Construction of a Dirichlet form on metric measure spaces of controlled geometry

Given a compact doubling metric measure space $X$ that supports a $2$-Poincar\'e inequality, we construct a Dirichlet form on $N^{1,2}(X)$ that is comparable to the upper gradient energy form on $N^{1,2}(X)$. Our approach is based on the approximation of $X$ by a family of graphs that is doubling and supports a $2$-Poincar\'e inequality. We construct a bilinear form on $N^{1,2}(X)$ using the Dirichlet form on the graph. We show that the $\Gamma$-limit $\mathcal{E}$ of this family of bilinear forms (by taking a subsequence) exists and that $\mathcal{E}$ is a Dirichlet form on $X$. Properties of $\mathcal{E}$ are established. Moreover, we prove that $\mathcal{E}$ has the property of matching boundary values on a domain $\Omega\subseteq X$. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form $\mathcal{E}$) on a domain in $X$ with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.


Introduction
The idea of approximating a compact doubling metric space by graphs now is wellestablished, with the work of Christ [13] containing a prototype of such an idea.This idea blossomed into the construction of hyperbolic fillings of compact doubling spaces to obtain a Gromov hyperbolic space whose boundary is (homeomorphic to) the given space (see for instance [7,8,10,18]).Under the additional condition that the compact doubling metric space is equipped with a doubling measure that supports a Poincaré inequality, much can be said about the approximating graphs, see [18].In the non-smooth setting of metric measure spaces equipped with a doubling measure supporting a Poincaré inequality, it is now known N.S.'s work is partially supported by the NSF (U.S.A.) grant DMS #2054960.This work was begun during the residency of L.L. and N.S. at the Mathematical Sciences Research Institute (MSRI, Berkeley, CA) as members of the program Analysis and Geometry in Random Spaces which is supported by the National Science Foundation (NSF U.S.A.) under Grant No. 1440140, during Spring 2022.They thank MSRI for its kind hospitality.The authors thank Patricia Alonso-Ruiz and Fabrice Baudoin for sharing with us their early manuscript [1], which helped us work out the nature of the tool of Mosco convergence.
that there is a rich theory of harmonic functions (see for instance [5,6,29]).What is missing in the current literature is a construction of a Dirichlet form which is compatible with the upper gradient structure on the underlying metric measure space and, in addition, a way of approximating the Dirichlet form via its discrete couterparts defined on graphs approximating the metric space so that solutions to a Dirichlet boundary value problem for functions that are harmonic with respect to the limit Dirichlet form can be approximated by solutions to Dirichlet boundary value problem for the approximating sequence of graph-Dirichlet forms.The goal of this present note is to demonstrate one way of constructing such an approximation.
The notion of Dirichlet forms has its roots in the work of Beurling and Deny [3,4].The theory of Dirichlet forms has been fleshed out in the textbook [17], and an excellent exposition of the notions of Γ-convergence and Mosco-convergence of Dirichlet forms can be found for instance in [9,15,25,26,27].Of these expositions, the works of [9,15,27] consider sequences of Dirichlet forms on Euclidean spaces, while the works [25,26] consider sequences of Dirichlet forms, each defined on a (potentially) different metric measure space, converging in a sense akin to measured Gromov-Hausdorff convergence (see [12] for more on Gromov-Hausdorff convergence) of Dirichlet forms.
The paper [25] discusses convergence of graphs to graphs and the convergence of corresponding Dirichlet forms, while [26] studies Gromov-Haudorff convergence of function classes and Dirichlet forms.The papers [24,1] construct a Dirichlet form that is equivalent to the Korevaar-Schoen energy, but their approximations are not discrete and hence do not lend themselves to numerical schemes.Neither of them work for us as we consider graphs that approximate a potentially non-graph metric measure space, and we do not know Gromov-Hausdorff convergence of corresponding Sobolev classes.Hence we have instead used graphs X r , r > 0, to construct a family of Dirichlet forms E r on N 1,2 (X), see (3.2).We then show that this family of Dirichlet forms has a subsequence that Γ-converges to a Dirichlet form on X so that the domain of this limit form coincides with the Newton-Sobolev class N 1,2 (X), but with Dirichlet energy that is comparable to the upper gradient energy therein.The definition of Γ-convergence is given in Definition 3.15 below.Theorem 1.1.Let µ be a doubling measure on the complete metric space (X, d) so that (X, d, µ) supports a 2-Poincaré inequality.Then for forms E r defined by (3.2), the following is true for every u ∈ N 1,2 (X) and its minimal 2-weak upper gradient g u ∈ L 2 (X): (1) There is a constant (2) There is a sequence ( The form E is a positive-definite symmetric Markovian closed bilinear form on N 1,2 (X).
