On the Laplacean transfer across fractal mixtures

R. CapitanelliDipartimento di Scienze di Base e Applicate per l’Ingegneria,

“Sapienza” Universita di Roma,Via A. Scarpa 16, 00161 Roma, Italyemail raffaela.capitanelli@uniroma1.it

M.A. VivaldiDipartimento di Scienze di Base e Applicate per l’Ingegneria,

“Sapienza” Universita di Roma,Via A. Scarpa 16, 00161 Roma, Italyemail maria.vivaldi@sbai.uniroma1.it

Abstract

Laplacean transport across and towards irregular interfaces havebeen used to model many phenomena in nature and technology. Thepeculiar aspect is that these phenomena take place in domains withsmall bulk and large interfaces in order to produce rapid and efficienttransport. In this paper we perform the asymptotic homogenizationanalysis of Robin problems in domains with a fractal boundary.

Keywords Fractals, Variational methods for second-order elliptic equa-tions, Boundary value problems for second-order elliptic equations, Homog-enization.

2010 Mathematics Subject Classification. 28A80, 35J20, 35J25,35B27.

GRANT: P.U. “Sapienza” Universita di Roma C26A109989.

1 Introduction

Robin problems for the Laplace equation in irregular domains have beenused to model many phenomena in physics, physiology, electrochemistryand chemical engineering. The peculiar aspect is that these phenomenatake place in domains with small bulk and large interfaces in order to pro-duce rapid and efficient transport (see, for a detailed discussion and othermotivations, [18], [19], and [37]). From this point of view, domains with afractal boundary provide a natural setting for studying these phenomena.

Robin problems in fractal domains have been studied by R.F. Bass, K.Burdzy, Z. Chen [5] using a probabilistic approach. We are interested in ananalytical approach.

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Here we propose to face the fractal problem via the homogenizationtheory. We think that this approach provides a sound physical basis to themodel.

The homogenization approach for reinforcement problems, in the classi-cal setting of regular domains, was widely used from the 1970s in conjunc-tion with numerous applications. The works include those by P. H. Hung, E.Sanchez-Palencia [20], J. R. Cannon, G. H. Meyer [8], H. Brezis, L.Caffarelli,A.Friedman [6], E. Acerbi, G. Buttazzo [1], and G. Buttazzo, G. Dal Maso,U. Mosco [7], as well as the book by H. Attouch [3], whose bibliographycontains an exhaustive list.

The use of the homogenization approach for reinforcement problems indomains with a fractal (or prefractal) boundary is recent, the first papersbeing, to our knowledge, [33], [25], [34], and [35]. These works actuallyconcern singular homogenization results. More precisely, the fractal layeris approximated by a two dimensional highly conductive thin layer withvanishing thickness and increasing conductivity. The scaling assumptionscan be roughly expressed by the condition that the thickness of the fibertends to zero and the conductivity diverges to +∞, while the product tendsto a positive number, as n → +∞. From the PDE’s point of view, in thelimit problem, highly conductive layers bring out a second order transmissioncondition on the fractal layer.

In this paper instead we deal with insulating layers: both thickness andconductivity of the fiber tend to zero. A technical aspect, which is peculiarto insulating layers, is that there is a loss of coerciveness as n → +∞,and uniform H1−estimate fails. We overcome this difficulty by establishingPoincare type inequalities (see Theorem 6.1 and Theorem 7.3) and by usingsome delicate tools such as the extension theorem for (ε,∞) domains dueto Jones (Theorem 1 in [21]). As in the classical case, the limit problemdepends on the asymptotic behavior of the ratio between the conductivityand the thickness. If the ratio tends to a positive number, as n → +∞,then a first order condition on the fractal boundary - the Robin condition- appears in the limit problem. Here we consider also the cases in whichthe previous condition is not satisfied. More precisely, if the ratio betweenthe conductivity and the thickness tends to zero, then a different first ordercondition on the fractal boundary -the Neumann condition- appears in thelimit problem. If the ratio diverges, then the Dirichlet condition appears inthe limit problem.

Now we describe the layout and the main results of this paper.In the second section, we fix notation and preliminary results. We con-

sider a simple-connected domain Ω(ξ) whose boundary is a fractal curve and

reinforced (approximating) domains Ω(ξ),nε which are constructed by adding

ε−thin, polygonal, 2−dimensional fibers Σ(ξ),nε to the prefractal approxi-

mating domains Ω(ξ),n. In these domains, we consider suitable energy forms

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an(·, ·) (defined in (2.6)) that are the sum of a “weighted” Dirichlet integral

on Ω(ξ),nε and a volume term of zero order with a positive parameter δn. The

coefficients anε are equal to one on Ω(ξ),n, and equal to the product of the

weight wnε and two positive constants cn and σn on the fiber Σ(ξ),nε . The

weight wnε is equal to one on the prefractal domains Ω(ξ),n and behaves like

the thickness of the fiber on each fiber Σ(ξ),nε , while the constants cn describe

the conductivity of the material and σn are normalizing factors that dependon the intrinsic parameters of the limit mixture. We state the existence andthe uniqueness of the functions un that realize the minimum of the completeenergy functionals that are the sum of the energy forms an(·, ·), a volumeterm and a layer contribution on the boundary ∂Ω(ξ),n (see Theorem 2.1).Moreover we study the limit problems and we state the existence and theuniqueness of the relative variational solutions (see Theorems 2.2 and 2.3).

In Section 3, we state the main results. We consider the Dirichlet prob-lems for elliptic operators introduced in the previous section and we showthat the asymptotic behavior of the solutions un depends on the conductiv-ity parameters cn. More precisely, we prove that the sequence un convergesto the variational solution of Robin problem, Neumann problem, or Dirich-let problem on Ω(ξ) when the sequence cn tends to a positive number, tozero, or it diverges (see Theorems 3.1 and 3.3). Moreover, we state the con-vergence of the spectral structures associated with the above operators (seeTheorems 3.2 and 3.4).

In Section 4 we recall the definition and main properties of M-convergence,and we prove M-convergence of the functionals corresponding to the rein-forcement problems (see Theorems 4.1 and 4.2).

Section 5 concerns the convergence of spectral structures. As in ourframework the operators act in different Hilbert spaces, we introduce thedefinition and main properties of M-K-S-convergence (that is a generaliza-tion of M-convergence to variable Hilbert spaces) and we prove Theorems3.2 and 3.4.

In Section 6 we give the proof of Theorems 3.1 and 3.3 where sharpestimates in the trace theorems on polygonal boundaries, extension resultsand a suitable Poincare inequality (Theorem 6.1) play an important role.In particular, given that the constant C∗P appearing in Poincare inequalitydoes not depend on the increasing number of sides we are able to provethat suitable extensions u∗n of the minimizers un are equi-bounded in theH1−norm with respect to n and ε.

Section 7 concerns Laplacean transfer across fractal mixtures. The math-ematical model of the so-called Laplacean transport and of some other phe-nomena involves mixed Dirichelet-Robin conditions on the boundary. Infact, species characterized by their concentration diffuses in the bulk from asource (Dirichlet condition) towards a semi-permeable interface (Robin con-dition) on which it disappears at a given rate (either by transferring across

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the interface, or by reacting in the case of a catalytic cell) (see [19]). Ir-regular semi-permeable interfaces give rise to rapid and efficient flows. Inthe present setting the geometry of the bulk is represented by the domain

Ω(ξ) = Ω(ξ) \ B(P0,18), where B(P0,

18) is the ball with center P0 = (1

2 ,−12)

and radius 18 . In a similar way, we construct the prefractal domains Ω(ξ),n =

Ω(ξ),n \ B(P0,18), and the reinforced domains Ω

(ξ),nε = Ω

(ξ),nε \ B(P0,

18).

Existence and uniqueness results for the variational solution of the mixedDirichelet-Robin problem modeling Laplacean transport (on Ω(ξ)) can beproved as in [9] and [10] by making some natural changes due to the differentgeometry of the domain. In the above mentioned papers, prefractal mixedDirichlet-Robin problems (where the fractal curve is replaced by polygonalcurves approximating the fractal) have been also considered and existence,uniqueness and regularity results were established. The convergence of thesolutions of approximating problems to the solution of the fractal problemhas been proved in [11] and [12], where a suitable choice of the coefficientsin the Robin condition on the polygonal curves is a crucial condition.

In the framework of the homogenization approach, the results proved inthe previous sections can be adapted to the geometry of Ω(ξ). Actually thepresence of the Dirichlet condition (in a part of the boundary) allows us toimprove the previous results (see Theorems 7.1, 7.2 and 7.4). In particular,we establish a Poincare inequality where the constant CP does not dependon the increasing number of sides (see Theorems 7.3) and we prove that suit-able extensions u∗n of the minimizers un are equi-bounded in the H1−normwith respect to n and ε. In a previous paper ([13]) we have already addressedmixed Dirichelet-Robin problems with the homogenization approach. How-ever Theorem 4.2 in [13] concerns only the case in which the sequence cnis constant and positive. In particular the case in which the sequence cndiverges as well as the results concerning the spectral structures are new.Moreover the variational problems considered in Theorems 7.2 and 7.4 differfrom those in [13] because of the presence of the layer terms in the data.We remark that these terms appear in the models of Laplacean transportbeing due to the source terms (see [9]). From the mathematical point ofview these terms can be managed by means of trace results (see Theorem8.1).

