sup β ∈A ng sup x ∈ X β \ V ng | g ( x ) − e g ( x ) | < ε P α ∈A n E ( h α )and therefore(96) (cid:12)(cid:12)(cid:12)(cid:12) X α ∈A n Z X α \ V ng g | X ( m ) dν ( m ) h α | X ( m ) − Z X α \ V ng e g | X ( m ) dν ( m ) h α | X ( m ) (cid:12)(cid:12)(cid:12)(cid:12) < ε m and also(97) (cid:12)(cid:12)(cid:12)(cid:12) X α ∈A n Z X α \ V ng g dν h α − Z X α \ V ng e g dν h α (cid:12)(cid:12)(cid:12)(cid:12) < ε . The energy measures ν h α are nonatomic, hence by (50) and the Portmanteau lemma we can find a positiveinteger m ε ≥ n g so that for all m ≥ m ε and all α ∈ A n we have(98) ν ( m ) h α | X ( m ) ( V n g ) < ε |A n | k g k and(99) (cid:12)(cid:12)(cid:12)(cid:12) ν ( m ) h α | X ( m ) ( X β \ V n g ) − ν h α ( X β \ V n g ) (cid:12)(cid:12)(cid:12)(cid:12) < ε |A n | k g k . Since (99) implies (cid:12)(cid:12)(cid:12)(cid:12) X α ∈A n X β ∈A ng g ( x β ) ν ( m ) h α | X ( m ) ( X α ∩ X β ∩ V cn g ) − X α ∈A n X β ∈A ng g ( x β ) ν h α ( X α ∩ X β ∩ V cn g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k g k X β ∈A ng (cid:12)(cid:12)(cid:12)(cid:12) ν ( m ) h α | X ( m ) ( X β \ V n g ) − ν h α ( X β \ V n g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, we can use (96) and (97) to obtain(100) (cid:12)(cid:12)(cid:12)(cid:12) X α ∈A n Z X α \ V ng g | X ( m ) dν ( m ) h α | X ( m ) − X α ∈A n Z X α \ V ng g dν h α (cid:12)(cid:12)(cid:12)(cid:12) < ε . On the other hand, we have k g | X ( m ) · b | X ( m ) k H ( m ) = X α ∈A n Z X α g | X ( m ) dν ( m ) h α | X ( m ) + X α,α ′ ∈A n ,α ′ = α Z X α ∩ X α ′ g | X ( m ) dν ( m ) h α | X ( m ) ,h α ′ | X ( m ) . By (98), the Cauchy-Schwarz inequality for energy measures and Definition 5.1 (vi) we see that the secondsummand on the right hand side is bounded by X α ∈A n ν ( m ) h α | X ( m ) ( V n g ) / ! < ε , and using (98) once more, we obtain(101) (cid:12)(cid:12)(cid:12)(cid:12) k g | X ( m ) · b | X ( m ) k H ( m ) − X α ∈A n Z X α \ V ng g | X ( m ) dν ( m ) h α | X ( m ) (cid:12)(cid:12)(cid:12)(cid:12) < ε . Combining (100), (101) and the fact that k g · b k H = P α ∈A n R X α g dν h α , we arrive at (95). (cid:3) ppendix A. Generalized strong resolvent convergence
The notation in this section is different from that in the main text. We review a special case of thenotion of convergence for bilinear forms as studied in [98] (and, among more general results, also in [99]). Itcovers in particular the case of coercive closed forms, [79]. The results in [98] are generalization of resultsin [38, Section 3] to the framework of varying Hilbert spaces in [77].In [77, Subsections 2.2 - 2.7] a concept of convergence H m → H of Hilbert spaces H m to a Hilbert space H was introduced, including a suitable notion of generalized strong resolvent convergence for self-ajointoperators, cf. [77, Definition 2.1]. A basic tool of the method in [77] is a family of identification operatorsΦ m defined on a dense subspace C of the limit space H , each mapping C into one of the spaces H m . Let H , H , H , ... be separable Hilbert spaces. The sequence ( H m ) m is said to converge to H in KS-sense ,lim m H m = H , if there are a dense subspace C of H and operators(102) Φ m : C → H m such that(103) lim m k Φ m w k H m = k w k H , w ∈ C . We recall [77, Definitions 2.4, 2.5 and 2.6].