This above theorem enables us to have a way of numerically computing Dirichlet energies on a compact doubling metric measure space supporting a 2-Poincaré inequality, and in sodoing, we improve on the goal set in [16].In this goal, we rely on the type of Poincaré inequality on graphs as found in [18].This forms the first part of the present note.While our construction of approximating Dirichlet forms is based on the work of [18], there are alternate constructions based on the notion of Korevaar-Schoen energy, see [1].As the construction of [1] is not immediately related to the notion of Poincaré inequalities (via averages of functions on balls), and as numerical approximations are not relevant to the construction found in [1], we do not consider their construction in the present paper.
The primary reason for considering this specific construction of a Dirichlet form E on X is to have a notion of harmonic functions on X (related to this form) that is easily approximated by graph-harmonic functions.Since X is compact, each approximating graph for X is a finite graph, and then given boundary data for each of these graphs, it is an exercise in computational linear algebra together with the mean value property for harmonic functions on graphs [22,23] to obtain harmonic functions on an approximating domain in the graph by considering the adjacency matrix of the domain in the graph.Here, given the boundary data f ∈ N 1,2 (X) for the Dirichlet problem for E on a domain Ω in X, we may consider a discrete mollification of f as the boundary data for the Dirichlet form E r with respect to the approximating domain Ω r ⊂ X. Observe that given a function v ∈ N 1,2 (X), the Γ-convergence gives us some sequence However, in solving the Dirichlet boundary value problem, we would need the approximating functions v k to have the same boundary values as u.We establish this in the second main theorem of this paper.Theorem 1.2.Let u ∈ N 1,2 (X) and Ω be a domain.If there exist v j → u in N 1,2 (X) and E[u] = lim j→∞ E r j [v j ], then there exists a sequence u j ∈ N 1,2 (X) such that and u − u j ∈ N 1,2 0 (Ω).The rest of the paper is organized as follows.The next section, Section 2, will provide the background information regarding Sobolev spaces N 1,2 (X) and the related notions.Towards the beginning of this section we also provide the description of approximating grapns X r of the metric space X.In Section 3 we give the construction of the approximating forms E r on X.Sections 3 and 4 contain the proofs of Theorem 1.1 and Theorem 1.2.
Theorem 1.2 will be proved near the end of Section 3. Theorem 1.1 will be proved in stages.The first claim of Theorem 1.1 will follow from Theorem 3.13, and the existence of the Γ-limit will be established in Proposition 3.16.The last part of Theorem 1.1, listing the properties of the Γ-limit form E, will be established via the lemmas in Section 4. Since the approximating forms E r are not known to be Markovian on N 1,2 (X), the fact that E is Markovian does not follow directly from the theory of Γ-limits; instead, we establish this property in Proposition 4.8.

Background
In this note (X, d, µ) will denote a complete geodesic metric space (X, d), equipped with a Radon measure µ supported on X, such that µ is doubling; that is, there is a constant C d ≥ 1 such that for all x ∈ X and r > 0 we have Here, B(x, r) := {y ∈ X : d(x, y) < r} is the open metric ball, and B(x, r) := {y ∈ X : d(x, y) ≤ r} is the closed ball; note that the topological closure of B(x, r) is contained in B(x, r), but need not equal.Definition 2.1.With D ⊂ L 2 (X) a vector subspace, a positive-definite symmetric bilinear form [17] this property identified as the closedness property of E, see Definition 4.3 below), (3) E(u T , u T ) ≤ E(u, u) whenever u ∈ D and u T := max{0, min{u, 1}} (in [17] this property is identified as the Markovian property).
Associated to the Dirichlet form there is an energy form, also denoted E, with E[u] := E(u, u) for each u ∈ D.