The last section includes definitions and properties of the scale irregularKoch curves and some trace and extension results stated in the specificgeometry of the domains.

2 Notation and preliminary results

In this section we introduce the notation and some preliminary results.Let Ω0 be the square (x, y) : 0 < x < 1,−1 < y < 0 with vertices

A = (0, 0), B = (1, 0), C = (1,−1), and D = (0,−1). On each of the 4 sides

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we construct a scale irregular Koch curve (for definition and main properties,see Section 8). More precisely, we consider the set Ω(ξ) bounded by 4 scale

irregular Koch curves K(ξ)j j = 1, 2, 3, 4 with endpoints, respectively, A and

B, B and C, C and D, D and AWe consider the sets Ω(ξ),n bounded by 4 approximating prefractal curves

K(ξ),nj j=1,2,3,4 starting from the segments Kj with endpoints A and B, B

and C, C and D, D and A respectively, and the reinforced domain Ω(ξ),nε

constructed by an iteration procedure and starting from a suitable initialfiber Σε. More precisely, let Ω0 be the square introduced before and let K1

be the interval with end-points A and B. For every 0 < ε ≤ ε0 <12 , we

define the fiber Σ1,ε ε-neighborhood of K1, to be the (open) polygon whosevertices are the points A,B, P2, P1, where

P1 = (ε, ε), P2 = (1− ε, ε).

We then subdivide Σ1,ε as the union of the rectangle R1,ε and the two trian-gles T1,h,ε, h = 1, 2. Here, R1,ε is the rectangle with vertices P1, P2, P3, P4;T1,1,ε is the triangle with vertices A,P1, P4 and T1,2,ε is the triangle withvertices B,P3, P2 where

P4 = (ε, 0), P3 = (1− ε, 0).

For every n and ε as above, we define the fiber Σ(ξ),n1,ε , ε-neighborhood of

K(ξ),n1 , to be the (open) set

Σ(ξ),n1,ε =

⋃i|n

Σ(ξ),i|n1,ε ,

whereΣ

(ξ),i|n1,ε = ψi|n(Σ1,ε)

(see Figures 2.1, 2.2, 2.3, 2.4).

We proceed in a similar way in order to construct the fiber Σ(ξ),nj,ε ,

ε-neighborhood of K(ξ),nj (j = 2, 3, 4), and we define the fiber Σ

(ξ),nε , ε-

neighborhood of ∂Ω(ξ),n,

Σ(ξ),nε =

4⋃j=1

Σ(ξ),nj,ε

andΩ(ξ),nε = Ω(ξ),n

⋃Σ(ξ),nε

(see Figure 2.5).We define a weight wnε as follows.

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Figure 2.1: The “ε-neighborhood” Σ1,ε.

Figure 2.2: The “ε-neighborhood” Σ(ξ),1j,ε .

Figure 2.3: The “ε-neighborhood” Σ(ξ),2j,ε .

Figure 2.4: The “ε-neighborhood” Σ(ξ),3j,ε .

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Let P – for some i|n – belong to the boundary ∂(Σ(ξ),i|n1,ε ) of Σ

(ξ),i|n1,ε and

let P⊥ be the orthogonal projection of P on K(ξ),i|n1 . If (x, y) belongs to the

segment with end-points P and P⊥, we set, in our current notation,

wn1,ε(x, y) = |P − P⊥|,

where |P − P⊥| is the (Euclidean) distance between P and P⊥ in R2.We proceed in a similar way in order to construct the weights wnj,ε on

Σ(ξ),nj,ε (j = 2, 3, 4) and we define wnε on Ω

(ξ),nε

wnε (x, y) =

wnj,ε(x, y) if (x, y) ∈ Σ

(ξ),nj,ε

1 if (x, y) ∈ Ω(ξ),n

.(2.1)

Associated with the weight wnε , we consider the Sobolev spacesH1(Ω(ξ),nε ;wnε )

and H10 (Ω

(ξ),nε ;wnε ), defined as the completion of C∞(Ω

(ξ),nε ) and C∞0 (Ω

(ξ),nε ),

respectively, in the norm

‖u‖H1(Ω

(ξ),nε ;wnε )

= ∫

Ω(ξ),nε

u2dxdy +

∫Ω

(ξ),nε

|∇u|2wnε dxdy12 . (2.2)

We stress the fact that the gradient is uniquely defined as the weight wnεand its reciprocal 1

wnεbelong to L1(Ω

(ξ),nε ) (see, e.g., [16]).

We define the coefficients

anε (x, y) =

cnσnw

nε (x, y) if (x, y) ∈ Σ

(ξ),nε

1 if (x, y) ∈ Ω(ξ),n

,(2.3)

wherecn > 0 (2.4)

and

σn =`(ξ)(n)

4n, `(ξ)(n) =

n∏i=1

`ξi . (2.5)

Here `ξi denotes the reciprocal of the contraction factor of the family Ψ(ξi)

(see Section 8).

From now on, when it does not give rise to misunderstanding, we denoteby C positive, possibly different constants that do not depend on n and onε.

The following theorem states the existence and the uniqueness of thevariational solution of the reinforcement problem. We consider the bilinearform associated with the reinforcement problem

an(u, v) :=

∫Ω

(ξ),nε

anε∇u∇v dxdy + δn

∫Ω

(ξ),nε

u v dxdy (2.6)

where anε is defined in (2.3), (2.5), (2.4), and δn > 0.

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Figure 2.5: The domain Ω(ξ),nε .

Theorem 2.1 Let σn be as in (2.5) and dn ∈ R. Then, for any fn ∈L2(Ω

(ξ),nε ), there exists one and only one solution un of the following problem

find un ∈ H10 (Ω

(ξ),nε ;wnε ) such that

an(un, v) =∫

Ω(ξ),nε

fn v dxdy + σndn∫∂Ω(ξ),n v ds ∀ v ∈ H1

0 (Ω(ξ),nε ;wnε ),

(2.7)where an(·, ·) is defined in (2.6). Moreover, un is the only function thatrealizes the minimum of the energy functional

minv∈H1

0 (Ω(ξ),nε ;wnε )

an(v, v)− 2

∫Ω

(ξ),nε

fn v dxdy − 2σndn

∫∂Ω(ξ),n

v ds. (2.8)

Proof. We use Lax-Milgram Theorem. By Theorems 8.1 and 8.3, wehave

|σndn∫∂Ω(ξ),n

v ds| ≤ σn|dn|||v||L2(∂Ω(ξ),n)

√|∂Ω(ξ),n| ≤ 2|dn|

√C1CJ ||v||H1(Ω(ξ),n)

(where here and in the following |∂Ω(ξ),n| denotes the arc length measure ofthe polygonal curve ∂Ω(ξ),n) and

|∫

Ω(ξ),nε

fn v dxdy| ≤ ||fn||L2(Ω(ξ),nε )||v||

L2(Ω(ξ),nε )

.

Moreover the bilinear form an(u, v) satisfies

an(v, v) ≥ min(δn, cnσn, 1)||v||2H1

0 (Ω(ξ),nε ;wnε )

(2.9)

and

|an(u, v)| ≤ max(δn, cnσn, 1)||u||H1

0 (Ω(ξ),nε ;wnε )

||v||H1

0 (Ω(ξ),nε ;wnε )

.

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In the following theorems, we state the existence and uniqueness of thevariational solution of the Robin, Neumann, and Dirichelet problems on thedomain Ω(ξ).

We consider the bilinear form associated with the Robin problem

ac0(u, v) :=

∫Ω(ξ)

∇u∇v dxdy + δ0

∫Ω(ξ)

u v dxdy + c0

∫∂Ω(ξ)

γ0u γ0v dµ(ξ)

(2.10)

where µ(ξ) is the measure on ∂Ω(ξ) that coincides, on each K(ξ)j j = 1, 2, 3, 4,

with the Radon measure defined in (8.4) and γ0u denotes the trace of thefunction u on the boundary of Ω(ξ) (see (8.5)). From now on, we suppress γ0

in the notation, when it does not give rise to misunderstanding, by writingsimply v instead of γ0v and similar expressions.

