Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 19 (2000), No. 1, 203–225

Relaxation for Dirichlet Problems Involving a Dirichlet Form

M. Biroli and N. Tchou

Abstract. For a fixed Dirichlet form, we study the space of positive Borel measures (possibly infinite) which do not charge polar sets. We prove the density in this space of the set of the measures which represent varying domains. Our method is constructive. For the Laplace operator, the proof was based on a pavage of the space. Here, we substitute this notion by that of homogeneous covering in the sense of Coiffman and Weiss.

Keywords: Dirichlet spaces, Asymptotic behavior, Variational methods AMS subject classification: 31 C 25, 35 B 40, 35 A 15

1. Introduction

Let X be a connected, locally compact, separable Hausdorff space, X_{0} a relatively compact open connected subset of X and Ω an open subset of X_{0}. Let m be a positive Radon measure on X. In this paper, we consider a strongly local regular Dirichlet-Poincaré form a on X whose domain relative to Ω is denoted by D_{o}(a, Ω) ⊂ L^{2}(X, m) (see [2],[3]). Let M_{o}^{a} be the space of the non-negative Borel measures on Ω, which do not charge the polar sets with respect to a (i.e. sets whose a-capacity is 0). The space M_{0}^{a} is compact with respect to the γ^{a}-convergence: see [17] for strongly eliptic operators,

[29] for symmetric Dirichlet forms and [28] for non-symmetric Dirichlet forms. Consider a sequence (C_{h}) of closed subsets of Ω and denote Ω_{h} = Ω\C_{h}; we shall

define

μ_{h} = ∞_{C_{h}}

in a suitable way (see Lemma 3.1) such that μ_{h} ∈ M_{o}^{a}. The γ^{a}-convergence of the sequence (μ_{h}) means that there exists a limit measure μ in M_{o}^{a} such that the solution u_{h} of the variational problem:

Find u_{h} ∈ D_{o}(a, Ω_{h}) such that for any v ∈ D_{o}(a, Ω_{h}),

a(u_{h}, v) = ∫_{Ω\C_{h}} fv dm.  (1.1)

M. Biroli: Politecnico di Milano, Dept. Math., Piazza L. da Vinci 32, 20133 Milano, Italy N. Tchou: Université de Rennes 1, IRMAR, Beaulieu, 35042 Rennes Cedex, France This research has been supported by Progetto di ricerca MURST 9801262841 ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

or, equivalently, find u_{h} ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ_{h}) such that for any v ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ_{h}),

a(u_{h}, v) + ∫_{Ω} u_{h}v dμ_{h} = ∫_{Ω} fv dm,  (1.2)

converges strongly in L^{2}(Ω, m) to the solution u of the problem: u ∈ D_{o}(a, Ω)∩L^{2}(Ω, μ) and, for any v ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ),

a(u, v) + ∫_{Ω} uv dμ = ∫_{Ω} fv dm.  (1.3)

Dal Maso and Mosco in [17] proved that in the strongly elliptic case for X = ℝ^{n} (n ≥ 2) the space M_{o}^{a} is the closure of the Dirichlet problems of type (1.1) with respect to the γ^{a}-convergence, i.e. for all μ ∈ M_{o}^{a}, there exists a sequence of open sets Ω_{h} (and C_{h} = Ω\Ω_{h}) such that the solutions u_{h} of (1.1) converge strongly in L^{2}(Ω, m) to the solution u of problem (1.3). For this reason, (1.3) has been called a relaxed Dirichlet problem.

Other density results have been proved:

• In the case where a corresponds to a strongly elliptic symmetric operator, Dal Maso and Mosco [17] have proved the density in M_{o}^{a} of the measures with very regular density: μ_{h} = q_{h}L where L is the Lebesgue measure, and q_{h} ∈ C_{0}^{∞}(Ω).

• Dal Maso and Garroni [14] have proved the density in M_{0}^{a} of the Radon measures, when a corresponds to a strongly elliptic non-symmetric operator.

• The same density result holds when a is a possibly non-symmetric strongly local Dirichlet-Poincaré form (see [28]).

The aim of this paper is on the one hand to prove that, for a Dirichlet-Poincaré form, the problems of type (1.3) are the relaxed Dirichlet problems (at least under suitable assumption on the form a), i.e. the space M_{o}^{a} is the closure with respect to the γ^{a}-convergence of the set of measures of type ∞_{C} , and on the other hand, for a given a measure μ ∈ M_{o}^{a}, to provide an explicit construction of the subsets Ω_{h} for which the solutions of (1.2) with μ_{h} = ∞_{C_{h}} converge to the solution of (1.3). The last question has been solved by Dal Maso and Malusa [15] in the case of strongly elliptic symmetric operators.

Before defining more precisely the framework for our study, we wish to describe the method used in [15]. We take X = ℝ^{n} (n ≥ 2) (this hypothesis on the dimension yields that the points in ℝ^{n} have zero a-capacity). Let μ be a Radon measure. For any h ∈ ℕ, we fix a pavage of ℝ^{n} made of the cubes Q_{h}^{i} of sides \frac{1}{h} and centers x_{i}. We denote by I_{h} the sets of integers such that Q_{h}^{i} ⊂ Ω. For i ∈ I_{h} let B_{h}^{i} be the Euclidean ball with center x^{i} and radius \frac{1}{2h}. Let E_{h}^{i} be a concentric ball whose radius is chosen so that

cap^{a}(E_{h}^{i}, B_{h}^{i}) = μ(Q_{h}^{i}).

Observe that such an E_{h}^{i} always exists by virtue of the continuity and the monotonicity of the a-capacity, and because

cap^{a}({x_{i}}, B_{h}^{i}) = 0  and  cap^{a}(B_{h}^{i}, B_{h}^{i}) = +∞.  (1.4)

Let C_{h} be the set ∪_{i∈I_{h}}E_{h}^{i}. Dal Maso and Malusa [15], generalizing the method of Cioranescu and Murat [9], and using a special Poincaré inequality, proved that the sequence Ω_{h} = Ω\C_{h} has the desired property.

This method can be used for some degenerate operators, for instance for the Heisenberg operator

Δ_{H} = \frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} + (4y^{2} + 4x^{2})\frac{∂^{2}}{∂z^{2}} + (4y)\frac{∂^{2}}{∂z∂x} − (4x)\frac{∂^{2}}{∂z∂y}  (1.5)

naturally associated with the homogeneous group of the translations of Heisenberg. The ingredients for generalizing the previous technique are

1. a special pavage (using the above mentioned translations) which has been studied in [6].

2. the notion of balls in a metric associated with the operator, which is different from the Euclidean metric because the operator (1.5) is degenerate.

However, it is not clearly possible to construct a suitable pavage for general degenerate elliptic operators, even if they are associated with homogeneous groups of translations.

In this paper, we shall investigate the possibility of replacing the disjoint covering (pavage) by a covering with a small overlapping. The method that we shall use consists of taking an homogeneous covering by balls B(x_{i}, r) (i = 1, …, q), in the metric associated with the operator (see [10]). We shall see that this covering has the property d(x_{i}, x_{j}) ≥ \frac{r}{2} for i ≠ j. This implies that the balls B(x_{i}, \frac{r}{2}) are disjoint and that any point x belongs at most to M balls B(x_{i}, r) (where M is an intrinsic constant). Moreover, a property similar to (1.4) (connected with the behavior of the Green functions for the Dirichlet form a (see [3]) allows us to define the sets E_{h}^{i}. More precisely, if μ ∈ M_{o}^{a} is a Radon measure and if we suppose that N = {x_{0} : a − cap({x_{0}}) > 0} ∩ Ω = ∅, we can construct a finite covering of Ω: B(x_{i}, r) (i ∈ I) such that B(x_{i}, \frac{r}{2}) are pairwise disjoint. Let I^{∗} ⊂ I such that for all i ∈ I^{∗}, B(x_{i}, \frac{r}{2}) ⊂ Ω. Defining

A(x_{1}, r) = B(x_{1}, r), …, A(x_{i}, r) = B(x_{i}, r)\ ∪_{j<i} B(x_{j}, r), …,

let E_{r}^{i} = E(x_{i}, r) ⊆ A(x_{i}, r) be a ball centered in x_{i}, contained in B(x_{i}, \frac{r}{4}) with radius such that

cap^{a} (E(x_{i}, r), B (x_{i}, \frac{r}{4})) = μ(A(x_{i}, r)),

then if E_{r} = ∪_{i∈I^{∗}}E(x_{i}, r), the sequence ∞_{E_{r}} γ-converges to μ.

This result is then generalized for measures in M_{o}^{a}. Note that the assumptions N = ∅ can be weakened (see [7]), but not suppressed, because in the case of the dimension 1 (the points have non-zero capacity) this density result is false.

The γ^{a}-convergence of the measures is proved thanks to a generalized Poincaré inequality for positive Kato measures (see Theorem 4.1 which has an interest for itself.