While [17] sets a Dirichlet form to be E : and then sets D to be the collection of all functions u ∈ L 2 (X) for which E(u, u) < ∞, this is not necessary and recent papers on Dirichlet forms do not require E to act on all of L 2 (X), see for instance [25,Section 5], [11], or [30, page 7, Section 2C].
Since the doubling measure is supported on X, it follows that (X, d) is itself a doubling metric space; that is, there is some positive integer N d such that for each x ∈ X and r > 0, there are at most N d number of points in B(x, r) that are separated by a distance at least r/2 from each other; that is, whenever A ⊂ B(x, r) such that for x, y ∈ A we have d(x, y) ≥ r/2 or x = y, we must have that A has at most N d number of elements.It follows that for each r > 0 we can find a set X r ⊂ X such that The set X r can be given the structure of a graph, with elements of X r being the vertices of the graph and two vertices x, ȳ ∈ X r neighbors if x = ȳ and B(x, 2r) ∩ B(ȳ, 2r) = ∅.We denote the neighborhood relationship by x ∼ ȳ.By gluing an interval of length r to the two neighboring vertices x, ȳ, we obtain a metric graph, equipped with the path-metric d r .Note that because X is a connected metric space, necessarily we must also have X r be a connected metric space.We refer the interested reader to [20] for details regarding this construction.
Note that x ∈ X r now has two identities; one as a point in the graph X r , and one as a point in the metric space X.The following lemma compares the two metrics on X r , namely the metrics d r and d.Lemma 2.2.Let (X, d) be a doubling geodesic metric space and (X r , d r ) be defined as above.Then for all x, ȳ ∈ X r .
Proof.For x, ȳ ∈ X r with x = ȳ, we have that where N is the smallest number for which there is a collection of points {x i } N i=0 so that x0 = x, xN = ȳ and xi−1 ∼ xi for i = 1, . . ., N .Such a collection is called an X r -chain between x and ȳ.Given the shortest such chain, we see that Since X is a geodesic space, we also have a reverse relationship between d and d r as well.
Let x, ȳ ∈ X r be two distinct points, connected in X by a geodesic γ.If d r (x, ȳ) = r, then r ≤ d(x, ȳ) ≤ 4r.If d r (x, ȳ) ≥ 2r, then d(x, ȳ) ≥ 4r, and we can find points z 0 := x, z 1 , . . ., z N −1 , z N := ȳ on γ with the property that , and so d r (z i−1 , zi ) ≤ r (with the value either 0, if zi−1 = zi , or r otherwise).Thus The graph metric space X r plays the role of a discrete approximation of the metric space (X, d) at scale r.By equipping X r with a "lift" of the measure µ, given by whenever A ⊂ X r , we obtain a metric measure space (X r , d r , µ r ) with the measure µ r a doubling Radon measure on X r .For more on the approximation nature of (X r , d r , µ r ) we refer the reader to [18].
Given a function u : X → R, we say that a non-negative Borel measurable function g on X is an upper gradient of u if for every non-constant, compact, rectifiable curve γ in X we have where x and y denote the two end points of γ.If u is a locally Lipschitz-continuous function on X, then the local Lipschitz-constant function Lip u, given by is an upper gradient of u.For 1 < p < ∞, we say that u ∈ N 1,p (X) if u is measurable with ´X |u| p dµ < ∞ such that u has an upper gradient g ∈ L p (X).We refer the reader to [21] for more on upper gradients and the Sobolev class N 1,p (X).
In this note we will also assume that (X, d, µ) supports a p-Poincaré inequality, that is, there is a constant C P ≥ 1 such that for each ball B ⊂ X and each function-upper gradient pair (u, g) on X we have A more general version of Poincaré inequality would have the existence of a constant Θ ≥ 1 such that the right-hand side of the above inequality has ΘB rather than B. However, when X is a geodesic space (and a biLipschitz change in the metric does provide a geodesic space), it is possible to choose Θ = 1 at the expense of increasing the value of C P , and this is what we choose to do here.We refer the interested reader to [19,21] for more on this.A theorem of Cheeger [12] tells us that there is a linear differential structure on such X, commensurate with a notion of Taylor approximation by Lipschitz charts on X, and a measurable inner product structure on this differential structure, so that the inner product norm of the differential of a Lipschitz function u on X is comparable to Lip u.However, this proof, as given in [12], is that of existence.