We assume that

c0 ≥ 0, δ0 ≥ 0, and max(c0, δ0) > 0. (2.11)

Theorem 2.2 Let us assume (2.11) and d ∈ R. Then, for any f ∈ L2(Ω(ξ)),there exists one and only one solution u of the following problem

find u ∈ H1(Ω(ξ)) such that

ac0(u, v) =∫

Ω(ξ) f v dxdy + d∫∂Ω(ξ) v dµ(ξ) ∀ v ∈ H1(Ω(ξ))

(2.12)

where ac0(·, ·) is defined in (2.10). Moreover, u is the only function thatrealizes the minimum of the energy functional

minv∈H1(Ω(ξ))

ac0(v, v)− 2

∫Ω(ξ)

f v dxdy − 2d

∫∂Ω(ξ)

v dµ(ξ). (2.13)

Proof. By Theorems 8.2 and 8.4, we have that

|d∫∂Ω(ξ)

v ds| ≤ 2|d|||v||L2(∂Ω(ξ)) ≤ 2√C∗1CR|d|||v||H1(Ω(ξ))

and

|∫

Ω(ξ)

f v dxdy| ≤ ||f ||L2(Ω(ξ))||v||L2(Ω(ξ)).

The bilinear form ac0(u, v) is continuous. Indeed, by using Theorem 8.2again,

|ac0(u, v)| ≤ (max(δ0, 1) + c0C∗1CR)||u||H1(Ω(ξ))||v||H1(Ω(ξ)).

Moreover, the form is coercive. In fact, if δ0 > 0 we obtain trivially

ac0(v, v) ≥ min(δ0, 1)||v||2H1(Ω(ξ))

.

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Instead, if δ0 = 0 and c0 > 0 by using generalized Poincare inequality (seeLemma 3.1.1 in [28]), we obtain that

ac0(v, v) ≥ C min(c0, 1)||v||2H1(Ω(ξ))

.

In a similar way, we prove the following result.We consider the bilinear form associated with the Dirichlet problem

a∞(u, v) :=

∫Ω(ξ)

∇u∇v dxdy + δ0

∫Ω(ξ)

u v dxdy. (2.14)

We assume thatδ0 ≥ 0. (2.15)

Theorem 2.3 Let us assume (2.15). Then, for any f ∈ L2(Ω(ξ)), thereexists one and only one solution u of the following problem

find u ∈ H10 (Ω(ξ)) such that

a∞(u, v) =∫

Ω(ξ) f v dxdy ∀ v ∈ H10 (Ω(ξ))

(2.16)

where a∞(·, ·) is defined in (2.14). Moreover, u is the only function thatrealizes the minimum of the energy functional

minv∈H1

0 (Ω(ξ))

a∞(v, v)− 2

∫Ω(ξ)

f v dxdy. (2.17)

3 Main results

In this section we state the main results.

Let Ω∗ be an open regular domain such that Ω∗ ⊃ Ω(ξ),nε , for ∀n : in

order to fix notation we choice as Ω∗ the ball with the center in the pointP0 = (1

2 ,−12) and radius 1.

We use an extension operator ExtJ from H1(Ω(ξ),n) to the space H1(R2)whose norm is independent of the (increasing) number of sides (see Theorem8.3 in Section 8).

We setu∗n = (ExtJ un|Ω(ξ),n)|Ω∗ . (3.1)

where un is the solution of the problem (2.7).In order to study the asymptotic behaviour of the functions un, we fix

the further assumptions

fn, f ∈ L2(Ω∗), and fn → f ∈ L2(Ω∗), asn→ +∞, (3.2)

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δn > 0 and δn → δ0 asn→ +∞, (3.3)

cn > 0 and cn → c0 asn→ +∞, (3.4)

dn, d ∈ R, and dn → d asn→ +∞. (3.5)

Theorem 3.1 Let us assume conditions (3.4), (3.5), (3.3), (2.11), and(3.2). Let ε = ε(n) be an arbitrary sequence such that ε(n)→ 0 as n→ +∞.Then the sequence of the restrictions to Ω(ξ) of the functions u∗n (defined in(3.1)) converges to the function u (defined in (2.12)) weakly in H1(Ω(ξ)).

For a fixed n, the second order operator associated with the form (2.6)An(u) = −div(anε(n)(x, y)∇u) + δnu (with homogeneous Dirichlet condi-

tion) has a compact resolvent in L2(Ω(ξ),nε ) and the spectrum consists of

a increasing sequence of real eigenvalues λnε,k such that λnε,k → +∞ whenk → +∞. We denote the associated spectral resolution of An with Pn(λ). Ifc0 > 0 the second order operator associated with the form (2.10) Ac0(u) =−div(∇u) + δ0u (with Robin condition) has a compact resolvent in L2(Ω(ξ))and the spectrum consists of a increasing sequence of real eigenvalues λksuch that λk → +∞ when k → +∞ (see e.g. [15]). If c0 = 0 the second or-der operator associated with the form (2.10) Ac0(u) = −div(∇u)+δ0u (withNeumann condition) has a compact resolvent in L2(Ω(ξ)) and the spectrumconsists of a increasing sequence of real eigenvalues λk such that λk → +∞when k → +∞ (see e.g. [27]). In both cases, we denote by Pc0(λ) thespectral resolution associated with Ac0 .

We recall that for every interval (λ, µ] of R, Pn((µ−λ]) = Pn(µ)−Pn(λ)

is a projector operator in the Hilbert space L2(Ω(ξ),nε ) and Pc0((µ − λ]) =

Pc0(µ)− Pc0(λ) is a projector operator in the Hilbert space L2(Ω(ξ)).

Theorem 3.2 Let ε = ε(n) be an arbitrary sequence such that ε(n)→ 0 asn → +∞. Let us assume conditions (3.4) and (3.3). Then ∀ 0 < µ < λ,points of continuity for the spectral resolution Pc0 of the operator Ac0, wehave that the projector operators Pn((µ− λ]) of the spectral resolutions Pn

of the operators An in L2(Ω(ξ),nε ) converge to the projector Pc0((µ − λ]) of

the spectral resolution Pc0 strongly as n → +∞, in the sense of Definition5.5 of the Section 5.

Theorems 3.1 and 3.2 concern the case in which the conductivity of thethin fibers vanishes at the same rate of the thickness of the fiber and the

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case in which the conductivity of the thin fibers vanishes faster than thethickness of the fiber.

Now we consider the case when the conductivity of the thin fibers van-ishes slower than the thickness of the fiber: more precisely, we suppose

cnwn → 0, cn → +∞, ε(n) ≥ 1

`(n). (3.6)

Theorem 3.3 Let ε = ε(n) be a sequence such that ε(n) → 0 as n →+∞. Let us assume (3.5), (3.3), (3.6), and (3.2). Then the sequence of therestrictions to Ω(ξ) of the functions u∗n (defined in (3.1)) converges to thefunction u (defined in (2.16)) weakly in H1(Ω(ξ)).

The second order operator associated with the form (2.14) AD(u) =−div(∇u)+δ0u (with Dirichlet condition) has a compact resolvent in L2(Ω(ξ))and the spectrum consists of a increasing sequence of real eigenvalues λk suchthat λk → +∞ when k → +∞ (see e.g. [27]). We denote by PD(λ) thespectral resolution associated with AD.

Theorem 3.4 Let ε = ε(n) be a sequence such that ε(n)→ 0 as n→ +∞.Let us assume conditions (3.3) and (3.6). Then ∀ 0 < µ < λ, pointsof continuity for the spectral resolution PD of the operator AD, we havethat the projector operators Pn((µ − λ]) of the spectral resolutions Pn of

the operators An in L2(Ω(ξ),nε ) converge to the projector PD((µ − λ]) of the

spectral resolution PD strongly as n → +∞, in the sense of Definition 5.5of the Section 5.

The proofs of Theorems 3.1, 3.3, 3.2, 3.4 are postponed in Sections 6and 5. More precisely, in Section 4 we recall the definition and the mainproperties of M-convergence of forms in a fixed Hilbert space and we provethe convergence of the functionals corresponding to the forms (2.6) and(2.10). In Section 5 we recall the definition and the main properties of M-K-S-convergence for the case of variable Hilbert spaces according to K. Kuwaeand T. Shioya and we prove Theorems 3.2 and 3.4. Theorems 3.1 and 3.3will be proved in Section 6.

4 Mosco convergence

We recall the notion of M−convergence of functionals, introduced in [29],(see also [30]).

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Definition 4.1 A sequence of functionals Fn : H → (−∞,+∞] is said toM−converge to a functional F : H → (−∞,+∞] in a Hilbert space H, if

(a) For every u ∈ H there exists un converging strongly to u in H suchthat

lim supFn[un] ≤ F [u], as n→ +∞. (4.1)

(b) For every vn converging weakly to u in H

lim inf Fn[vn] ≥ F [u], as n→ +∞. (4.2)

We consider the sequence of weighted energy functionals in L2(Ω∗)

Fnε [u] =

∫Ω

(ξ),nε

anε (x, y)|∇u|2dxdy + δn∫

Ω(ξ),nε

u2dxdy ifu|Ω

(ξ),nε∈ H1

0 (Ω(ξ),nε ;wnε )

+∞ otherwise in L2(Ω∗)

(4.3)(the coefficients anε are defined in (2.3), (2.5), (2.4), δn > 0) and

Fc0 [u] =

∫Ω(ξ) |∇u|2dxdy + δ0

∫Ω(ξ) u2dxdy + c0

∫∂Ω(ξ) u2dµ(ξ) ifu|Ω(ξ) ∈ H1(Ω(ξ))

+∞ otherwise in L2(Ω∗).