Before giving the precise hypothesis on the Dirichlet Poincaré form we want to point out that our results apply for instance to the following cases:

• forms connected with strongly elliptic operators for X = ℝ^{n} and n ≥ 2, • forms connected with degenerate elliptic operators with a weight w in the A_{2}

Muckenhoupt class such that N = ∅; let us remember that in the model case X = ℝ^{n} (n ≥ 2) and w(x) = |x|^{α} the requirement w ∈ A_{2} means that −n < α < n, but if we want that cap^{a}({0}, ℝ^{n}) = 0, we have to add the hypothesis −n + 2 ≤ α (see [20]).

• forms connected with vector fields satisfying a Hörmander condition X = ℝ^{n} and ν ≥ 2 where ν is the intrinsic dimension (in X_{0}) (see [3, 21, 23, 27]).

2. Hypothesis on the space, the measure and the Dirichlet form

Let X be a connected, locally compact, separable Hausdorff space. We fix a positive Radon measure m on X, with supp m = X, which is called the “volume” measure on X. For the general theory of Dirichlet forms we refer to [22]. Let us consider a strongly local regular symmetric Dirichlet form a(u, v) on the Hilbert space L^{2}(X, m) with domain D(a); we recall that D(a) is a Hilbert space with the intrinsic norm

\sqrt{a(u, v) + ∫_{X} uv dm}. It is possible to associate with a(u, v) a Radon-measure-valued non-negative definite symmetric bilinear form α(u, v)(dx), called the energy measure of a, such that

a(u, v) = ∫_{X} α(u, v)(dx)  (2.1)

for u, v ∈ D(a). We refer to [3] for the definition and for the main properties of the energy measure, and to [26] for the proofs in a more general context. Since the form a(u, v) is regular, there exists a core C ⊂ C_{c}(X) ∩ D(a) which is dense in C_{c}(X) with the uniform norm, and in D(a) with the intrinsic norm. We assume that C is an mseparating core, that is, for every x, y ∈ X, with x ≠ y, there exists ϕ ∈ C such that ϕ(x) ≠ ϕ(y) and α(ϕ, ϕ) ≤ m, where the last inequality is taken in the sense of Radon measures on X. Let X_{0} ≠ X be a connected relatively compact open subset of X and Ω an open subset of X_{0}, the closure of C_{c}(Ω) ∩ D(a) in D(a) for the intrinsic norm is denoted by D_{o}(a, Ω). We define D(a, Ω) as the set of all restrictions u|_{Ω} to Ω of the functions u ∈ D(a). By the strong local property of the form a(u, v), the restriction to Ω of the energy measure α(u, v) depends only on the restrictions of u and v to Ω. For the properties of the space D(a, Ω) we refer to [3,8]. By using the energy measure of the Dirichlet form a(u, v) we can introduce a metric on X, called the intrinsic metric, defined by

d(x, y) = sup {φ(x) − φ(y) : φ ∈ C, α(φ, φ) ≤ m in X}.  (2.2)

By B(x, r) we denote the intrinsic ball centered at x with radius r, i.e.

B(x, r) = {y ∈ X : d(x, y) < r}.  (2.3)

We assume that the topology induced by this metric coincides with the given topology of X.

We also assume that the measure m on X satisfies the following doubling condition with respect to the intrinsic metric: there exist two constants R_{o} > 0 and C_{o} > 1 such that

0 < m(B(x, 2r)) ≤ C_{o} m(B(x, r)) < +∞  (2.4)

for every x ∈ X_{0} and for every 0 < r ≤ R_{o}. Moreover, we suppose that

B(x, 2r) ≠ B(x, r)  (2.5)

for every x ∈ X_{0} and for every 0 < r ≤ \frac{R_{o}}{2} . Let us suppose that the metric space (X, d) is complete.

Let us recall that (X, d) together with this doubling measure m is a space of homogeneous type or, for brevity, a homogeneous space in the sense of Coifmman and Weiss [10].

We remark that if X is the union of a sequence of balls of radius R_{o}, then the separability of X is a consequence of the homogeneity.

Remark 2.1. We recall the following property of homogeneous spaces which is a fundamental tool for our method (see [10: pp. 66 – 71] and [8: Proof of Lemma 2.5]): there exist two constants R_{o} > 0 and M such that for every 0 < r ≤ \frac{R_{o}}{4} there exists a covering of Ω: B(x_{i}, r) (i = 1, …, q) such that d(x_{i}, x_{j}) ≥ r. This implies that the balls B(x_{i}, \frac{r}{2}) are disjoint and that any point x belongs at most to M balls B(x_{i}, r) (M is an intrinsic constant) and at most to K(k)M balls B(x_{i}, kr) where k ≥ 1.

Let us state here an easy lemma on the inverse doubling property of measure:

Lemma 2.1. There exist a constant C_{∗} > 1 such that for every ball B(x, 2r) ⊂⊂ X_{0}, 0 < r ≤ \frac{R_{o}}{2} , we have

m(B(x, 2r)) ≥ C_{∗}m(B(x, r)).  (2.6)

Proof. Using the connexity of X and the continuity of the distance, it is easy to see that there exists y_{0} ∈ B(x, 2r) such that d(x, y_{0}) = \frac{3}{2}r. Let us consider a ball centered in y_{0} with radius \frac{1}{2}r, that is B(y_{0}, \frac{1}{2}r). Easily,

B(y_{0}, \frac{1}{2}r) ⊂ B(x, 2r)  (2.7)

and

B(y_{0}, \frac{1}{2}r) ⊂ B(x, r)^{c}.  (2.8)

On the other hand,

B(x, 2r) ⊂ B(y_{0}, 4r).  (2.9)

Indeed, for z ∈ B(x, 2r),

d(z, y_{0}) ≤ d(z, x) + d(x, y_{0}) ≤ 2r + \frac{3}{2}r < 4r,  (2.10)

so we have

m(B(y_{0}, \frac{1}{2}r)) ≥ \frac{1}{C_{o}^{3}} m(B(y_{0}, 4r)) ≥ \frac{1}{C_{o}^{3}} m(B(x, 2r)) ≥ \frac{1}{C_{o}^{3}} m(B(x, r))  (2.11)

and

m(B(x, 2r)) − m(B(x, r)) = m(B(x, 2r)\B(x, r)) ≥ m(B(y_{0}, \frac{1}{2}r)) ≥ \frac{1}{C_{o}^{3}} m(B(x, r)),

(2.12)

and the statement follows for C_{∗} = 1 + \frac{1}{C_{o}^{3}}

We assume that the following Poincaré inequality holds: there exist two constants C_{1} > 0 and k ≥ 1 such that for every 0 < r ≤ R_{o}, y ∈ X_{0} and for every u ∈ D(a) we have

∫_{B(y,r)} |u − u_{y,r}|^{2}m(dx) ≤ C_{1}r^{2} ∫_{B(y,kr)} α(u, u)(dx),  (2.13)

where u_{y,r} is the average of u on B(y, r) with respect to the measure m. Let us remember that the embedding of D_{o}[a, Ω] into L^{2}(Ω, m) is compact

D_{o}(a, Ω) ⊂⊂ L^{2}(Ω, m)  (2.14)

when Ω ⊂⊂ X_{0} (this fact can be proved as in Lemma 2.5 of [8]).

3. Review on the definitions of measure space

We recall here some definitions and properties on measures not charging polar sets and on Kato measures (see [3, 5, 11, 12, 13, 16, 22]; all these definitions and properties are introduced and studied in previous papers, but for the sake of completeness, we describe them hereafter.

The capacity of a set E ⊂ X associated with the Dirichlet form a(u, v) is denoted by cap^{a}(E) (see [22: Section 3.1]). The capacity cap^{a}(E, Ω) of a set E with respect to an open set Ω ⊂ X is defined in a similar way, replacing X by Ω and D(a) by D_{o}(a, Ω).

A function u defined on an open set Ω ⊂ X is said to be quasi-continuous if for every ε > 0 there exists an open set G_{ε} with capacity less than ε (cap^{a}(G_{ε}) ≤ ε) such that the restriction of u to Ω \ G_{ε} is continuous.

Every u ∈ D(a, Ω) admits a quasi-continuous representative, that is, there exists a quasi-continuous function ũ on Ω, unique up to modifications on a set of capacity zero (with respect to X), such that u = ũ m-almost everywhere on Ω (see [22: Chapter 3]). In this paper, when considering pointwise values of functions in D(a, Ω), we shall tacitly consider their quasi-continuous representatives.

We say that a set U ⊂ X is quasi-open if for every ε > 0 there exists an open set U_{ε} such that U ⊂ U_{ε} and cap^{a}(U_{ε} \ U) < ε.

Definition 3.1. For every open set Ω ⊂ X, let M_{o}^{a}(Ω) be the set of all nonnegative Borel measures μ on Ω which are absolutely continuous with respect to cap^{a}, i.e. μ(E) = 0 for every Borel set E ⊂ Ω with cap^{a}(E) = 0.

We introduce now an equivalence relation on the class M_{o}^{a}(Ω).

Definition 3.2. We say that two non-negative measures μ and λ belonging to M_{o}^{a}(Ω) are equivalent (and we write μ ≃ λ) if ∫_{Ω} u^{2}dμ = ∫_{Ω} u^{2}dλ for every u ∈ D_{o}(a, Ω).