In the rest of the paper we consider only the case p = 2.The goal of this note is to give a construction of a Dirichlet form on X, by using the ideas of [16,18], together with a modification of the notion of Mosco convergence, by considering the natural Dirichlet forms on the approximating graphs.
Remark 2.3.It was shown in [18] that (X, d, µ) is doubling and supports a 2-Poincaré inequality if and only if the family of approximating spaces (X r , d r , µ r ) also is doubling and supports a 2-Poincaré inequality, with the relevant doubling constant and the Poincaré constant independent of r.We will make use of this result in our paper.
Proposition 2.4 (Gill-Lopez [18]).Let (X, d, µ) be a doubling metric measure space and (X r , d r , µ r ) be defined as above.If (X, d, µ) admits a Poincare inequality with constants C P I > 0 and λ ≥ 1, then (X r , d r , µ r ) admits a Poincare inequality in the Holopainen-Soardi sense: there are constants C HS > 0 and θ ≥ 1 independent of r such that for any x0 ∈ X r , R ≥ 0 and ū defined on X r , we have Given the graph X r as above, the graph energy of a function u r : X r → R is given by We straddle three realms here; the graph energy on the graph X r , the energy E r induced on X by the graph energy (see (3.2) below), and the Γ-limit energy E on X.The graph energy is directly seen to be derived from a Dirichlet form, and we will see in the last section that the Γ-limit energy E is also derived from a Dirichlet form.The energy E r , however, is not known to be from a Dirichlet form, though it is derived from a positive-definite symmetric bilinear form.

Constructing approximating energy forms E r
In this section we use the structure of X r to construct an approximating Dirichlet form and set the Dirichlet form by the formula whenever u, v ∈ N 1,2 (X).It is clear that E r is a bilinear form on the Hilbert space N 1,2 (X).
We set E r [u] := E r (u, u).
Remark 3.3.The form E r is a bilinear energy form but is not known to satisfy the Markov property, and so should, according to the definition of [3,17], not be called a Dirichlet form.Moreover, E r need not be closed as it acts only on N 1,2 (X), but it is closable in L 2 (X).The point here is that we can have a sequence However, E r is induced by a bilinear energy form on the graph X r which is indeed a Dirichlet form, and so we allow ourselves the liberty of calling E r also a Dirichlet form as this does not give rise to any confusion.The Γ-limit energy form on N 1,2 (X) obtained from the forms E r will indeed be a Dirichlet form, see Proposition 4.8 and Lemma 4.6 below.
We are also concerned with the natural energy form on X inherited from the upper gradient structure; for u ∈ N 1,p (X) we set where the infimum is over all upper gradients g of u.From the results in [21] we know that there is a minimal 2-weak upper gradient, g u , associated with each u ∈ N 1,2 (X), such that When u is locally Lipschitz on X, it follows from [12] together with [21] that g u = Lip u.
For functions u ∈ L 2 (X) that do not have (a modification on a set of measure zero) representative in N 1,2 (X) we set E[u] := ∞.
The following proposition gives us a comparison between the family E r and E.