(4.4)

Theorem 4.1 Let us assume (3.4) and (3.3). Let ε = ε(n) be an arbitrarysequence such that ε(n)→ 0 as n→ +∞. Then, the sequence of functionalsFnε(n), defined in (4.3), M−converges in L2(Ω∗) to the functional Fc0 defined

in (4.4) as n→ +∞.

In order to prove Theorem 4.1 we state the following convergence result.

Proposition 4.1 Let σn be as in (2.5). Then, for every sequence gn ∈H1(Ω(ξ)) weakly converging towards g∗ in H1(Ω(ξ)), we have

σn

∫∂Ω(ξ),n

gnds→∫∂Ω(ξ)

g∗ dµ(ξ) , as n→ +∞. (4.5)

Proof. From Theorems 8.1 and 8.4,

√σn||g||L2(∂Ω(ξ),n) 6 CT ||g||Hα(Ω(ξ)) ∀g ∈ H

α(Ω(ξ)), (4.6)

where α is a fixed number in the interval (1/2, 1) and the constant CT =√CRCα does not depend on n (not even on g). On the other hand, by the

compactness of the embedding of the space Hα(Ω(ξ)) with α ∈ (1/2, 1) inthe space H1(Ω(ξ)) (see, for instance, Theorem 1.4.6.2 in [27] and Theorem

13

16.2 in [26]), the sequence gn strongly converges towards g∗ in Hα(Ω(ξ)).Then

√σn||gn − g∗||L2(∂Ω(ξ),n) 6 CT ||gn − g

∗||Hα(Ω(ξ)) → 0 , as n→ +∞ (4.7)

and also

|σn∫∂Ω(ξ),n

(gn − g∗)ds| 6√σn||gn − g∗||L2(∂Ω(ξ),n)

√σn

√|∂Ω(ξ),n|, (4.8)

hence

σn

∫∂Ω(ξ),n

(gn − g∗)ds→ 0 , as n→ +∞. (4.9)

By an approximation result for the measure µ(ξ) (see Theorem 2.1 in [12]and also Lemma 8.4 in [17]), we have that

σn

∫∂Ω(ξ),n

ϕds−∫∂Ω(ξ)

ϕdµ(ξ) → 0 , as n→ +∞ (4.10)

∀ϕ ∈ C1(Ω(ξ)). Lastly, we use the fact that the space C1(Ω(ξ)) is a densesubspace of the space in H1(Ω(ξ)) and Theorem 8.2 to complete the proofof Proposition 4.1.

From now on, we simply denote ε(n) by ε and in order to further simplifynotation, when it does not give rise to misunderstanding, we write Fn inplace of Fnε(n), w

n in place of wnε(n), as well as an in place of anε(n). Moreover

we suppress the index (ξ) in the notation of the fibers.

Proof of Theorem 4.1 First we proceed with the proof of condition(a) in Definition 4.1. We consider a given function u as in condition (a)and we observe that, without loss of generality, we can assume that u|Ω(ξ) ∈H1(Ω(ξ)), otherwise the inequality (4.1) becomes trivial.

We assume, in addition, that u|Ω

(ξ) ∈ C1(Ω(ξ)

). For every n and ε as

above, we define u1,n on Σn1,ε =

⋃i|n Σ

i|n1,ε

u1,n(x, y) = G1,ε(u ψi|n) ψ−1i|n (x, y) if (x, y) ∈ Σ

i|n1,ε (4.11)

where G1,ε is the operator from C1(Σ1,ε) to Lip(Σ1,ε) defined in the followingway. For every ζ ∈ (0, 1), we define P+ = P+(ζ) = (ζ, η+(ζ)) ∈ ∂Σ1,ε to bethe intersection of ∂Σ1,ε with the vertical line through the point (ζ, 0) ∈ K1.Then, for a given g ∈ C1(Σ1,ε) we put

G1,ε(g)(ζ, η) =

g(0, 0) if (ζ, η) = (0, 0)

g(ζ, 0) η+−ηη+if (ζ, η) ∈ Σ1,ε \ A,B

g(1, 0) if (ζ, η) = (1, 0).

(4.12)

14

We construct, in a similar way, Gj,ε on Σj,ε and uj,n on Σnj,ε for j = 2, 3, 4.

We set

u =

u on Ω

(ξ)

0 on Ω∗ \ Ω(ξ)

and for every n and ε as above, we define

un(x, y) =

u(x, y) if (x, y) ∈ Ω∗ \ Σn

ε

uj,n(x, y) if (x, y) ∈ Σnj,ε.

(4.13)

We note that un tend to u in L2(Ω∗), supΣnε|un| 6 sup

Ω(ξ) |u| and the re-

strictions on Ω(ξ),nε of the functions un defined in (4.13) belong to H1

0 (Ω(ξ),nε ).

For each n, we split the integral Fn[un] in three terms, taking into ac-count the definitions of an and un,

Fn[un] =

∫Ω(ξ),n

|∇u|2dxdy + δn

∫Ω

(ξ),nε

u2ndxdy + σncn

∫Σnε

|∇un|2wndxdy

Since the sets Ω(ξ),n tend to the set Ω(ξ) as n→ +∞, we get

limn→+∞

∫Ω(ξ),n

|∇u|2dxdy =

∫Ω(ξ)

|∇u|2dxdy, (4.14)

limn→+∞

δn

∫Ω(ξ),n

u2dxdy = δ0

∫Ω(ξ)

u2dxdy, (4.15)

limn→+∞

δn

∫Σnε

u2ndxdy = 0. (4.16)

Now we prove

limn→+∞cnσn

∫Σnε

|∇un|2wndxdy = c0

∫∂Ω(ξ)

u2dµ(ξ). (4.17)

We split the integral on Σnε as the sum of the 4 integrals on the sets Σn

j,ε,

j = 1, 2, 3, 4. We now consider Σi|n1,ε : in all the other cases we proceed in a

similar manner.We decompose each Σ

i|n1,ε as the union of one rectangle and two triangles

and we evaluate the corresponding integrals by making use of the coordinateschange provided by the map ψi|n. Thus, we write

cnσn

∫Σi|n1,ε

|∇un|2wndxdy ≡ R1,0 +

2∑h=1

X1,h

where

R1,0 = cnσn

∫ψi|n(R1,ε)

|∇un|2wndxdy, X1,h = cnσn

∫ψi|n(T1,h,ε)

|∇un|2wndxdy , h = 1, 2 .

15

We note that for (x, y) ∈ Σi|n1,ε, (x, y) = ψi|n(ζ, η) and wn(x, y) = `ε(ζ)

l(n) ,where

`ε(ζ) =

ζ 0 < ζ < εε ε < ζ < 1− ε1− ζ 1− ε < ζ < 1 .

Put g(ζ, η) := (un ψi|n)(ζ, η) and set M = ||u||2C1(Ω

(ξ)).

We have

R1,0 = cnε

4n

∫ ε

0dy

∫ 1−ε

ε|gζ(ζ, 0)(1−η

ε)|2dζ+cn

ε

4n

∫ ε

0dη

∫ 1−ε

ε|g(ζ, 0)

1

ε|2dζ ≡ An+Bn

Now

An =cnε

2

4n3

1

l(n)

∫ψi|n([ε,1−ε])

|∇τun|2ds ≤Mcnε

2

4nl(n)2. (4.18)

Moreover,

Bn =cnl(n)

4n

∫ψi|n([ε,1−ε])

|un(x, y)|2ds. (4.19)

Now we evaluate the terms

X1,h = cnσn

∫ψi|n(T1,h,ε)

|∇un|2wndxdy , h = 1, 2 :

we have

X1,1 = cnσn

∫ψi|n(T1,1,ε)

|∇un|2wndxdy =

=cn4n

∫ ε

0dζ

∫ ζ

0|gζ(ζ, 0)(1−η

ζ)+g(ζ, 0)

η

ζ2|2ζdη+

cn4n

∫ ε

0dζ

∫ ζ

0|g(ζ, 0)

1

ζ|2ζdη ≡

≡ Cn +Dn.

The first term can be estimated as follows

Cn ≤ CcnMε

4n. (4.20)

The second term can be estimated as before

Dn =cnl(n)

4n

∫ψi|n([0,ε])

u2nds. (4.21)

As X1,2 is analogous, we obtain

cnσn

∫Σn1,ε

|∇un|2wndxdy =∑i|n

(cnl(n)

4n

∫ψi|n(K1)

u2nds+ C

cnMε

4n) (4.22)

16

and

limn→+∞cnσn

∫Σn1,ε

|∇un|2wndxdy = limn→+∞cnl(n)

4n

∫Kn

1

u2nds. (4.23)

In a similar way, we can estimate the other terms for j = 2, 3, 4, hence

limn→+∞cnσn

∫Σnε

|∇un|2wndxdy = limn→+∞cnl(n)

4n

∫∂Ω(ξ),n

u2nds. (4.24)

As un = u on ∂Ω(ξ),n, by Proposition 4.1 (for gn = u) we obtain

limn→+∞cnσn

∫Σnε

|∇un|2wndxdy = c0

∫∂Ω(ξ)

u2dµ(ξ). (4.25)

We complete the proof of part (a) of the Theorem, by making use of thediagonal formula of Corollary 1.16 of [3].