Definition 3.3. Let Ω be an open subset of X, let μ ∈ M_{o}^{a}(Ω), and let E ⊂⊂ Ω be a μ-measurable set. The set E is μ-admissible if there exsits a function u ∈ D_{o}(a, Ω) with (u − 1) ∈ L^{2}(E, μ). If E is μ − admissible, the variational μ-capacity of E in Ω relative to the Dirichlet form a(u, v) is defined by

cap_{μ}^{a}(E, Ω) = min {a(u, u) + ∫_{E}(u − 1)^{2}dμ : u ∈ D_{o}(a, Ω)} .  (3.1)

The minimum above is attained taking into account the lower semi-continuity of the functional in the weak topology of D_{o}(a, Ω); if E ⊂⊂ Ω, then the unique minimum point u_{E} of (3.1) is called the μ-capacitary potential of E in Ω. If E is not μ − admissible, we define cap_{μ}^{a}(E, Ω) = +∞.

Remark 3.1. Arguing as in [11: Theorem 2.9] we obtain that cap_{μ}^{a}(·, Ω) is an increasing countably sub-additive set function, cap_{μ}^{a}(∅, Ω) = 0, cap_{μ}^{a}(E, Ω) ≤ μ(E), and cap_{μ}^{a}(E, Ω) ≤ cap^{a}(E, Ω). Moreover, cap_{μ}^{a}(E_{h}, Ω) ↗ cap_{μ}^{a}(E, Ω) whenever E is the union of an increasing sequence (E_{h}) of μ-measurable sets contained in Ω. In particular, for every open set U ⊂ Ω we have

cap_{μ}^{a}(U, Ω) = sup {cap_{μ}^{a}(K, Ω) : K compact, K ⊂ U}.  (3.2)

Example 3.1 (see [13: Example 1.6]). Let E ⊂ Ω and let ∞_{E} be the Borel measure on Ω defined by

∞_{E}(B) = \{ \begin{matrix}0 & if cap^{a}(B ∩ E) = 0 \\ +∞ & otherwise. \end{matrix}

Then ∞_{E} belongs to M_{o}^{a}(Ω). We notice that a Borel function defined on Ω belongs to L^{2}(Ω, ∞_{E}) if and only if u = 0 on E, except on a subset of capacity zero. Therefore, if E is closed in Ω, we have D_{o}(a, Ω \ E) = D_{o}(a, Ω) ∩ L^{2}(Ω, ∞_{E}) (see [22: Theorem 4.4.2/(i)]). It is not difficult to see that cap_{∞_{E}}^{a}(B, Ω) = cap^{a}(E ∩ B, Ω) for every Borel set B ⊂ Ω.

The Green function for the form a(u, v) in an open set Ω ⊂⊂ X_{0} is denoted by G_{Ω}(x, y). We refer to [2,3] for its definition and main properties. In particular, we shall use the following estimate for balls whose radius is sufficiently small with respect to the constant R_{o} which appears in the doubling condition (2.4).

Proposition 3.1. Assume that 20R < R_{o} and that B(x, 40R) ⊆ X_{0} is different from X. If 0 < d(x, y) < qR, q ∈ (0, 1) then

\frac{1}{c} ∫_{d(x,y)}^{R} \frac{s^{2}}{m(B(x, s))} \frac{ds}{s} ≤ G_{B(x,R)}(x, y) ≤ c ∫_{d(x,y)}^{R} \frac{s^{2}}{m(B(x, s))} \frac{ds}{s}.  (3.3)

The constant c depends only q and on the constants occurring in the doubling condition and in the Poincaré inequality.

Let us remark that this implies (see [3: Remark 6.1]) that if 0 ≤ r ≤ qR, then

cap^{a}(B(r), B(R)) ∼ (∫_{r}^{R} \frac{s^{2}}{m(B(s))} \frac{ds}{s})^{−1} ,

where the constants depends on q; this implies that if 0 ≤ q ≤ 1, then

C_{1}(q)\frac{m(B(r))}{r^{2}} ≤ cap^{a}(B(qr), B(r)) ≤ C_{2}(q)\frac{m(B(r))}{r^{2}}.  (3.4)

From the definition of capacity we have also

lim_{ρ→r} cap^{a}(B(ρ), B(r)) = +∞.

Let us also remark that the hypothesis N = {x_{0} : a − cap(x_{0}) > 0} ∩ Ω = ∅ means that

lim_{r→0} ∫_{r}^{R} \frac{s^{2}}{m(B(s))} \frac{ds}{s} = ∞.  (3.5)

Following [5], we recall the notion of Kato measure associated with a regular Dirichlet form:

Definition 3.4. Let Ω be a relatively compact open subset of X with 2 diam(Ω) = R < R_{o}, where R_{o} is the constant which appears in the doubling condition (2.4). Assume that there exists x_{o} ∈ Ω with B(x_{o}, 4R) ⊂⊂ X and B(x_{o}, 4R) ≠ X. We say that μ is a Kato measure on Ω if μ is a Radon measure on Ω such that

lim_{r↓0} sup_{x∈Ω} ∫_{Ω∩B(x,r)} (∫_{d(x,y)}^{R} \frac{s^{2}}{m(B(x, s))} \frac{ds}{s}) |μ|(dy) = 0,  (3.6)

where |μ| denotes the total variation of the measure μ. The space of Kato measures is denoted by K(Ω), while K_{+}(Ω) indicates the cone of non-negative elements of K(Ω).

In the following we assume that Ω is a relatively compact open subset of X_{0} with Ω ⊂ B(R) ⊂ B(40R) ⊂ X_{0}, where B(R) denotes a ball of radius R andR_{o} is the constant which appears in the doubling condition (2.4), if it is not the case we decompose Ω into open sets satisfying the previous conditions.

We remark that a Kato measure μ is diffuse, i.e. μ({x}) = 0 for every x ∈ Ω; hence we have lim_{r↓0} |μ|(B(x, r)) = 0 for every x ∈ Ω. From (3.6) we have that for every compact set K ⊂ Ω and for every ε > 0 there exists δ > 0 such that |μ|(E) < ε for every Borel set E ⊂ K with diam(E) < δ. Moreover, if μ ∈ K(Ω) and g is a bounded Borel function, then it is not difficult to see that gμ ∈ K(Ω). We recall that if μ is a Kato measure and if u ∈ D_{o}(a, Ω) is the solution of a(u, v) = ⟨μ, v⟩ for any v ∈ D_{o}(a, Ω), then u has the representation

u(x) = ∫_{Ω} G_{Ω}(x, y) dμ(y)  q.e. in Ω  (3.7)

(see [19: Proposition 37] and [5: Proposition 3.2]). Moreover, u is continuous in Ω (see [5: Proposition 4.1]).

Finally, we recall a decomposition result for diffuse measures in M_{o}^{a}(Ω) proven in [12] by using an extension to Dirichlet spaces of the Evans-Vasilesco Theorem and a result on reconstruction of a measure using capacities proven in [13].

Theorem 3.1. Let Ω be an open relatively compact subset of X and let μ ∈ M_{o}^{a}(Ω) be a diffuse measure. Then there exist a Borel function g: Ω → [0, +∞] and a measure λ ∈ K_{+}(Ω) such that μ ≃ gλ.

Theorem 3.2. Let Ω an open relatively compact subset of X, let μ be a measure in M_{o}^{a}(Ω), and let

E_{μ} = {x ∈ Ω : cap_{μ}^{a}({x}, Ω) > 0}.  (3.8)

Then for every Borel set B ⊂ Ω we have

μ(B) = sup ∑_{i∈I} cap_{μ}^{a}(B_{i}, Ω) + ∑_{x∈B∩E_{μ}} \frac{(cap_{μ}^{a}(\{x\}, Ω))^{2}}{cap^{a}(\{x\}, Ω) − cap_{μ}^{a}(\{x\}, Ω)}  (3.9)

where the supremum is taken over all finite Borel partitions (B_{i})_{i∈I} of B (as usual if E_{μ} is not countable the sum for x ∈ B ∩ E_{μ} denotes the supremum of the sums on all finite subfamilies of B ∩ E_{μ}).

For every μ ∈ M_{o}^{a}(Ω) we consider the functional F_{μ}: L^{2}(Ω) → [0, +∞] defined by

F_{μ}(u) = a(u, u) + ∫_{Ω} u^{2}dμ.  (3.10)

We recall a variational convergence in the class M_{o}^{a}(Ω) (see also [8, 12, 29]).

Definition 3.5. We say that a sequence (μ_{h}) of measures in M_{o}^{a}(Ω) γ^{a}-converges to a measure μ ∈ M_{o}^{a}(Ω) if the sequence of functionals F_{μ_{h}} Γ-converges to the functional F_{μ} in L^{2}(Ω, m) in the sense of [18].