Proposition 3.4.Let X be a doubling metric measure space admitting a 2-Poincaré inequality.Then there exists C > 0 such that E r [u] ≤ CE[u] for all r > 0 and u ∈ N 1,2 (X).Moreover, whenever Lip u is continuous and belongs to L 2 (X, dµ) ,we have The constant of comparison, C, is dependent solely on the doubling constant C d associated with µ, the metric doubling constant N d associated with the doubling metric space (X, d), and the scaling constant θ from Proposition 2.4; in particular, it is independent of u and r.This proposition will be proved using Lemma 3.5 below.Lemma 3.5.Let u ∈ N 1,2 (X) be a locally Lipschitz function such that Lip u is a continuous function on X.Then for any r 0 > 0 and any x ∈ X there exists a ball B x , centered at x and of radius r(B x ) ≤ r 0 , such that for all ε ≤ rad(B x )/36 we have where we set Proof.We fix r 0 > 0 and x ∈ X.Without loss of generality we may assume that Lip u(x) > 0. Then by the continuity of Lip u, there is some positive number r 1 ≤ r 0 such that Next, by the definition of Lip u(x), there exists a positive number r 2 ≤ r 1 /2 and a point y ∈ B(x, r 2 ) such that To see this claim, we argue as follows.By the definition of Lip u(x) we know that there is some y ∈ B(x, r 1 /2) such that y = x and Setting r 2 = d(x, y), we obtain (3.9).By (3.7) we know that and as for each x ′ ∈ B(x, r 2 ) we know that B(x ′ , r 2 ) ⊂ B(x, r 1 ), we know that u is We set B x := B(x, r 2 ) and will show that (3.6) holds for ε ≤ r 2 /36.In order to do this, consider the balls B 1 = B(x, r 2 /9) and B 2 = B(y, r 2 /9).Then by the choice of r 2 and by (3.8), for all x ′ ∈ B 1 we have Similarly, because X is a geodesic space and Lip u is an upper gradient of the Lipschitz function u (see [20], [21]) and by (3.7) we see that for all Note that d(x, y) = r 2 , and so For x ∈ E(x, r 2 /9) and ȳ ∈ E(y, r 2 /9) we have Hence by (3.9), for all x ∈ E(x, r 2 /9) and ȳ ∈ E(y, r 2 /9), As the balls B(x, ε/2) are disjoint in X and the measure µ is doubling, we have Similarly we obtain µ ε (E(y, r 2 /9)) µ(B 2 ).On the other hand, as ε ≤ r 2 /36 and X ε is an ε-net, we have that Combining the above two sets of comparisons, we obtain and as r(B 1 ) = r(B 2 ) = r(B x )/9 with the centers x and y, of B 1 and B 2 respectively, in B x , we obtain from the doubling property of µ that µ ε (E(x, r 2 /9)) ≈ µ(B x ) and µ ε (E(y, r 2 /9)) ≈ µ(B x ).
Finally, by (3.10) we obtain Now we are ready to prove Proposition 3.4.
Proof of Proposition 3.4.Let u ∈ N 1,2 (X) be a locally Lipschitz continuous function on X such that its local Lipschitz constant function Lip u is also continuous on X.Without loss of generality, we assume that ´X(Lip u) 2 dµ is positive, for otherwise u is constant and all the relevant Dirichlet energies are zero.Now we introduce a large parameter λ > 1 that we will fix at the end (indeed, it can be any choice of λ > 12θ + 1 with θ from the Poincaré inequality given in Proposition 2.4 above).Let K be a closed ball of sufficiently large radius so that Note that as X is complete and doubling, it is proper; hence K is compact.By uniform continuity of Lip u on 10λK, we can find r 1 ∈ (0, 1) small enough so that 2r 1 is less than the radius of K and Here we have fixed a number Λ > 2, to be chosen later.
For each x ∈ K we find a ball B x satisfying the conclusions of Lemma 3.5 with the choice of r 0 = r 1 /(5λ).Such a family of balls is a cover of the compact set K, and so we can obtain a a finite subcover of K.Moreover, by the 5r-covering lemma (see e.g.[20, Theorem 1.2] or [21, page 60]), we can further find a subcollection of balls {B i } N i=1 such that With this subcover, we get where the balls B i satisfy the conclusions of Lemma 3.5.We denote the centers of these balls by x i .Since the radii of each 5λB i is no more than r 1 and the center of 5λB i lies in K, it follows from (3.11) that We now obtain from the above observations that The last few steps in the above argument was arrived at by using the fact that the balls B i are pairwise disjoint and are contained in 2K.The comparison constant associated with the above series of inequalities, C * , depends solely on the doubling constant of µ and the choice of λ.We choose Λ ≥ 2C * .It follows that C * /Λ ≤ 1/2, and so Recall that in Lemma 3.5, the parameter ε 0 depended on the radius of the ball B x .We will indicate this dependence by denoting ε 0 by ε 0 (x).Therefore, choosing any ε ≤ min 1≤i≤N ε 0 (x i ), we get where, as in Lemma 3.5, we have and By the doubling property of µ again, we have that µ ε (G(x i , r(B i ))) ≈ µ(B i ).Now we apply Proposition 2.4, to obtain where we used Tonelli's theorem to obtain the last inequality.Since λ > 12θ + 1 and the balls λB i , i = 1, • • • , N are pairwise disjoint, and as for each i we have that G(x i , θr(B i )) ⊂ 2θB i ⊂ λB i , we have that The above holds for all sufficiently small ε > 0, and so we have that which verifies the claimed latter inequality in the statement of the proposition.We now complete the proof of the proposition by verifying the first inequality stated in the claim.To this end, let u ∈ N 1,2 (X).We fix r > 0 such that r < diam(X)/2.Recall that for distinct x, ȳ ∈ X r with x ∼ ȳ we have d(x, ȳ) ∈ [r, 4r).Thus B(ȳ; r) ⊂ B(x; 5r) ⊂ B(ȳ; 9r).