Now we prove condition (b) of Definition 4.1. Let vn be a sequence as in(b) that is,

vn u in L2(Ω∗). (4.26)

In order to prove the inequality (4.2), it is not restrictive to assume that

lim infFn[vn] ≤ C∗ < +∞. (4.27)

Then, from (4.26) and (4.27), up to passing to a subsequence, we deducethat

||vn||H1(Ω(ξ),n) ≤ C

where C is a constant independent of n. By Theorem 8.3, there exists abounded linear extension operator ExtJ : H1(Ω(ξ),n) → H1(R2), whosenorm is independent of n, that is,

||ExtJ vn||H1(R2) 6 CJ ||vn||H1(Ω(ξ),n) (4.28)

with CJ independent of n. We put

vn = (ExtJ vn|Ω(ξ),n)|Ω(ξ) , (4.29)

then there exist v ∈ H1(Ω(ξ)) and a subsequence of vn, denoted by vn again,weakly converging to v in H1(Ω(ξ)). By a direct calculation, we can provethat the sequence vn weakly converges to u in L2(Ω(ξ)) hence

vn u|Ω(ξ) in H1(Ω(ξ)), (4.30)

lim infn→+∞

∫Ω(ξ),n

|∇vn|2 dxdy ≥∫

Ω(ξ)

|∇u|2 dxdy (4.31)

17

and

limn→+∞

δn

∫Ω(ξ),n

v2ndxdy = δ0

∫Ω(ξ)

u2dxdy. (4.32)

In order to conclude the proof, we only have to show that (if c0 > 0)

lim infn→+∞

cnσn

∫Σnε

|∇vn|2wndxdy ≥ c0

∫∂Ω(ξ)

u2dµ(ξ). (4.33)

Let us consider a sequence of functions um ∈ C1(Ω∗) such that

limm→+∞

‖um − u‖L2(Ω∗) = 0

andlim

m→+∞‖um − u|Ω(ξ)‖H1(Ω(ξ)) = 0. (4.34)

As in the proof of (a) (lim sup condition), for any (fixed) m, we constructthe functions um,n = (um)n – starting with um– as in (4.13) and we obtainthat

limn→+∞cnσn

∫Σnε

|∇um,n|2wndxdy = c0

∫∂Ω(ξ)

u2mdµ

(ξ). (4.35)

Now we prove that

limn→+∞cnσn

∫Σnε

(∇um,n,∇vn)wndxdy = c0

∫∂Ω(ξ)

umudµ(ξ). (4.36)

We proceed as in the proof of condition (a) and we obtain∑i|n

cnσn

∫Σi|n1,ε

(∇um,n,∇vn)wndxdy =∑i|n

cnσn

∫ψi|n([0,1])

um,nvnds+C√C∗cnMε

(4.37)where we have used (4.27).

As the terms on the other sides can be evaluated in the same way, weobtain

limn→+∞cnσn

∫Σnε

(∇um,n,∇vn, )wndxdy = limn→+∞cnσn

∫∂Ω(ξ),n

um,nvnds.

(4.38)From Proposition 4.1 (with gn = vnum, see (4.30) and (4.29))

limn→+∞σn

∫∂Ω(ξ),n

um,nvnds =

∫∂Ω(ξ)

umudµ(ξ) (4.39)

and so (4.36) is proved.

18

Combining (4.36) and (4.35),

lim infn→+∞cnσn

∫Σnε

|∇vn|2wndxdy ≥

≥ limn→+∞cnσn

(2

∫Σnε

(∇um,n,∇vn)wndxdy −∫

Σnε

|∇um,n|2wndxdy

)=

= 2c0

∫∂Ω(ξ)

umudµ(ξ) − c0

∫∂Ω(ξ)

u2mdµ

(ξ) :

letting m→ +∞ we conclude the proof.

Now we consider the case where the layer is weakly insulating (see (3.6))and we introduce the following functional (4.40) in L2(Ω∗)

F∞[u] =

∫Ω(ξ) |∇u|2dxdy + δ0

∫Ω(ξ) u2dxdy if u|Ω(ξ) ∈ H1

0 (Ω(ξ))+∞ otherwise in L2(Ω∗)

.(4.40)

Theorem 4.2 Let ε = ε(n) be a sequence such that ε(n)→ 0 as n→ +∞.Let us assume (3.6) and (3.3). Then the sequence of functionals Fnε(n), de-

fined in (4.3), M−converges in L2(Ω∗) as n→ +∞ to the energy functionalF∞[u] defined in (4.40).

Proof. We now proceed with the proof of condition (a) in Definition4.1. We consider a given function u as in condition (a) and we observe that,without loss of generality, we can assume that u|Ω(ξ) ∈ H1

0 (Ω(ξ)), otherwisethe inequality (4.1) becomes trivial. We assume, in addition, that u|

Ω(ξ) ∈

C10 (Ω

(ξ)). For every n and ε as above, we define u1,n on Σn

1,ε

u1,n(x, y) = G1,ε(u ψi|n) ψ−1i|n (x, y) if (x, y) ∈ Σ

i|n1,ε (4.41)

where G1,ε is the operator from C1(Σ1,ε) to Lip(Σ1,ε) defined in the followingway. For every ζ ∈ (0, 1), we define P+ = P+(ζ) = (ζ, η+(ζ)) ∈ ∂Σ1,ε to bethe intersection of ∂Σ1,ε with the vertical line through the point (ζ, 0) ∈ K1.Then, for a given g ∈ C1(Σ1,ε) we put

G1,ε(g)(ζ, η) =

0 if (ζ, η) = (0, 0)

g(ζ, 0) η+−ηη+if (ζ, η) ∈ Σ1,ε \ A,B

0 if (ζ, η) = (1, 0).

(4.42)

We construct, in a similar way, Gj,ε on Σj,ε and uj,n on Σnj,ε for j = 2, 3, 4.

We set

u =

u on Ω

(ξ)

0 on Ω∗ \ Ω(ξ) (4.43)

19

and for every n and ε as above, we define

un(x, y) =

u(x, y) if (x, y) ∈ Ω∗ \ Σn

ε

uj,n(x, y) if (x, y) ∈ Σnj,ε.

(4.44)

We note that un tend to u in L2(Ω∗), supΣnε|un| 6 sup

Ω(ξ) |u| and the restric-

tion on Ω(ξ),nε of the functions un defined in (4.44) belong to H1

0 (Ω(ξ),nε , wn).

We proceed as in the proof of previous Theorem 4.1 and we note that theonly difference is that we have to prove

limn→+∞cnσn

∫Σnε

|∇un|2wndxdy = 0 (4.45)

hence we need to evaluate the terms like (4.19) and (4.21) by taking intoaccount the assumptions of Theorem 4.2. By using the definition (4.42), weobtain

supψi|n([0,1])

|un| ≤√M

l(n)(4.46)

where M = ||u||2C1(Ω

(ξ)). Starting from formula (4.23), we evaluate the term

on the right end side by taking into account (4.46): more precisely

cnl(n)

4n

∫Kn

1

u2nds ≤M

cnl(n)2

. (4.47)

By using (3.6) we conclude

limn→+∞cnl(n)

4n

∫Kn

1

u2nds ≤ limn→+∞M

cnl(n)2

= 0. (4.48)

In a similar way, we can evaluate the terms on Σnj,ε, j = 2, 3, 4, and the

proof (4.45) is concluded. We complete the proof of part (a) of the Theorem,by making use of the diagonal formula of Corollary 1.16 of [3].

Now we prove condition (b) of Definition 4.1. Let vn be a sequence as in(b) of Definition 4.1, that is,

vn u in L2(Ω∗). (4.49)

In order to prove the inequality (4.2), it is not restrictive assume that the

condition (4.27) holds and (up to passing to a subsequence) vn ∈ H10 (Ω

(ξ),nε , wn).

As the sequences cn in the definition of the coefficients anε (see (2.3)) tendto +∞, for any k > 0, there exists n such that for any n > n, cn > k. Then,

Fn[vn] =

∫Ω(ξ),n

|∇vn|2dxdy + cnσn

∫Σnε

|∇vn|2wndxdy + δn

∫Ω

(ξ),nε

v2ndxdy >

20

>

∫Ω(ξ),n

|∇vn|2dxdy + kσn

∫Σnε

|∇vn|2wndxdy + δn

∫Ω

(ξ),nε

v2ndxdy

for any n > n. By (4.31) and (4.33) in the proof of Theorem 4.1, we obtain

lim infn→+∞Fn[vn] ≥

∫Ω(ξ)

|∇u|2 dxdy + k

∫∂Ω(ξ)

u2dµ(ξ) + δ0

∫Ω(ξ)

u2dxdy,

for any k > 0. Then we have

lim infn→+∞Fn[vn] ≥

∫Ω(ξ)

|∇u|2 dxdy + δ0

∫Ω(ξ)

u2dxdy,

and, for any k > 0,

1

klim infn→+∞F

n[vn] ≥∫∂Ω(ξ)

u2dµ(ξ),

that is, u = 0 on ∂Ω(ξ).