Definition A.1. (i)
A sequence ( u m ) m with u m ∈ H m is said to converge KS-strongly to u ∈ H if there is a sequence ( e u m ) m ⊂ C such that (104) lim n →∞ lim m →∞ k Φ m e u n − u m k H m = 0 and lim n →∞ k e u n − u k H = 0 . (ii) A sequence ( u m ) m with u m ∈ H m is said to converge KS-weakly to u ∈ H if lim m h u m , v m i H m = h u, v i H for every sequence ( v m ) m KS-strongly convergent to v . (iii) A sequence ( B m ) m of bounded linear operators B m : H m → H m is said to converge KS-strongly to a bounded linear operator B : H → H if for any sequence ( u m ) m with u m ∈ H m convergingKS-strongly to u ∈ H the sequence ( B m u m ) m converges KS-strongly to Bu .Remark A.1 . In the classical case where H m ≡ H and Φ m ≡ id H for all m the strong convergence ofbounded linear operators B m defined in (iii) differs from the classical definition of strong convergence ofbounded linear operators on Hilbert spaces, as pointed out in [77, Section 2.3]. However, a sequence ( B m ) m of bounded linear operators B m : H → H admitting a uniform bound in operator norm sup m k B m k < + ∞ converges KS-strongly to a bounded linear operator B : H → H if and only if it converges strongly to B inthe usual sense, [77, Lemma 2.8 (1)].Now suppose that ( A m ) m is a sequence of linear operators A m : H m → H m each of which generates a C -semigroup and also A : H → H is the generator of a C -semigroup. Suppose that there exist constants ω ∈ R and M > A m and of A contain ( ω, + ∞ ) and for any positiveinteger n and any λ > ω we have sup m k ( λ − A m ) − n k ≤ M ( λ − ω ) − n and k ( λ − A ) − n k ≤ M ( λ − ω ) − n . Inthis situation we say that the A m converge to A in KS-generalized strong resolvent sense if for some (henceall) λ > ω the λ -resolvent operators R A m λ = ( λ − A m ) − of the A m converge KS-strongly to the λ -resolventoperator R Aλ = ( λ − A ) − of A . Remark
A.2 . For any λ > ω the sequence ( R A m λ ) m satisfies sup m (cid:13)(cid:13) R A m λ (cid:13)(cid:13) < M ( λ − ω ) − . In the classicalcase where H m ≡ H and Φ m ≡ id H for all m we therefore observe that the sequence of operators ( A m ) m asin (iv) converges to A as in (iv) in the KS-generalized strong resolvent sense if and only if it converges to A in the usual strong resolvent sense, see [59, Section 8.1] (or [89, Section VIII.7] for the self-adjoint case).One can also introduce a generalization of Mosco convergence for coercive closed forms (not necessarilysymmetric). The following definition is a shorted version for coercive closed forms, [79], of [99, Definition7.14] (see also [98, Definition 2.43]) sufficient for our purposes. We use notation (22) to denote the symmetricpart of a bilinear form. Definition A.2.
A sequence (( Q ( m ) , D ( Q ( m ) ))) m of coercive closed forms ( Q ( m ) , D ( Q ( m ) )) on H m , respec-tively, with uniformly bounded sector constants, sup m K m < + ∞ , is said to converge in the KS-generalized osco sense to a coercive closed form ( Q , D ( Q )) on H if there exists a subset C ⊂ D ( Q ) , dense in D ( Q ) ,and the following two conditions hold: (i) If ( u m ) m KS-weakly converges to u in H and satisfies lim m e Q ( m )1 ( u m ) < ∞ , then u ∈ D ( Q ) . (ii) For any sequence ( m k ) k with m k ↑ ∞ , any w ∈ C , any u ∈ D ( Q ) and any sequence ( u k ) k , u k ∈ H m k , converging KS-weakly to u and such that sup k e Q ( m k )1 ( u k ) < ∞ , there exists a sequence ( w k ) k , w k ∈ H m k , converging KS-strongly to w and such that lim k Q ( m k ) ( w k , u k ) ≤ Q ( w, u ) . In [38, 98, 99] one can find further details. The next Theorem is a special case of [99, Theorem 7.15,Corollary 7.16 and Remark 7.17] (see also [98, Theorem 2.4.1 and Corollary 2.4.1]), which generalize [38,Theorem 3.1].
Theorem A.1.
For each m let ( Q ( m ) , D ( Q ( m ) )) be a coercive closed form on H m and assume that thecorresponding sector constants are uniformly bounded, sup m K m < + ∞ . Let (cid:0) G Q ( m ) α (cid:1) α> , (cid:0) T Q ( m ) t (cid:1) t> and ( L Q ( m ) , D ( L Q ( m ) )) be the associated resolvent, semigroup and generator on H m . Suppose that ( Q , D ( Q )) isa coercive closed form on H with resolvent (cid:0) G Q α (cid:1) α> , semigroup (cid:0) T Q t (cid:1) t> and generator ( L Q , D ( L Q )) . Thenthe following are equivalent: (1) The sequence of forms ( Q ( m ) , D ( Q ( m ) )) m converges to ( Q , D ( Q )) in the KS-generalized Mosco sense. (2) The sequence of operators (cid:0) G Q ( m ) α (cid:1) m converges to G Q α KS-strongly for any α > . (3) The sequence of operators (cid:0) T Q ( m ) t (cid:1) m converges to T Q t KS-strongly for any t > . (4) The sequence of operators ( L Q ( m ) , D ( L Q ( m ) )) converges to ( L Q , D ( L Q )) in the KS-generalized strongresolvent sense.Remark A.3 . Theorem A.1 and Definition A.2 provide a characterization of convergence in the (KS-generalized) strong resolvent sense in terms of the associated bilinear forms. In the case of symmetric formsthese conditions differ from those originally used in [81, Definition 2.1.1 and Theorem 2.4.1] and [77, Defini-tion 2.11 and Theorem 2.4], see [38, Remark 3.4]
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E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
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