As µ is doubling, it follows that Hence, if x ∼ ȳ, then where N d is the doubling constant of the metric space (X, d).Next, applying the Poincaré inequality, we get where we have used the fact that two balls B(x, 5r) and B(x ′ , 5r) intersect with x = x′ if and only if r ≤ d(x, x′ ) < 5r.The constants of comparison in the above inequalities all depend only on the doubling constant C d of the measure µ, and in particular are independent of u and r.
Theorem 3.13.Let X be a doubling metric measure space admitting a 2-Poincaré inequality.Then there is a constant C > 0 such that for each u ∈ N 1,2 (X) we have Proof.The first inequality follows immediately from Proposition 3.4, and so only the second inequality needs verification.That is the focus of this proof.
Let u ∈ N 1,2 (X), and (u k ) k be a sequence of compactly supported Lipschitz functions on X with compactly supported Lipschitz upper gradients, such that u k → u in N 1,p (X); see [12,Theorem 5.3] (note that in [12], the continuous upper gradient vk,j(k) used to construct the approximating function u k = fk,j(k) , by the construction of u k , necessarily has the property that Lip u k = vk,j(k) ).By passing to a subsequence if necessary, we may assume that u − u k N 1,2 (X) ≤ 1/k.Then by the first part of Proposition 3.4 again, we have that Note that the constant C independent of r.Therefore, for each ε > 0 we can find a positive integer k ε such that whenever k ε ≤ k ∈ N, for all r > 0 we have 1/k < ε and Now by the last inequality in the statement of Proposition 3.4, there is some r ε > 0 such that for each 0 < r < r ε , we have For such r and k, we have by triangle inequality, The desired conclusion follows from letting r → 0 + and then ε → 0. We now show that if (u k ) k is a sequence in N 1,2 (X) such that u k → u in N 1,2 (X), then for each r > 0 we have that E r [u k − u] → 0 as k → ∞.We fix r > 0 and consider the balls B j , j ∈ N, related to the construction of X r ; namely the balls of the form B(x, r), x ∈ X r .Note that x ∼ ȳ if and only if Note that if x ∼ ȳ, then B(ȳ; r) ⊂ B(x; 5r); moreover, there is at most N such ȳ ∈ X r that are neighbors of x, with N independent of x and r (in fact, it depends solely on the metric doubling constant N d ).Therefore by the doubling property of µ we have where in the last step we have used the bounded overlap property x∈Xr χ B(x;5r) ≤ K on X with K depending solely on the doubling constant This completes the proof of the theorem; though note that the convergence of E r [u k − u] to zero is not uniform across r > 0 as we have r 2 in the denominator of the above estimate.
Definition 3.15.A sequence E r k of Dirichlet forms on the Hilbert space N 1,2 (X) is said to Γ-converge to a Dirichlet form E if the following two conditions are satisfied: Given the first condition above, the second condition is equivalent to knowing the existence of a sequence Proposition 3.16.There exists a Γ-convergent subsequence E r k of E r .