5 M-K-S-convergence

We recall that M-convergence of forms implies the convergence of semi-groups, resolvents and spectral structures relative to the associated opera-tors (see [30]). As in the framework of the present paper the operators actin different Hilbert spaces we need to introduce the definition and the mainproperties of M-K-S-convergence that is a generalization of M-convergenceto the case of variable Hilbert spaces.

Definition 5.1 A sequence of Hilbert spaces Hn is said to converge to theHilbert space H (Hn → H) if there exists a dense subspace C ⊂ H and asequence of linear maps Φn : C → Hn such that

||Φnu||Hn → ||u||H , (5.1)

as n→ +∞, for every u ∈ C.

Lemma 5.1 The Hilbert spaces Hn = L2(Ω(ξ),nε ) converge to the Hilbert

space H = L2(Ω(ξ)) in the sense of Definition 5.1 as n→ +∞.

Proof. We choose C = C(Ω(ξ)

) and, for every n, we define the map Φn

Φnu =

u in Ω(ξ) ∩ Ω

(ξ),nε

0 in Ω(ξ),nε \ Ω(ξ).

21

The convergence of the spaces in the sense of Definition 5.1 follows from the

convergence of the sets Ω(ξ),nε to the set Ω(ξ) as n→ +∞.

We recall the notion of convergence for functionals and operators givenin [29], see also [30], as adapted by K.Kuwae - T.Shioya and A. Kolesnikovin the papers [24], [23] to variable Hilbert spaces as in Definition 5.1.

We start by giving the definitions of strong convergence and of weakconvergence of vectors in variable Hilbert spaces according to K. Kuwae -T. Shioya (see [24]).

Definition 5.2 Let Hn → H according to Definition 5.1. A sequence ofvectors un ∈ Hn converges K-S-strongly to a vector u ∈ H (un → u K-S-strongly) if there exists a sequence um ∈ C such that both conditions belowhold:

||um − u||H → 0 (5.2)

as m→ +∞,limm

lim supn||Φnum − un||Hn = 0. (5.3)

as n→ +∞ and m→ +∞.

Definition 5.3 Let Hn → H according to Definition 5.1. A sequence ofvectors un ∈ Hn converges K-S-weakly to a vector u ∈ H (un → u K-S-weakly), if

(un, vn)Hn → (u, v)H (5.4)

for every vn → v K-S-strongly, as n→ +∞ .

Definition 5.4 Let Hn → H according to Definition 5.1. Let Fn : Hn →(−∞,+∞] for every n and let F : H → (−∞,+∞]. The functionals Fn

M-K-S-converge to F as n→ +∞ if both conditions below hold:

(a) For every u ∈ H there exists un ∈ Hn converging K-S-strongly to usuch that

lim supFn[un] ≤ F [u], as n→ +∞. (5.5)

(b) For every vn ∈ Hn converging K-S-weakly to u ∈ H

lim inf Fn[vn] ≥ F [u], as n→ +∞. (5.6)

Convergence of operators is defined as follows.

Definition 5.5 Let Hn → H according to Definition 5.1. A sequenceof bounded linear operators Bn ∈ L(Hn) converges strongly to the linearbounded operator B ∈ L(H) if for every un ∈ Hn converging K-S-stronglyto u ∈ H, the sequence Bn(un) ∈ Hn converges K-S-strongly to B(u) ∈ H.

22

The following Proposition 5.1 collects some convergence properties from[23] (see also [24]).

Proposition 5.1 Let Hn → H according to Definition 5.1. Then the fol-lowing properties hold:

(i) For every u ∈ H there exist vectors un ∈ Hn that converge K-S-strongly to u.

(ii) The vectors un ∈ Hn converge K-S-strongly to u ∈ H if and only if

||un||Hn → ||u||H , and (un,Φnϕ)Hn → (u, ϕ)H

for every ϕ ∈ C.

We are now in position to prove Theorem 3.2 and Theorem 3.4.

Proof of Theorem 3.2. We note that Theorem 3.2 and Theorem 4.1have the some hypotheses. Then the functionals Fnε M-converge in L2(Ω∗)to the functional Fc0 . We show that M-convergence of the functionals Fnεimplies M-K-S-convergence of the functionals Enε to the functional Ec0 where

Enε [u] =

∫Ω

(ξ),nε

anε (x, y)|∇u|2dxdy + δn∫

Ω(ξ),nε

u2dxdy if u ∈ H10 (Ω

(ξ),nε ;wnε )

+∞ if u ∈ L2(Ω(ξ),nε ) \H1

0 (Ω(ξ),nε ;wnε )

(5.7)(the coefficients anε are defined in (2.3), (2.5), (2.4), δn > 0) and

Ec0 [u] =

∫Ω(ξ) |∇u|2dxdy + δ0

∫Ω(ξ) u2dxdy + c0

∫∂Ω(ξ) u2dµ(ξ) ifu ∈ H1(Ω(ξ))

+∞ ifu ∈ L2(Ω(ξ)) \H1(Ω(ξ)).

(5.8)If u ∈ L2(Ω(ξ)) \ H1(Ω(ξ)) then condition (a) in Definition 5.4 follows

from property (i) of Proposition 5.1. If instead u ∈ H1(Ω(ξ)), we put

u =

u in Ω(ξ)

0 in Ω∗ \ Ω(ξ)(5.9)

and by condition (a) in Definition 4.1 there exists a sequence u∗n ∈ L2(Ω∗)

strongly convergent to u (in L2(Ω∗)) such that un := u∗n|Ω(ξ),nε∈ H1(Ω

(ξ),nε ;wnε )

and

lim supEnε [un] = lim supFnε [un] ≤ Fc0 [u] = Ec0 [u] < +∞.

We show now that the sequence un converges K-S-strongly to u by prov-ing the two conditions in item (ii) of Proposition 5.1. In fact for any fixedn0 we have

23

∫Ω(ξ),n0

u2ndxdy →

∫Ω(ξ),n0

u2dxdy, asn→ +∞.

As the increasing sequence of the sets Ω(ξ),n0 converges to the set Ω(ξ)

in the Hausdorff metric we deduce, passing to the limit on n0, that

lim infn→+∞

∫Ω

(ξ),nε

u2ndxdy ≥ lim

n0→+∞lim

n→+∞

∫Ω(ξ),n0

u2ndxdy =

∫Ω(ξ)

u2dxdy.

On the other hand,

lim supn→+∞

∫Ω

(ξ),nε

u2ndxdy ≤ lim

n→+∞

∫Ω∗

(u∗n)2dxdy =

∫Ω∗u2dxdy =

∫Ω(ξ)

u2dxdy.

hence the first condition in item (ii) of Proposition 5.1 is proved. Consider

now ϕ ∈ C0(Ω(ξ)) and set

ϕ =

ϕ in Ω

(ξ)

0 in Ω∗ \ Ω(ξ) (5.10)

and

ϕn =

ϕ in Ω

(ξ),nε

0 in Ω∗ \ Ω(ξ),nε .

(5.11)

It is easy to note that the sequence of the functions ϕn is weakly con-vergent to the function ϕ in L2(Ω∗).

Then we have

∫Ω

(ξ),nε

unΦnϕdxdy =

∫Ω

(ξ),nε ∩Ω(ξ)

unϕdxdy =

∫Ω(ξ)

u∗nϕndxdy →∫

Ω(ξ)

uϕdxdy

as n→ +∞ and also the second condition in item (ii) of Proposition 5.1 isproved.

Let us consider condition (b) in Definition 5.4. We assume that thesequence vn converges K-S-weakly to u and lim inf Enε [vn] 6 C∗ then, up topassing to a subsequence still denoted by vn, the function vn belongs to the

space H10 (Ω

(ξ),nε ;wnε ) and limEnε [vn] = limFnε [vn] ≤ C∗.

We set

vn =

vn in Ω

(ξ),nε

0 in Ω∗ \ Ω(ξ),nε

(5.12)

and we prove that the sequence of the functions vn is weakly convergent (inL2(Ω∗)) to the function u defined in (5.9). In fact for any g ∈ C0(Ω∗) weset

gn =

g in Ω

(ξ),nε

0 in Ω∗ \ Ω(ξ),nε

(5.13)

24

and by using Proposition 5.1 we prove that the sequence gn converges K-S-strongly to g.

Hence∫Ω∗

(vn − u)gdxdy =

∫Ω

(ξ),nε

vngndxdy −∫

Ω(ξ)

ugdxdy → 0

and so condition (b) in Definition 5.4 follows from condition (b) in Definition4.1.