Let us remember that this convergence is equivalent to the L^{2}-strong convergence of the minima, more precisely:

A sequence (μ_{h}) of measures in M_{o}^{a}(Ω) γ^{a}-converges to a measure μ ∈ M_{o}^{a}(Ω) if and only if for every f ∈ D′_{o}(a, Ω) (D′_{o}(a, Ω) is the dual space of D_{o}(a, Ω)) the sequence of weak solutions u_{h} ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ_{h}) of the problems

a(u_{h}, v) + ∫_{Ω} u_{h}v dμ_{h} = ⟨f, v⟩

for any v ∈ D_{o}(a, Ω)∩L^{2}(Ω, μ_{h}), converges weakly in D_{o}(a, Ω) and strongly in L^{2}(Ω, m) to the weak solution u ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ) of the limit problem

a(u, v) + ∫_{Ω} uv dμ = ⟨f, v⟩

for any v ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ).

The following result (see [28, 29]) shows that the space M_{o}^{a}(Ω) is sequentially compact under the γ^{a}-convergence.

Proposition 3.2. Every sequence of measures of M_{o}^{a}(Ω) has a subsequence which γ^{a}-converges to a measure of M_{o}^{a}(Ω).

4. Poincaré inequality

In this section, when dealing with balls, we shall omit the dependence on the center. So we shall denote by B(t) any ball with radius t centered in a given point x_{o} ∈ X_{0}. We shall also omit the dependence on the form, which will be fixed. For instance, cap(B(qr), B(r)) = cap^{a}(B(qr), B(r)).

Our main result of this section is a Poincaré inequality on an intrinsic ball B(r). It consists of estimating the L^{2}(μ)-norm of a function u with zero outer capacitary mean (where μ is a Kato measure, possibly different from m) by the energy of the form times a term tending to 0 as the radius tends to 0. This method can be easily extended to other bounded projections. See also [1, 15, 31].

First we define the outer capacitary measure:

Definition 4.1. Let U and V two open sets such that V ⊂⊂ U ⊂⊂ Ω, and w the capacitary potential of V with respect to U (see also (3.1) with μ = ∞_{V} and Ω = U), i.e. the minimum point :

min {∫_{U} α(v, v)(dx) : v ∈ D_{o}(a, U), v ≥ 1 q.e. on V }
= cap^{a}(V, U) = ∫_{U} α(w, w)(dx).

(4.1)

Let us extend w by 0 on Ω\U. Then there exist two positive measures ν, λ ∈ D′_{o}(a) (D′_{o}(a) is the dual space of D_{o}(a) = D_{o}(a, X)) such that

a(w, v) = ⟨ν − λ, v⟩  for any v ∈ D_{o}(a, Ω),  (4.2)

supp(ν) ⊂ ∂V, supp(λ) ⊂ ∂U and ν(Ω) = λ(Ω) = cap^{a}(V, U). We call ν and λ the inner and outer a-capacitary distribution, respectively, of V with respect to U.

The proof of the existence of such measures is similar to that of [15: Lemma 3.1], where the essential tools are Stampacchia’s techniques (see [30: Theorems 2.1 and 3.8] and [25: Theorem 2.6.4]) which can be easily extended in our framework.

Theorem 4.1. Let μ ∈ K_{+}(Ω) be a Kato measure and let P be defined by

P (u) = \frac{1}{λ_{r}^{ρ}(B(2r))} ∫_{B(2r)} u dλ_{r}^{ρ} = \frac{1}{λ_{r}^{ρ}(B(r))} ∫_{B(r)} u dλ_{r}^{ρ}  (4.3)

where λ_{r}^{ρ} is the outer a-capacitary distribution of B(ρ) with respect to B(r) for ρ < r. There exists r_{o} > 0 such that for any r ≤ r_{o}, if B(r) ⊂ 40B(r) ⊂⊂ Ω and u ∈ D(a, B(2kr)), then

∫_{B(r)}(u − P (u))^{2}dμ ≤ Cη(r) ∫_{B(2kr)} α(u, u)(dx)  (4.4)

where C is a constant independent of r and η is a function independent of the center of the ball B(r) with η(r) → 0 as r → 0.

We begin with an abstract result estimating the L^{2} norm with respect to a Kato measure in B(s) by an energy norm in B(t) with t > s.

Proposition 4.1. Let u ∈ D(a, B(t)) and μ be a Kato measure, μ ∈ K_{+}(B), where B(t) ⊂ B ⊂ 40B ⊂ Ω. There exists t_{0} > 0 such that, for any s < t ≤ t_{0},

∫_{B(s)} u^{2}dμ ≤ Cη(t) (∫_{B(t)} α(u, u)(dx) + \frac{1}{(t − s)^{2}} ∫_{B(t)\B(s)} u^{2}dm)  (4.5)

where C is a constant independent of t and η(t) → 0 when t → 0.

Proof. We assume, without loss of generality, that u ∈ C(B(t)). Let W be the solution of the problem to find W ∈ D_{o}(a, B(t)) such that

∫_{B(t)} α(W, v)(dx) = ∫_{B(t)} v dμ  for any v ∈ D_{o}(a, B(t)).  (4.6)

Let us denote by G_{B(t)}^{x} the Green function with singularity in x with respect to B(t). Then we have

W (x) = ∫_{B(t)} G_{B(t)}^{x}(y) dμ ≤ ∫_{B(t)} G_{B}^{x}(y) dμ ≤ η(t) = sup_{x∈B} ∫_{B(t)} G_{B}^{x}(y) dμ.  (4.7)

Using the definition of Kato measure and (3.3) we obtain that η(t) → 0 when t → 0 and W ∈ D_{o}(a, B(t)) ∩ C(B(t)) (see [5]). Let ϕ be the cut-off function between B(s) and B(\frac{t+s}{2}). Then u^{2}ϕ^{2} ∈ C_{0}(B(t)) and we obtain using (4.6)

∫_{B(t)} u^{2}ϕ^{2}dμ = ∫_{B(t)} α(W, u^{2}ϕ^{2})(dx)

= 2 ∫_{B(t)} uϕ^{2}α(W, u)(dx) + 2 ∫_{B(t)} u^{2}ϕα(W, ϕ)(dx)

≤ \frac{ε}{2} ∫_{B(t)} ϕ^{2}α(u, u)(dx) + \frac{2}{ε} ∫_{B(t)} u^{2}ϕ^{2}α(W, W)(dx)

+ \frac{ε}{2} ∫_{B(t)} u^{2}α(ϕ, ϕ)(dx) + \frac{2}{ε} ∫_{B(t)} u^{2}ϕ^{2}α(W, W)(dx)

= \frac{ε}{2} ∫_{B(t)}(ϕ^{2}α(u, u)(dx) + u^{2}α(ϕ, ϕ)(dx)) + \frac{4}{ε} ∫_{B(t)} u^{2}ϕ^{2}α(W, W)(dx).

Let us estimate the last term:

∫_{B(t)} u^{2}ϕ^{2}α(W, W)(dx)
= ∫_{B(t)} α(W, W u^{2}ϕ^{2})(dx) − 2 ∫_{B(t)} uϕW α(W, uϕ)(dx)
= ∫_{B(t)} W u^{2}ϕ^{2}dμ − 2 ∫_{B(t)} uϕW α(W, uϕ)(dx)
≤ η(t) ∫_{B(t)} u^{2}ϕ^{2}dμ + ∫_{B(t)} u^{2}2ϕ^{2}α(W, W)(dx) + 2 ∫_{B(t)} W^{2}α(uϕ, uϕ)(dx)
≤ η(t) ∫_{B(t)} u^{2}ϕ^{2}dμ + \frac{1}{2} ∫_{B(t)} u^{2}ϕ^{2}α(W, W)(dx)
+ 4η^{2}(t) ∫_{B(t)} (u^{2}α(ϕ, ϕ)(dx) + ϕ^{2}α(u, u)(dx))

and we obtain

\begin{matrix}∫_{B(t)} u^{2}ϕ^{2}α(W, W)(dx) \\ ≤ 2η(t) ∫_{B(t)} u^{2}ϕ^{2}dμ + 8η^{2}(t) ∫_{B(t)} (u^{2}α(ϕ, ϕ)(dx) + ϕ^{2}α(u, u)(dx)). \end{matrix}

But

\begin{aligned}∫_{B(t)} u^{2}ϕ^{2}dμ & ≤ \frac{ε}{2} ∫_{B(t)} (ϕ^{2}α(u, u)(dx) + u^{2}α(ϕ, ϕ)(dx)) + \frac{8}{ε}η(t) ∫_{B(t)} u^{2}ϕ^{2}dμ \\ & + \frac{32}{ε}η^{2}(t) ∫_{B(t)} (u^{2}α(ϕ, ϕ)(dx) + ϕ^{2}α(u, u)(dx)) \\ & = \frac{8}{ε}η(t) ∫_{B(t)} u^{2}ϕ^{2}dμ \\ & + (\frac{ε}{2} + \frac{32η^{2}(t)}{ε}) ∫_{B(t)} (u^{2}α(ϕ, ϕ)(dx) + ϕ^{2}α(u, u)(dx)). \end{aligned} (4.8)