Proof.From [2] we know that N 1,2 (X) is separable.Therefore, by [14, Theorem 8.5], there exists a Γ-convergent subsequence E r k of E r .From [14, Proposition 6.8] we know that the Γ-limit is a closable bilinear energy form.We denote the Γ-limit of E r k from the previous proposition by E. We now show that the energy form induced by the Γ-limit E is comparable to the Newton-Sobolev energy form, thus proving Condition (2) of Theorem 1.1.Proposition 3.17.There is a constant C > 0 such that the following is true for every u ∈ N 1,2 (X) and its minimal 2-weak upper gradient g u ∈ L 2 (X): Proof.If u k , k ∈ N, is the sequence guaranteed by the Γ-limit so that By [14, Proposition 6.8], we know that the Γ-limit is lower semicontinuous.Then by Theorem 3.13 we have that and so E[u] ≤ C ´X g 2 u dµ, which gives the left side of (3.18).On the other hand, by Theorem 3.13, we have , where we have used the fact that (Again from Theorem 3.13), This proves Claim (2) of Theorem 1.1.
One of the key properties needed with respect to the Γ-convergence is that if the function being approximated vanishes outside a bounded domain Ω ⊂ X with µ(X \ Ω) > 0, then the sequence approximating this function in the Γ-convergence condition (ii) also vanishes outside the approximating domain.This property is essential in the discussion on approximating solutions to the Dirichlet problem on Ω, and is called the matching boundary values in [9, Section 4.2.1].However, the method of De Giorgi, as outlined in [9], uses a decomposition of the domain Ω into suitable annular rings and constructing a partition of unity subordinate to this decomposition.An alternate method is to construct different Γ-limits for different boundary data as in [14,Chapter 21], but this again is computationally involved (see [14,Theorem 21.1]).In the second main theorem, Theorem 1.2, we give a more direct proof of the existence of optimal sequences with matching boundary values in the sense of [9].We now prove this theorem. Then, and therefore u j − u ∈ N 1,2 0 (Ω).Further, note that as w ≥ 0, when x ∈ X we have that u j (x) − u(x) > 0 if and only if v j (x) − u(x) > 0, and in which case we also have To see the N 1,2 -convergence, we use [21,Proposition 6.3.23].Using this proposition and the fact that g w ≤ χ Ω , we get Note that for any ǫ > 0, ˆX g 2 where Since v j → u in L 2 (X), it follows that lim sup j→∞ µ({|v j − u| > ǫ}∩ Ω ǫ ) = 0, and moreover, as v j → u in the Hilbert space N 1,2 (X), it also follows that lim sup j→∞ ´Ω g 2 v j −u dµ = 0. Therefore we have lim sup As ǫ > 0 can be chosen arbitrarily small, and ǫ>0 Ω \ Ω ǫ is empty, it follows that lim sup ǫ→0 + µ(Ω \ Ω ǫ ) = 0, and so lim j→∞ ˆX g 2 u j −u dµ = 0.
Thus we have that u j → u in N 1,2 (X).
Finally, note that |E . By Theorem 3.13 we therefore have that

Properties of E
In this section we will consider some key basic properties of the limit form E. There is a property of E that would be essential in order to call E a Dirichlet form; namely, that E is Markovian.In this section we will also prove that E is Markovian and that it is local.The first lemma follows from [14, Proposition 11.9(e) and Theorem 11.10].
In our setting, we have chosen the Hilbert space H = N 1,2 (X), and consequent to Proposition 3.17 (or Theorem 1.1), we also have that N 1,2 (X) = D(E).This is because the comparability of E and the energy form on N 1,2 (X) (Proposition 3.17) guarantees that E should always be finite on N 1,2 (X).
We can choose the core C to be the collection of all Lipschitz functions on X.Since the measure µ on X is doubling and supports a 2-Poincaré inequality, it follows that Lipschitz functions are dense in N 1,2 (X), see for instance [21,Section 8.2].
The above lemma follows from [14, Proposition 11.9 and Theorem 11.10], where symmetric bilinear forms are called quadratic forms.
Lemma 4.5.The form E is a local form on N 1,2 (X).Moreover, for each u ∈ N 1,2 (X) we have that Proof.We first prove the locality property of E. Let u, v ∈ N 1,2 (X) such that the support K u of u and the support K v of v are disjoint.Then, as X is compact, there is some ρ > 0 such that dist(K u , K v ) ≥ 10ρ.By the bilinearity and symmetry properties of E, we have To show that E(u, v) = 0 (the definition of locality for a symmetric bilinear form), it suffices to show that E X) be an approximating sequence for u + v as guaranteed by the Γ-convergence.Then, thanks to Theorem 1.2, we can also assume that each f k = 0 on X \ (K u ∪ K v ).Then for each positive integer k, the functions f k = f k χ Ku and h k = f k χ Kv both belong to N 1,2 (X) and f k = f k + h k .Indeed, as there is a positive distance between K u and K v , we can find two compactly supported Lipschitz functions ϕ u and ϕ v so that ϕ u = 1 on K u , ϕ v = 1 on K v , and On the other hand, when r k < ρ, we have that and so as Replacing v with −v in the above argument also gives that E and so we have that E By Theorem 3.13 and by the fact that , from which the desired conclusion now follows.