From Theorem 2.4 of [24], we deduce that ∀ 0 < µ < λ, points ofcontinuity for the spectral resolution Pc0 of the operator Ac0 , the projectoroperators Pn((µ− λ]) of the spectral resolutions Pn of the operators An in

L2(Ω(ξ),nε ) converge to the projector Pc0((µ − λ]) of the spectral resolution

Pc0 strongly as n→ +∞, in the sense of Definition 5.5. .

Proof of Theorem 3.4. In the same way we obtain that from Theorem4.2 it follows the convergence of the functionals Enε defined in (5.7) to thefunctional E∞ according the Definition (5.4) where

E∞[u] =

∫Ω(ξ) |∇u|2dxdy + δ0

∫Ω(ξ) |u|2dxdy if u ∈ H1

0 (Ω(ξ))

+∞ if u ∈ L2(Ω(ξ)) \H10 (Ω(ξ)).

(5.14)From Theorem 2.4 of [24], we deduce that ∀ 0 < µ < λ, points of

continuity for the spectral resolution PD of the operator AD, the projectoroperators Pn((µ− λ]) of the spectral resolutions Pn of the operators An in

L2(Ω(ξ),nε ) converge to the projector PD((µ − λ]) of the spectral resolution

PD strongly as n→ +∞, in the sense of Definition 5.5. .

6 Asymptotics

In this section we prove Theorems 3.1 and 3.3. We note that the coercivenessestimate (2.9) is not sharp enough for our aim and we need to improveit: therefore, before proving Theorem 3.1, we establish a Poincare typeinequality where the relevant fact is that the constant C∗P is independent ofn.

Theorem 6.1 There exists a constant C∗P independent of n, such that,

||u||L2(Ω

(ξ),nε )

≤ C∗P (||∇u||2L2(Ω(ξ),n)

+ σn

∫Σ

(ξ),nε

|∇u|2wnε dxdy)1/2 (6.1)

for all u ∈ H10 (Ω

(ξ),nε ;wnε ).

25

Proof. Suppose the statement to be proved is false: for every n in N,

there exists vn ∈ H10 (Ω

(ξ),nε ;wnε ), such that

||vn||2L2(Ω

(ξ),nε )

> n2(||∇vn||2L2(Ω(ξ),n)+ σn

∫Σ

(ξ),nε

|∇vn|2wnε dxdy). (6.2)

Setun =

vn||vn||L2(Ω

(ξ),nε )

.

We extend un from the set Ω(ξ),nε to Ω∗

un =

un in Ω

(ξ),nε

0 in Ω∗ \ Ω(ξ),nε .

(6.3)

Therefore, there exists a function u∗ in L2(Ω∗) and a subsequence (stilldenoted by un) that converges to u∗ weakly in L2(Ω∗).

From (6.2), we obtain

||∇un||2L2(Ω(ξ),n)+ σn

∫Σ

(ξ),nε

|∇un|2wnε dxdy <1

n2(6.4)

and, in particular,

||un||2H1(Ω(ξ),n)6 1 +

1

n2.

We extend un from the set Ω(ξ),n to R2 (see Theorem 8.3): we set

u∗∗n = (ExtJ un|Ω(ξ),n)|Ω(ξ) (6.5)

and we have

||u∗∗n ||H1(Ω(ξ)) ≤√CJ ||un||H1(Ω(ξ),n) 6 C

√CJ . (6.6)

Therefore, there exists a function u∗∗ in H1(Ω(ξ)) and a subsequencestill denoted by u∗∗n that converges to u∗∗ weakly in H1(Ω(ξ)) and stronglyin L2(Ω(ξ)).

We fix n0 : then, as u∗∗n = un = un on Ω(ξ),n0 , we deduce u∗∗ = u∗ inΩ(ξ),n0 .

As the increasing sequence of the sets Ω(ξ),n converges to the set Ω(ξ) inthe Hausdorff metric we deduce, passing to the limit on n0, that u∗∗ = u∗

on Ω(ξ). Hence the sequence un converges to u∗ weakly in H1(Ω(ξ),n0) : then

||∇u∗||L2(Ω(ξ),n0 ) ≤ lim infn||∇un|Ω(ξ),n0 ||L2(Ω(ξ),n0 ) = 0, (6.7)

i.e. the function u∗ is constant on Ω(ξ),n0 and, passing to the limit on n0

again, u∗ is constant on Ω(ξ).

26

Since the functions un converge to u∗ weakly in L2(Ω∗) and un|Ω(ξ),nε

=

un ∈ H10 (Ω

(ξ),nε ;wnε ), we can proceed as in the proof of Theorem 4.1 (see

(4.33)) and we obtain∫∂Ω(ξ)

(u∗)2dµ(ξ) 6 lim infn→+∞

σn

∫Σ

(ξ),nε

|∇un|2wnε dxdy. (6.8)

As u∗ is constant on Ω(ξ), from (6.4) and (6.8) we deduce that u∗ = 0 onΩ(ξ).

By a direct calculation, we obtain

||un||2L2(Σ

(ξ),nε )

≤ σn∫

Σ(ξ),nε

|∇un|2wnε dxdy 61

n2, (6.9)

therefore, as

1 = ||un||2L2(Ω

(ξ),nε )

= ||un||2L2(Ω(ξ),n)+ ||un||2

L2(Σ(ξ),nε )

≤ ||u∗∗n ||2L2(Ω(ξ))+

1

n2,

passing to the limit, we obtain a contradiction.

We now prove Theorem 3.1.

Proof. Let un be the solution of the problem (2.7). Then

an(un, un) ≤ ||fn||L2(Ω(ξ),nε )||un||L2(Ω

(ξ),nε )

+ C|dn|||un||H1(Ω(ξ),n) (6.10)

where we have used Theorems 8.3 and 8.1.Suppose first that δ0 > 0. Then, from (6.10), by the definition of the

form an(·, ·) (see (2.6)) we deduce

δn||un||2L2(Ω

(ξ),nε )

+||∇un||2L2(Ω(ξ),n)6 ||fn||L2(Ω

(ξ),nε )||un||L2(Ω

(ξ),nε )

+C|dn|||un||H1(Ω(ξ),n)

(6.11)and hence

||un||2H1(Ω(ξ),n)6 C∗, (6.12)

where the constant C∗ does not depend on n.By assumption (2.11), if δ0 = 0, then c0 > 0. From inequality (6.1) we

derive

||un||2L2(Ω

(ξ),nε )

6 (C∗P )2(||∇un||2L2(Ω(ξ),n)+ σn

∫Σ

(ξ),nε

|∇un|2wnε dxdy). (6.13)

From (6.10) and by the definition of the form an(·, ·), we deduce that

||∇un||2L2(Ω(ξ),n)+ σncn

∫Σ

(ξ),nε

|∇un|2wnε dxdy 6 (6.14)

27

||fn||L2(Ω(ξ),nε )||un||L2(Ω

(ξ),nε )

+ C|dn|||un||H1(Ω(ξ),n)

and, as a consequence of (6.13) and (6.14),

||un||2H1(Ω(ξ),n)6 C∗∗, (6.15)

where the constant C∗∗ does not depend on n.In any case, we shall consider the function u∗n, which, as before, is the

extension of the function un from the set Ω(ξ),n to the set Ω∗, and in anycase for every n, from either (6.12) or (6.15) ve derive

||u∗n||H1(Ω∗) ≤√CJ ||un||H1(Ω(ξ),n) ≤ C. (6.16)

Therefore, there exists a subsequence still denoted by u∗n that weakly con-verges to a function u∗ in H1(Ω∗). Now we prove that u∗|Ω(ξ) = u. By usingcondition (b) of M-convergence we obtain that

F [u∗]− 2

∫Ω(ξ)

f u∗ dxdy − 2d

∫∂Ω(ξ)

u∗ dµ(ξ) ≤ (6.17)

≤ lim inf

(Fnε [u∗n]− 2

∫Ω

(ξ),nε

fn u∗n dxdy − 2σndn

∫∂Ω(ξ),n

u∗n ds

).

By using condition (a) of M-convergence there exists vn ∈ L2(Ω∗),

vn|Ω(ξ),nε

∈ H10 (Ω

(ξ),nε ;wnε ) converging strongly in L2(Ω∗) to u (defined in

(4.43)) such thatlimFnε [vn] = F [u] = F [u],

as n → +∞. Then by Proposition 4.1 (where gn is a suitable extension ofvn) we obtain

lim

(Fnε [vn]− 2

∫Ω

(ξ),nε

fn vn dxdy − 2σndn

∫∂Ω(ξ),n

vn ds

)= (6.18)

= F [u]− 2

∫Ω(ξ)

f u dxdy − 2d

∫∂Ω(ξ)

u dµ(ξ).