Let us choose ε = 16 η(t) in the preceding inequality. Then

\begin{matrix}∫_{B(t)} u^{2}ϕ^{2}dμ ≤ \\ \frac{1}{2} ∫_{B(t)} u^{2}ϕ^{2}dμ + 10 η(t) ∫_{B(t)} (u^{2}α(ϕ, ϕ)(dx) + ϕ^{2}α(u, u)(dx)), \end{matrix}  (4.9)

and then

\begin{aligned}∫_{B(s)} u^{2}dμ & ≤ ∫_{B(t)} u^{2}ϕ^{2}dμ \\ & ≤ 20 η(t) ∫_{B(t)} (u^{2}α(ϕ, ϕ)(dx) + ϕ^{2}α(u, u)(dx)) \\ & ≤ 20 η(t) (∫_{B(t)} α(u, u)(dx) + \frac{C}{(t − s)^{2}} ∫_{B(t)\B(s)} u^{2}dm) \end{aligned}  (4.10)

for any u ∈ C(B(t)), and we conclude by density

Remark 4.1. Easily, from Proposition 4.1 we have that, if u ∈ D(a, B(2r)) and c ∈ ℝ,

∫_{B(r)}(u − c)^{2}dμ ≤ Cη(2r) (∫_{B(2r)} α(u, u)(dx) + \frac{1}{r^{2}} ∫_{B(2r)}(u − c)^{2}dm) .  (4.11)

Let H be the Hilbert space H = D(a, B(2r)) endowed with the scalar product

⟨u, v⟩ = C (∫_{B(2r)} α(u, v)(dx) + \frac{1}{r^{2}} ∫_{B(2r)} uv dm)  (4.12)

where C is a suitable constant . Let H_{1} = ℝ. The Hilbert projection on H_{1} is given by Π(u) = \frac{1}{m(B(2r))} ∫_{B(2r)} u dm. On the other hand, we denote by P a bounded linear operator P : H → H_{1} such that P(u_{1}) = u_{1} for any u_{1} ∈ H_{1}. We can use (4.11) with c = P(u). Then

\begin{matrix}∫_{B(r)}(u − P(u))^{2}dμ \\ ≤ Cη(2r) (∫_{B(2r)} α(u, u)(dx) + \frac{1}{r^{2}} ∫_{B(2r)}(u − P(u))^{2}dm) . \end{matrix}  (4.13)

In view of obtaining our Poincaré inequality (4.4), we have to prove that

∫_{B(2r)}(u − P(u))^{2}dm ≤ Cr^{2} ∫_{B(2kr)} α(u, u)(dx).  (4.14)

Using the Poincaré inequality (2.13) with Π(u) = \frac{1}{m(B(2r))} ∫_{B(2r)} u dm, it is enough to prove that

(Π(u) − P(u))^{2} ≤ \frac{Cr^{2}}{m(B(2r))} ∫_{B(2kr)} α(u, u)(dx),  (4.15)

but

(Π(u) − P(u))^{2} = (P(u − Π(u)))^{2} ≤ ∥P∥_{H′}^{2}∥u − Π(u)∥_{H}^{2}.  (4.16)

Thanks to

∥u − Π(u)∥_{H}^{2} ≤ C ∫_{B(2kr)} α(u, u)(dx)  (4.17)

we have Theorem 4.1 below for any P such that ∥P∥_{H′}^{2} ≤ \frac{Cr^{2}}{m(B(2r))}.

It remains to prove that for P defined as in (4.3)

P (u) = \frac{1}{λ_{r}^{ρ}(B(2r))} ∫_{B(2r)} u dλ_{r}^{ρ} = \frac{1}{λ_{r}^{ρ}(B(r))} ∫_{B(r)} u dλ_{r}^{ρ}  (4.18)

we have

∥P ∥_{H′}^{2} ≤ \frac{Cr^{2}}{m(B(2r))}

where λ_{r}^{ρ} is the outer a-capacitary distribution of B(ρ) with respect to B(r) for ρ < r.

Lemma 4.1. Let r ≤ \frac{R_{o}}{4} , 40B(r) ⊂⊂ Ω and P as defined in (4.3). If λ_{r}^{ρ} is the outer capacitary measure of B(ρ) in B(r) concentrated on the boundary ∂B(r) and if there exist q ∈ (0, 1) and a constant C such that for any ρ ∈ (0, qr) and for any 0 ≤ ϕ ∈ H

\frac{1}{λ_{r}^{ρ}(∂B(r))} ∫_{∂B(r)} ϕ dλ_{r}^{ρ} ≤ C \frac{1}{λ_{r}^{qr}(∂B(r))} ∫_{∂B(r)} ϕ dλ_{r}^{qr},  (4.19)

then

∥P ∥_{H′}^{2} ≤ \frac{Cr^{2}}{m(B(2r))}.  (4.20)

Proof. Using (4.19), for any ϕ ∈ H

|\frac{1}{λ_{r}^{ρ}(∂B(r))} ∫_{∂B(r)} ϕ dλ_{r}^{ρ}| ≤ C \frac{1}{λ_{r}^{qr}(∂B(r))} ∫_{∂B(r)} |ϕ| dλ_{r}^{qr}  (4.21)

and then

∥P ∥_{H′} = \frac{1}{λ_{r}^{ρ}(∂B(r))}∥λ_{r}^{ρ}∥_{H′} ≤ C \frac{1}{λ_{r}^{qr}(∂B(r))}∥λ_{r}^{qr}∥_{H′}.  (4.22)

We want to prove that

∥λ_{r}^{qr}∥_{H′}^{2} ≤ \frac{Cr^{2}}{m(B(2r))}(λ_{r}^{qr}(∂B(r)))^{2}.

Let us consider the function ζ = 1 − θ where θ is the cut-off function between B(qr) and B(r). Then ζ = 1 on ∂B(r), ζ = 0 in \overline{B(qr)} and α(ζ, ζ)(dx) ≤ \frac{c}{r^{2}} dm. Using the equality λ_{r}^{qr}(∂B(r)) = cap(B(qr), B(r)) and the definition of the capacitary potential w_{qr}^{r} of the ball B(qr) in B(r) (extended to zero in the complement of B(r)), we obtain for any ϕ ∈ H ∩ C(B(2r))

\begin{aligned}|λ_{r}^{qr}(ϕ)| & = |∫_{B(2r)} ϕ dλ_{qr}^{r}| \\ & = |∫_{B(2r)} ϕζ dλ_{qr}^{r}| \\ & = |∫_{B(2r)} α(w_{qr}^{r}, ϕζ)(dx)| \\ & ≤ C\sqrt{cap(B(qr), B(r))} (∫_{B(2r)} α(ϕ, ϕ)dx + \frac{1}{r^{2}} ∫_{B(2r)} ϕ^{2}dm)^{\frac{1}{2}} . \end{aligned} (4.23)

So we have

∥λ_{r}^{qr}∥_{H′}^{2} ≤ Ccap(B(qr), B(r)).  (4.24)

It follows that

cap(B(qr), B(r)) ≤ \frac{Cr^{2}}{m(B(2r))}(λ_{r}^{qr}(∂B(r)))^{2}  (4.25)

if and only if

cap(B(qr), B(r)) ≥ C \frac{m(B(2r))}{r^{2}}.  (4.26)

Using (3.4) and the previous inequalities we obtain the desired result

We end the proof of Theorem 4.1 by observing that inequality (4.19) can be proved using the same methods as in [15].

Remark 4.2. Using Lemma 4.1, it is also true that (4.4) holds for P(u) satisfying an inequality of the type (4.20), for instance for

P(u) = \frac{1}{λ_{r4}^{ρ}B(\frac{r}{4})} ∫_{B( \frac{r}{4} )} u dλρ_{\frac{r}{4}} .  (4.27)

5. Main results

Now we are able to prove the density result announced in the introduction. We shall prove it first for Kato measures in a Dirichlet space without polar points. This result gives a natural meaning to the notion of relaxed Dirichlet problems involving a form, i.e. any weak solution u of the ”relaxed Dirichlet problem” in Ω with homogeneous Dirichlet boundary data a(u, v) + ∫_{Ω} uv dμ = ∫_{Ω} fv dm (with v in a suitable space) is the L^{2}(Ω)-strong limit of solutions u_{h} of a(u_{h}, v) = ∫_{Ω_{h}} fv dm with homogeneous Dirichlet boundary data in varying domains Ω_{h}.

Different density results have been stated for instance by Mataloni and Tchou [28] proving that the relaxed Dirichlet problems are also the closure of Schrödinger type problems. These results have been already proved in the Laplacian case by Dal Maso and Mosco [16, 17] and in a more constructive way for general strongly elliptic operators by Dal Maso and Malusa [15]. In both articles, the Euclidean pavage associated with the Laplacian plays a fundamental role. The density result in the case of elliptic degenerated Heisenberg operator can be easily proven with the same methods as [16, 17 ] by using results stated in [6], and in particular the special pavage associated with the operator.

In a more general case, it is not possible to construct a pavage of the space associated to a general degenerated elliptic operator and enjoying the properties needed by the Dal Maso and Mosco theory. The method used here is issued from harmonic analysis on homogeneous spaces (see [10]) and consists of replacing the notion of ”pavage” (disjoint covering) associated to the operator, by the notion of ”homogeneous” covering associated to the operator. This notion is really weaker than the first one, and more suitable for general degenerated operators. Moreover, also in the classical strongly elliptic case, this method provides new approximating Dirichlet problems associated to homogeneous coverings different from the pavage.