In [27, page 372], it is shown that the Γ-limit of bilinear forms on L 2 (X) is closed.In our setting, we consider a Γ-convergence with the ambient Hilbert space itself being N 1,2 (X).Therefore for completeness of the exposition, we include our proof here.
Lemma 4.6.The symmetric bilinear form E is closed.
it follows from Theorem 3.13 (or more specifically, from its limit version given in (2) of Theorem 1.1), we have that ´X g 2 fn−fm dµ → 0 as n, m → ∞.It follows that (f n ) n is Cauchy in the Banach space N 1,2 (X), and so such modification of f exists.Now let X), and as N 1,2 (X) is a Banach space (see [28]), there exists By [14, Proposition 6.8], we know that the Γ-limit is lower semicontinuous.Therefore, for each fixed n, using the lower semicontinuity of E, we have Hence, we have Proof.To show that the form is Markovian, it suffices to show that if u ∈ N 1,2 (X), setting v = max{0, min{1, u}}, we have E d, µ) is doubling and supports a 2-Poincaré inequality, we know that Lipschitz functions form a dense subclass of N 1,2 (X); hence it suffices to prove this property for the case that u is Lipschitz continuous.
So let u ∈ N 1,2 (X) be Lipschitz on X, and set E − = {u < 0}, E + = {u > 1}, and E 0 = {0 ≤ u ≤ 1.Then as X is compact, and as E − and E + are disjoint, we have that ρ := dist(E − , E + ) > 0. So if 0 < r < ρ/16, then we have that for x ∈ X r and ȳ ∈ X r with ȳ ∼ x, the set B(x, r/4) ∪ B(ȳ, r/4) cannot intersect both E + and E r/4) ∪ B(ȳ, r/4) intersects both E − and E 0 (the case that B(x, r/4) ∪ B(ȳ, r/4) intersects both E + and E 0 will be handled in s very similar manner).In this case, we have where we have used the fact that for each x ∈ X r there are at most N number of vertices ȳ ∈ X r such that x ∼ ȳ, with N depending only on the doubling constant associated with µ; and we also used fact that there is at most bounded overlap property of the balls B(ȳ, r), ȳ ∈ X r , to obtain the last inequality.A similar argument also gives So if µ(∂E − ) + µ(∂E + ) = 0, then we have that The last equality follows from Lemma 4.5 above.The assumption of µ(∂E − )+µ(∂E + ) = 0 is not a serious obstacle, as we know that for almost every t ∈ R we have that µ({u = t}) = 0, and so we can approximate 0 from below by such t − < 0, and approximate 1 from above by such t + > 1, and consider v t − ,t + = max{t − , min{t + , u}} and obtain the Markov property for these, and then let t − → 0, t + → 1 and note that v t − ,t + → v in N 1,2 (X); we know that E is bounded (from Theorem 3.13 above) bilinear form on N 1,2 (X).The fact that E r [v t − ,t + ] converges to E r [v] can be obtained from the closability of E (lemma 4.6 above) and Theorem 3.13, together with the fact that v t − ,t + → v in N 1,2 (X) as t − → 0 and t + → 1.

Definition 4 . 1 .
Let E be a symmetric bilinear form from a Hilbert space H to [−∞, ∞].Let D(E) be a dense linear subspace of H such that E is a symmetric bilinear form mapping D(E) to R. Then we say D(E) is the domain of E. Definition 4.2.Let C c (X) be the collection of all real-valued continuous functions with compact support on X.A core of a symmetric biliear form E is a subset C of D(E) ∩ C c (X) such that C is dense in D(E) under the norm (•, •) H + E(•) and dense in C c (X) under the uniform norm.Definition 4.3.We say that E is closed if for any