Let us recall that, using the measures defined in Example 3.1 μ_{h} = ∞_{E_{h}} where Ω_{h} = Ω\E_{h}, it is possible to consider the sequence u_{h} of solutions of the homogeneous Dirichlet problem a(u_{h}, v) = ∫_{Ω_{h}} fv dm as a special case of a sequence of solutions of the relaxed Dirichlet problem

a(u_{h}, v) + ∫_{Ω} u_{h}v dμ_{h} = ∫_{Ω} fv dm.

Let us recall also that for H^{−1} positive measures, Cioranescu, Murat and Kacimi (see [9, 24]) proved that the existence of the Dirichlet approximating problem is a consequence of the existence of two positive measures satisfying special conditions. Indeed, we want also to use a similar approach and we start with the following

Proposition 5.1. Let μ ∈ D_{o}^{−1}(a, Ω). If there exist two positive measures ν_{h}, λ_{h} ∈ D_{o}^{−1}(a, Ω) and a sequence of functions w_{h} ∈ D(a, Ω) such that

w_{h} ⇀ 1_{w_{h} = 0}_{a(w_{h}, v) = ⟨ν_{h} − λ_{h}, v⟩} weakly in D(a, Ω)

q.e. in E_{h}
for any v ∈ D_{o}(a, Ω)

ν_{h} → μ_{λ_{h} ⇀ μ}_{⟨λ_{h}, v⟩ = 0} strongly in D_{o}^{−1}(a, Ω)

weakly in D_{o}^{−1}(a, Ω)
for any v ∈ D_{o}(a, Ω) with v = 0 in E_{h},

then the measures ∞_{E_{h}} γ-converge to the measure μ.

Proof. First we remark that μ ∈ D_{o}^{−1}(a, Ω) is a positive measure (cf. [24: Remarque 2.4]). To prove our result, let us consider for f ∈ D_{o}^{−1}(a, Ω) the sequence u_{h} of the solutions of the problems

\begin{matrix}a(u_{h}, v) & = ⟨f, v⟩  for any v ∈ D_{o}(Ω) with v = 0 in E_{h} \\ u_{h} & = 0  in E_{h}. \end{matrix} \}  (5.1)

We want to prove its strong L^{2}(Ω)-convergence to the solution u of the problem

a(u, v) + ∫_{Ω} uv dμ = ⟨f, v⟩  for any v ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ).  (5.2)

Let us take as test function in (5.1) w_{h}ϕ where ϕ ∈ C (the core of the form); the choice is possible since ϕ is continuous and α(phi, phi) Has a bounded density, so w_{h}ϕ ∈ D_{o}(a, Ω). Then

∫_{Ω} w_{h}α(u_{h}, ϕ)(dx) + ∫_{Ω} α(w_{h}, u_{h}ϕ)(dx) − ∫_{Ω} u_{h}α(w_{h}, ϕ)(dx) = ⟨f, w_{h}ϕ⟩  (5.3)

and, using the definition of w_{h},

∫_{Ω} α(w_{h}, u_{h}ϕ)(dx) = ⟨ν_{h} − λ_{h}, u_{h}ϕ⟩ = ⟨ν_{h}, u_{h}ϕ⟩.  (5.4)

By using the techniques used in [8], we can prove that passing to the limit in (5.3) yields

∫_{Ω} α(u, ϕ)(dx) + ⟨μ, uϕ⟩ = ⟨f, ϕ⟩  (5.5)

for any ϕ ∈ C and then

∫_{Ω} α(u, ϕ)(dx) + ∫_{Ω} uϕ dμ = ⟨f, ϕ⟩  (5.6)

for any v ∈ D_{o}(a, Ω) ∩ L^{2}(Ω, μ)

Proposition 5.2. Let us suppose that N = {x_{0} : a − cap({x_{0}}) > 0} ∩ Ω = ∅, μ ∈ K_{+}(Ω) and supp μ ⊂⊂ Ω.

(i) There exists a positive constant depending only on R_{o} and C(R_{o}) such that if 0 < r ≤ min(C(R_{o}), \frac{1}{40}d(supp μ, Ω^{c})), we can construct a finite covering B(x_{i}, r) (i ∈ I) of Ω such that the balls B(x_{i}, \frac{r}{2}) are pairwise disjoint.

(ii) Let I^{∗} ⊂ I such that B(x_{i}, \frac{r}{2}) ⊂ Ω for all i ∈ I^{∗}. Let us define

\begin{aligned}A(x_{1}, r) & = B(x_{1}, r) \\ A(x_{i}, r) & = B(x_{i}, r)\ ∪_{j<i} B(x_{j}, r) (i ≥ 1). \end{aligned}

Let E(x_{i}, r) be a ball centered in x_{i}, contained in B(x_{i}, \frac{r}{4}) and with radius such that

cap^{a}(E(x_{i}, r), B(x_{i}, \frac{r}{4})) = μ(A(x_{i}, r)).  (5.7)

Then if E_{r} = ∪_{i∈I^{∗}}E(x_{i}, r) and r_{h} → 0 the sequence ∞_{E_{r_{h}}} γ-converges to μ.

Proof. We shall use the previous result to study the asymptotic behavior of the sequence ∞_{E_{r}}. To this aim, we construct three sequences w_{r} ∈ D(a, Ω), ν_{r} and λ_{r} as in the previous remark.

Using Definition 4.1, let us define λ_{i,r} as outer capacitary distribution of E(x_{i}, r) with respect to B(x_{i}, \frac{r}{4}) and ν_{i,r} as inner capacitary distribution of E(x_{i}, r) with respect to B(x_{i}, \frac{r}{4}). We introduce

\begin{aligned}λ_{r} & = ∑_{i∈I^{∗}} λ_{i,r} \\ ν_{r} & = ∑_{i∈I^{∗}} ν_{i,r} \\ w_{r} & = \{ \begin{matrix}1 & in Ω\ ∪_{i∈I^{∗}} B(x_{i}, \frac{r}{4}) \\ \{1 − ψ_{i,r}\} & in B(x_{i}, \frac{r}{4})\E(x_{i}, r) (i ∈ I^{∗}) \end{matrix} \end{aligned}  (5.8)

where ψ_{i,r} is the capacitary potential of E(x_{i}, r) with respect to B(x_{i}, \frac{r}{4}). Thanks to the weak maximum principle we have 0 ≤ w_{r} ≤ 1 and then ∥w_{r}∥_{L^{2}(Ω)} ≤ C. Moreover,

∫_{Ω} α(w_{r}, w_{r})(dx) = = ∑_{i∈I^{∗}} ∫_{B(x_{i}, \frac{r}{4} )\E(x_{i},r)} α(w_{i,r}, w_{i,r})(dx)

= ∑_{i∈I^{∗}} cap^{a}(E(x_{i}, r), B(x_{i}, \frac{r}{4}) =

= ∑_{i∈I^{∗}} μ(A(x_{i}, r))

≤ μ(Ω)

and we obtain

∥w_{r}∥_{D(a,Ω)} ≤ C.  (5.9)

Using the compact embedding of D_{o}(a, Ω) into L^{2}(Ω) (see (2.14)) and remarking that w_{r} − 1 ∈ D_{o}^{a}(Ω) we have

\begin{matrix}w_{r} & ⇀ w  in D(a, Ω) \\ w_{r} & → w  in L^{2}(Ω). \end{matrix} \}  (5.10)

We wish to prove that w = 1. For this let us consider the characteristic function

χ_{r} = χ_{(∪_{i∈I^{∗}} B(x_{i}, \frac{r}{2} )\B(x_{i}, \frac{r}{4} ))}.  (5.11)

We have χ_{r}w_{r} = χ_{r}. Let us call χ the w ∗ −L^{∞}-limit of χ_{r}. Passing to the limit as r → 0,

χw = χ.  (5.12)

We want to prove that q.e. χ ≠ 0. Let Ω′ be an open subset of Ω. If I^{∗′} ⊂ I is defined by i ∈ I^{∗′} if and only if B(x_{i}, \frac{r}{2}) ⊂ Ω′, and if C_{∗} is the constant of the inverse doubling condition (see (2.6)), we have

\begin{aligned}∫_{Ω′} χ_{r} & ≥ ∑_{i∈I^{∗′}} m(B(x_{i}, \frac{r}{2})) − (B(x_{i}, \frac{r}{4})) \\ & ≥ ∑_{i∈I^{∗′}}(C_{∗} − 1)m(B(x_{i}, \frac{r}{4})) \\ & ≥ \frac{C_{∗} − 1}{C_{∗}^{2}} ∑_{i∈I^{∗′}} m(B(x_{i}, r)) \\ & ≥ \frac{C_{∗} − 1}{C_{∗}^{2}}(m(Ω′) − O(r)) \end{aligned}  (5.13)

where O(r) → 0 as r → 0. Letting r → 0 and using a density argument we obtain χ ≥ \frac{(C_{∗}−1)^{2}}{C_{∗}} and this implies that w = 1.

Now we shall study the weak convergence of the sequence λ_{r} = ∑_{i∈I^{∗}} λ_{i,r}. Let ϕ ∈ C, and let us define

ϕ_{r} = ∑_{i∈I^{∗}} M_{i,r}(ϕ)χ_{A(x_{i},r)}, M_{i,r}(ϕ) = \frac{1}{λ_{i,r}(B(x_{i}, \frac{r}{4}))} ∫_{B(x_{i}, \frac{r}{4})} ϕ dλ_{i,r}.  (5.14)

From now on, let us choose r such that \frac{3r}{2} < \frac{1}{40}dist(supp μ, Ω^{c}); then it is easy to see that ∑_{Ω\∪_{i∈I^{∗}} B(x_{i},r)} ϕ dμ = 0. We recall that λ_{i,r} is the outer capacitary distribution of E(x_{i}, r) with respect to B(x_{i}, \frac{r}{4}). Then

λ_{i,r}(B(x_{i}, \frac{r}{4})) = cap^{a}(E(x_{i}, r), B(x_{i}, \frac{r}{4})) = μ(A(x_{i}, r))

and

|⟨λ_{r}, ϕ⟩ − ⟨μ, ϕ⟩|

≤ |∫_{∪_{i∈I^{∗}} B(x_{i},r)} ϕ dλ_{r} − ∫_{∪_{i∈I^{∗}} B(x_{i},r)} ϕ dμ|

= |∫_{∪_{i∈I^{∗}} B(x_{i}, \frac{r}{4} )} ϕ dλ_{r} − ∫_{∪_{i∈I^{∗}} A(x_{i},r)} ϕ dμ|

= |∑_{i∈I^{∗}} ∫_{B(x_{i}, \frac{r}{4} )} ϕ dλ_{i,r} − ∑_{i∈I^{∗}} ∫_{A(x_{i},r)} ϕ dμ|  (5.15)

= |∑_{i∈I^{∗}} \frac{μ(A(x_{i}, r))}{λ_{i,r}(B(x_{i}, \frac{r}{4}))} ∫_{B(x_{i}, \frac{r}{4} )} ϕ dλ_{i,r} − ∑_{i∈I^{∗}} ∫_{A(x_{i},r)} ϕ dμ|

= |∑_{i∈I^{∗}} (\frac{μ(A(x_{i}, r))}{λ_{i,r}(B(x_{i}, \frac{r}{4}))} ∫_{B(x_{i}, \frac{r}{4} )} ϕ dλ_{i,r} − ∫_{A(x_{i},r)} ϕ dμ)|

= |∑_{i∈I^{∗}} ∫_{A(x_{i},r)}(ϕ − ϕ_{r}) dμ| + |∫_{Ω\∪_{i∈I^{∗}} B(x_{i},r)} ϕ dμ| .

Then

\begin{matrix}|⟨λ_{r}, ϕ⟩ − ⟨μ, ϕ⟩| \\ ≤ ∑_{i∈I^{∗}} ∫_{A(x_{i},r)} |ϕ − ϕ_{r}| dμ \\ ≤ μ(Ω)^{\frac{1}{2}} (∑_{i∈I^{∗}} ∫_{A(x_{i},r)}(ϕ − ϕ_{r})^{2}dμ)^{\frac{1}{2}} \\ = μ(Ω)^{\frac{1}{2}} (∑_{i∈I^{∗}} ∫_{A(x_{i},r)}(ϕ − M_{i,r}(ϕ))^{2}dμ)^{\frac{1}{2}} \end{matrix}  (5.16)

and using the Poincaré inequality stated in Theorem 4.1 and Remark 4.2, we obtain that

\begin{aligned}∫_{A(x_{i},r)}(ϕ − M_{i,r}(ϕ))^{2}dμ & ≤ ∫_{B(x_{i},r)}(ϕ − M_{i,r}(ϕ))^{2}dμ \\ & = ∫_{B(x_{i},r)}(ϕ − P (ϕ))^{2}dμ \\ & ≤ η(r) ∫_{B(x_{i},2kr)} α(ϕ, ϕ)(dx). \end{aligned}  (5.17)

Using (5.15) - (5.17) we deduce that, denoting by N (depending on k and C_{o}) the maximal number of balls B(x_{i}, 2kr) such that x ∈ B(x_{i}, 2kr),

\begin{aligned}|⟨λ_{r}, ϕ⟩ − ⟨μ, ϕ⟩| & ≤ η(r) (∑_{i∈I^{∗}} ∫_{B(x_{i},2kr)} α(ϕ, ϕ)(dx)) \\ & ≤ Nη(r) (∫_{Ω} α(ϕ, ϕ)(dx)) \end{aligned}  (5.18)

We conclude by density.

Now the other conditions stated in Proposition 5.1 are easily verified and we have proved our statement.

Proposition 5.3. Let μ be a measure in K_{+}(Ω) and Ω_{h} an increasing sequence of open sets with closure contained in Ω ans such that ∪_{h}Ω_{h} = Ω. Denote by mu_{h} the measure defined by μ_{h}(E) = μ(E ∩ Ω_{h}) for every μ-measurable set E; then μ_{h} γ converges to μ as h → +∞.

Proof. Consider the problems

a(u_{h}, v) + ∫_{Ω} u_{h}v dμ_{h} = ⟨f, v⟩  (5.19_{h})

for any v ∈ D_{o}(a, Ω), u_{h} ∈ D_{o}(a, Ω).

and

a(u, v) + ∫_{Ω} uv dμ = ⟨f, v⟩  (5.19)
for any v ∈ D_{o}(a, Ω), u ∈ D_{o}(a, Ω).

where f ∈ L^{∞}(Ω, m). We may assume without loss of generality that u_{h} converges weakly to u_{o} in D_{o}(a, Ω), then u_{h} converges strongly to u_{o} in L_{loc}^{2}(Ω, m), [8]. Moreover the functions u_{h} are locally equicontinuous in Ω,[5], so we may assume that u_{h} converges to u_{o} uniformly locally in Ω. We observe that

lim_{h→+∞} ∫_{Ω} u_{h}v dμ_{h} = lim_{h→+∞} ∫_{Ω} u_{h}vχ_{h} dμ = ∫_{Ω} u_{o}v dμ  (5.20)

for every v ∈ D_{o}(a, Ω) ∩ C_{c}(Ω), then for every v ∈ D_{o}(a, Ω), since μ is a Kato measure in Ω, [5]. From (5.20) we obtain easily that u_{o} is the solution of (5.19). Since L^{∞}(Ω, m) is dense in D′_{o}(a, Ω), the above result holds again for f ∈ D′_{o}(a, Ω), so μ_{h} γ-converges to μ.

We remark that the γ-convergence is metrizable, [16, 17], then by a diagonal argument and by the results in Proposition 5.2 and 5.3 we prove:

Theorem 5.1. Let us suppose that N = {x_{0} : a − cap({x_{0}}) > 0} ∩ Ω = ∅, μ ∈ K_{+}(Ω) and denote Ω_{r} = {x ∈ Ω; dist(x, Ω^{c}) > 3r}.

(i) There exists a positive constant depending only on R_{o} and C(R_{o}) such that if 0 < r ≤ min(C(R_{o}), \frac{1}{40}d(supp μ, Ω^{c})), we can construct a finite covering B(x_{i}, r) (i ∈ I) of Ω such that the balls B(x_{i}, \frac{r}{2}) are pairwise disjoint.

(ii) Let I^{∗} ⊂ I such that B(x_{i}, \frac{r}{2}) ⊂ Ω for all i ∈ I^{∗}. Let us define

\begin{aligned}A(x_{1}, r) & = B(x_{1}, r) \\ A(x_{i}, r) & = B(x_{i}, r)\ ∪_{j<i} B(x_{j}, r) (i ≥ 1). \end{aligned}

Let E(x_{i}, r) be a ball centered in x_{i}, contained in B(x_{i}, \frac{r}{4}) and with radius such that

cap^{a}(E(x_{i}, r), B(x_{i}, \frac{r}{4})) = μ(A(x_{i}, r) ∩ Ω_{r}).

Then if E_{r} = ∪_{i∈I^{∗}}E(x_{i}, r) and r_{h} → 0 the sequence ∞_{E_{r_{h}}} γ-converges to μ.

We shall extend our previous result in the case of Radon measures in M_{o}^{a}.

Theorem 5.2. Let us suppose that

N = {x_{0} : a − cap(x_{0}) > 0} ∩ Ω = ∅  (5.21)

and μ ∈ M_{o}^{a} is a Radon measure. Then if E_{r} is defined as in Theorem 5.1, the sequence ∞_{E_{r}} γ-converges to μ.

Proof. Using a general result due to Mosco (see [29]) we know that M_{o}^{a} is compact. So it is possible to extract a subsequence of ∞_{E_{r}} (that we shall call again ∞_{E_{r}}) which converges to π ∈ M_{o}^{a}. We want to prove that π = μ. Let us show first that π ≤ μ. For this, let us prove that cap_{π}^{a}(A, Ω) ≤ μ(A) for any open A ⊂⊂ Ω. Let A′ be an open set, A′ ⊂⊂ A and r be a real number small enough so that ∪_{E(x_{i},r)∩A′≠∅}B(x_{i}, r) ⊂ A. We have

cap_{π}^{a}(A′, Ω) = lim inf_{r→0} cap_{∞_{Er}}^{a} (A′, Ω)

= lim inf_{r→0} cap^{a}(A′ ∩ E_{r}, Ω)

≤ lim inf_{r→0} ∑_{E(x_{i},r)∩A′≠∅} cap^{a}(A′ ∩ E(x_{i}, r), Ω)

≤ lim inf_{r→0} ∑_{E(x_{i},r)∩A′≠∅} cap^{a}(E(x_{i}, r), B(x_{i}, \frac{r}{4}))

≤ lim inf_{r→0} ∑_{E(x_{i},r)∩A′≠∅} μ(A(x_{i}, r) ∪ Ω_{r})

≤ μ(A).

In this inequality we have used that A(x_{i}, r) ∩ A(x_{j}, r) = ∅ if i ≠ j. If A′ → A increasingly, we have

cap_{π}^{a}(A, Ω) ≤ μ(A).  (5.22)

If μ is a Radon measure, we can easily extend (5.20) to any Borel set B ⊂ Ω:

cap_{π}^{a}(B, Ω) ≤ μ(B).

Using Theorem 3.2, we have

π(B) = sup ∑_{i∈I} cap_{π}^{a}(B_{i}, Ω) ≤ sup ∑_{i∈I} μ(B_{i}) ≤ μ(B)  (5.23)

where the supremum is taken over all finite partitions B_{i} of B.

Now we want to prove that

π(B) ≥ μ(B).

Thanks to Theorem 3.1 there exist a Borel function g : Ω → [0, +∞] and a measure λ ∈ K_{+}(Ω) such that μ ≃ gλ. Let us define g_{K}(x) = min{g(x), K} and μ_{K} = g_{K}λ for K ∈ ℕ (then μ_{K} ∈ K_{+}(Ω)), and for any K let us construct the set E_{r,K} as in Theorem 5.2, such that ∞_{E_{r,K}} γ-converges to μ_{K}. But we have E_{r,K} ⊂ E_{r} for any r and K and this implies ∞_{E_{r,K}} ≤ ∞_{E_{r}}. Using the monotonicity properties of the γ-convergence we have μ_{K} ≤ π and, also letting K → ∞,

μ ≤ π.  (5.24)

We conclude using (5.23) and (5.24).

Let us now give a more abstract result on the density of measures ∞_{E} in the general case of μ ∈ M_{o}^{a}.

Theorem 5.3. Let us suppose that N = {x_{0} : a − cap(x_{0}) > 0} ∩ Ω = ∅ and μ ∈ M_{o}^{a}. Then there exists a sequence ∞_{A_{r}} γ-converging to μ.

Proof. We give only a sketch of the proof. We remark that the space M_{o}^{a} with the γ-convergence is metrizable (see [16, 17] for a similar proof in a particular case). Let μ ∈ M_{o}^{a}. We use the [28: Proposition 4.13] (where no result of type of Theorem 3.1 is used) to find a sequence of positive Radon measures μ_{h} ∈ M_{o}^{a} which γ-converges to μ. Then we use Theorem 5.2 to approximate μ_{h} and we conclude by a diagonal argument

Remark 5.1. Let us remark that also in the general case when μ ∈ M_{o}^{a}, the construction of the sequence ∞_{A_{r}} is explicit. Indeed, the construction of the measures μ_{h} ∈ M_{o}^{a} in [29: Proposition 4.13] is completely explicit and our preceding Theorem 5.2 and the standard diagonal argument are also constructive.

Remark 5.2. No representation theorem (like Theorem 3.1) for the measure μ ∈ M_{o}^{a} has been used in the proof of Theorem 5.3 for the approximation of μ by Radon measures.

References

[1] Biroli, M.: An embedding property of Schechter’s type for Dirichlet forms on an homogeneous space. Rend. Ist. Lombardo Acc. Sc. e Lettere A 128 (1994), 247 – 255.

[2] Biroli, M. and U. Mosco: Formes de Dirichlet et estimations strcturelles dans les miluex discontinus. C.R.A.S. Paris 313 (1991), 593 – 598.

[3] Biroli, M. and U. Mosco: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. 169 (1995), 125 – 181.

[4] Biroli, M. and U. Mosco: Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces. Atti Acc. Naz. Lincei, Rend. Cl. Sc. Mat. Fis. Naturali 6 (1995), 637 – 644.

[5] Biroli, M. and U. Mosco: Kato space for Dirichlet forms. Potential Analysis 10 (1999), 327 – 345.

[6] Biroli, M., Mosco, U. and N. Tchou: Homogenization by the Heisenberg group. Adv. Math. Sci. Appl. 7 (1997), 809 – 831.

[7] Biroli, M. and N. Tchou: Relaxed Dirichlet problems involving a Dirichlet form (in preparation).

[8] Biroli, M. and N. Tchou: Asymptotic behaviour of relaxed Dirichlet problems involving a Dirichlet-Poincaré form. Z. Anal. Anw. 16 (1997), 281 – 309.

[9] Cioranescu, D. and F. Murat: Un terme étrange venu d’ ailleurs. In: Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. II and III (Research Notes in Mathematics: Vol. 60 and 70; eds.: H. Brézis and J. L. Lions). London: Pitman 1982, pp. 98 – 138 (Vol. I) and 154 – 178 (Vol. II).

[10] Coifman, R. R. and G. Weiss: Analyse harmonique sur certaines espaces homogènes. Lect. Notes Math. 242 (1971).

[11] Dal Maso, G.: γ-convergence and μ-capacities. Ann. Sc. Norm. Sup. Pisa C. Sc. 14 (1987), 59 – 100.

[12] Dal Maso, G. and V. De Cicco: Evans-Vasilesco theorem on Dirichlet spaces. Rend. di Mat. (to appear).

Relaxation for Dirichlet Problems 225

[13] Dal Maso, G., De Cicco, V., Notarantonio, L. and N. Tchou: Limits of variational problems for Dirichlet forms in varying domains. J. Math. Pures Appl. 1 (1998), 89 – 116.

[14] Dal Maso, G. and A. Garroni: New results on the asymptotic behavior of Dirichlet problems in perforated domains. Math. Models Methods in Appl. Sc. 4 (1994), 373 – 407.

[15] Dal Maso, G. and A. Malusa: Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains. Proc. Royal Soc. Edinburgh (Sect. A) 125 (1995), 99 – 114.

[16] Dal Maso, G. and U. Mosco: Wiener criteria and energy decay for relaxed Dirichlet problems. Arch. Rat. Mech. An. 95 (1986), 345 – 387.

[17] Dal Maso, G. and U. Mosco: Wiener’s criterion and γ-convergence. Appl. Math. Opt. 15 (1987), 15 – 63.

[18] De Giorgi, E. and T. Franzoni: Su un tipo di convergenza variazionale. Parte I: Atti Acc. Naz. Lincei, Rend. Cl. Sc. Mat. Fis. Naturali 58 (1975), 842 – 850; Part II: Rend. Sem. Mat. Brescia 3 (1979) 63 – 101.

[19] De La Pradelle, A. and D. Feyel: Construction d’ un espace harmonique de Brelot associé à un espace de Dirichlet de type local vérifiant une hypothèse de hypoellipticité. Inv. Math. 44 (1978), 109 – 128.

[20] Fabes, E. B., Kenig, C. E. and R. Serapioni: The local regularity of solutions of degenerate elliptic equations. Comm. Part. Diff. Eq. 7 (1982), 77 – 116.

[21] Franchi, B., Lu, G. and R. L. Wheeden: Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations. Potential Analysis 4 (1995), 361 – 376.

[22] Fukushima, M.: Dirichlet Forms and Markov Processes. Amsterdam: North Holland 1980.

[23] Jerison, D.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53 (1986), 503 – 523.

[24] Kacimi, H. and F. Murat: Estimations de l’ erreur dans des problèmes de Dirichlet où apparait un terme étrange. In: Partial Differential Equations and the Calculus of Variations, Vol. II (eds.: R. Serapioni and F. Tomarelli). Boston: Birkhäuser 1989, pp. 661 – 696.

[25] Kinderlehrer, D. and G. Stampacchia: An Introduction to Variational Inequalities and Their Applications (Pure and Applied Mathematics: Vol. 88). New York: Academic Press 1980.

[26] Le Jan, Y.: Mesures associeés à une forme de Dirichlet et applications. Bull. Soc. Math. France 106 (1978), 61 – 112.

[27] Lu, G.: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander condition and applications. Riv. Mat. Iberoamer. 8 (1992), 367 – 440.

[28] Mataloni, S. and N. Tchou: Homogenization for nonsymmetric Dirichlet forms. Tecnical Report, Università di Roma 2 (1998).

[29] Mosco, U.: Compact families of Dirichlet forms (to appear).

[30] Stampacchia, G.: Équations elliptiques à coefficients discontinus. Montréal: Les presses de l’ Universitè de Montrèal 1987.

[31] Zamboni, P.: Some function spaces and elliptic differential equations. Le Matematiche 42 (1987), 171 – 178.

226 M. Biroli and N. Tchou Received 08.04.1999