As, by direct calculations, we obtain

lim inf

(Fnε [u∗n]− 2

∫Ω

(ξ),nε

fn u∗n dxdy − 2σndn

∫∂Ω(ξ),n

u∗n ds

)≤ (6.19)

≤ lim inf

(Fnε [un]− 2

∫Ω

(ξ),nε

fn un dxdy − 2σndn

∫∂Ω(ξ),n

un ds

)and

Fnε [un]− 2

∫Ω

(ξ),nε

fn un dxdy − 2σndn

∫∂Ω(ξ),n

un ds = (6.20)

28

= minv∈H1

0 (Ωnε ;w(ξ),nε )

(Fnε [v]− 2

∫Ω

(ξ),nε

fn v dxdy − 2σndn

∫∂Ω(ξ),n

v ds

)≤

≤ Fnε [vn]− 2

∫Ω

(ξ),nε

fn vn dxdy − 2σndn

∫∂Ω(ξ),n

vn ds,

combining (6.17), (6.19), (6.20), and (6.18) we obtain that

F [u∗]−2

∫Ω(ξ)

f u∗ dxdy−2d

∫∂Ω(ξ)

u∗ dµ(ξ) ≤ F [u]−2

∫Ω(ξ)

f u dxdy−2d

∫∂Ω(ξ)

u dµ(ξ).

By the uniqueness of the solution (2.12), we conclude that u∗|Ω(ξ) = u, andu∗n|Ω(ξ) converges to u weakly in H1(Ω(ξ)).

7 Laplacean transport

The mathematical model of the so-called Laplacean transport and of someother phenomena involve mixed Dirichelet -Robin conditions on the bound-ary.

In the present setting the geometry of the bulk could be represented by

the domain Ω(ξ) = Ω(ξ) \ B(P0,18), where B(P0,

18) is the ball with center

P0 = (12 ,−

12) and radius 1

8 where Ω(ξ) is defined in Section 2.

In a similar way, we consider the prefractal domains Ω(ξ),n = Ω(ξ),n \B(P0,

18), and the reinforced domains Ω

(ξ),nε = Ω

(ξ),nε \B(P0,

18),

We consider the bilinear form associated with the Dirichelet Robin prob-lem

a(u, v) :=

∫Ω(ξ)

∇u∇v dxdy + c0

∫∂Ω(ξ)

u v dµ(ξ), (7.1)

where µ(ξ) is the measure on ∂Ω(ξ) that coincides, on each K(ξ)j j = 1, 2, 3, 4,

with the Radon measure defined in (8.4).

Theorem 7.1 For any d ∈ R, c0 ≥ 0, there exists one and only one solutionu of the following problem

find u ∈ V (Ω(ξ)) := u ∈ H1(Ω(ξ)) : u = 0 on ∂B(P0,18) such that

a(u, v) = d∫∂Ω(ξ) v dµ(ξ) ∀ v ∈ V (Ω(ξ)),

(7.2)where a(·, ·) is defined in (7.1). Moreover, u is the only function that realizesthe minimum of the energy functional

minv∈V (Ω(ξ))

a(v, v)− 2d

∫∂Ω(ξ)

v dµ(ξ).

29

Figure 7.1: The domain Ω(ξ).

Theorem 7.1 can be proved via the Lax-Milgram Theorem, the principaltools being a trace result (Theorem 8.2) and a generalized Poincare inequal-ity (Lemma 3.1.1 in [28]). Since the proof is similar to the proof of Theorem2.2, but easier, we skip it.

The following theorem concerns the Dirichlet problems in the reinforceddomains.

Theorem 7.2 Let dn ∈ R and σn be as in (2.5). Then, there exists oneand only one solution un of the problem

find un ∈ H10 (Ω

(ξ),nε ;wnε ) such that

an(u, v) = σndn∫∂Ω(ξ),n v ds ∀ v ∈ H1

0 (Ω(ξ),nε ;wnε ),

(7.3)

where the bilinear form an(·, ·) is

an(u, v) :=

∫Ω

(ξ),nε

anε∇u∇v dxdy (7.4)

and the coefficients anε are defined in (2.3) (see also (2.5) and (2.4)).Moreover, un is the only function that realizes the minimum of the func-

tional

minv∈H1

0 (Ω(ξ),nε ;wnε )

an(v, v)− 2σndn

∫∂Ω(ξ),n

v ds. (7.5)

In order to prove Theorem 7.2, we state a generalized Poincare inequalityfor functions belonging to H1(Ω(ξ),n) where the relevant fact is that theconstant CP is independent of n.

30

Theorem 7.3 There exists a constant CP independent of n, such that,

||u||L2(Ω(ξ),n) ≤ CP (||∇u||L2(Ω(ξ),n) + ||u||L2(∂B(P0,18

))) (7.6)

for all u ∈ H1(Ω(ξ),n).

Proof. Set, for u ∈ H1(Ω(ξ),n),

G(u) = ||u||L2(∂B(P0,18

)). (7.7)

Suppose the statement to be proved is false: for every n in N, there existsvn ∈ H1(Ω(ξ),n), such that

||vn||L2(Ω(ξ),n) > n(||∇vn||L2(Ω(ξ),n) +G(vn)). (7.8)

Setun =

vn||vn||H1(Ω(ξ),n)

.

We extend un from Ω(ξ),n to R2 by using an extension operator whose normis independent of the (increasing) number of sides (see Theorem 8.3) and for

u∗n = ExtJ un|Ω(ξ) (7.9)

we have||u∗n||H1(Ω(ξ)) ≤

√CJ ||un||H1(Ω(ξ),n) =

√CJ . (7.10)

Then, there exists a function u∗ in H1(Ω(ξ)) and a subsequence (still denotedby u∗n) that converges to u∗ weakly in H1(Ω(ξ)) and strongly in Hs(Ω(ξ))for 0 ≤ s < 1 (see Theorem 1.4.6.2 in [27] and Theorem 16.2 in [26]). From(7.8), we obtain

||∇un||L2(Ω(ξ),n) <1

n. (7.11)

We fix n0 : for all n ≥ n0 as Ω(ξ),n0 ⊆ Ω(ξ),n

||∇un||L2(Ω(ξ),n0 ) <1

n(7.12)

and hence

||∇u∗||L2(Ω(ξ),n0 ) ≤ lim infn||∇u∗n|Ω(ξ),n0 ||L2(Ω(ξ),n0 ) = 0, (7.13)

i.e. the function u∗ is constant on Ω(ξ),n0 and, passing to the limit on n0,we obtain that u∗ is constant on Ω(ξ).

On the other hand, from (7.8), we obtain

G(un) <1

n: (7.14)

31

soG(u∗) = ||u∗||L2(∂B(P0,

18

)) = limn→∞

||u∗n|Ω(ξ),n0 ||L2(∂B(P0,18

)) = 0

and then u∗ = 0 on Ω(ξ). Therefore, as

1 = ||un||2H1(Ω(ξ),n)= ||un||2L2(Ω(ξ),n)

+ ||∇un||2L2(Ω(ξ),n)≤ ||u∗n||2L2(Ω(ξ))

+1

n2,

passing to the limit, we obtain a contradiction.

Now we prove Theorem 7.2.

Proof. We use the Lax-Milgram theorem. Indeed, by Theorems 8.1 and8.3, we have that

|σndn∫∂Ω(ξ),n

v ds| ≤ σn|dn|||v||L2(∂Ω(ξ),n)

√|∂Ω(ξ),n| ≤ 2

√C1CJ |dn|||v||H1(Ω(ξ),n) ≤

≤ 2√C1CJ |dn|||v||H1

0 (Ω(ξ),nε ;wnε )

.

Obviously, the bilinear form an(u, v) is continuous, i.e.

|an(u, v)| ≤ max(1, σncn)||u||H1

0 (Ω(ξ),nε ;wnε )

||v||H1

0 (Ω(ξ),nε ;wnε )

.

In order to prove the coerciveness, we first obtain by means of a directcalculation that

||u||2L2(Σ

(ξ),nε )

≤ σn∫

Σ(ξ),nε

|∇u|2wnε dxdy, (7.15)

for all u ∈ H10 (Ω

(ξ),nε ;wnε ).

From Poincare inequality (7.6), we deduce that

||u||L2(Ω(ξ),n) ≤ CP ||∇u||L2(Ω(ξ),n), (7.16)

for all u ∈ H10 (Ω

(ξ),nε ;wnε ).

Then, from (7.15) and (7.16), we obtain that the bilinear form is coercive,i.e.

an(u, u) ≥ 1

2min(1, σncn, cn,

1

C2P

)||u||2H1

0 (Ω(ξ),nε ;wnε )

. (7.17)

For any n we denote by un the trivial extension onB(P0,18), i.e. (un)|B(P0,

18

) =

0, we then extend the functions un from Ω(ξ),n to R2 by using an extensionoperator whose norm is independent of the (increasing) number of sides (seeTheorem 8.3) and we set

u∗n = (ExtJ un|Ω(ξ),n) |Ω∗ . (7.18)

The following theorem, that can be proved as Theorem 3.1, establishesthe asymptotic behavior of the functions u∗n (defined in (7.18)).

32

Theorem 7.4 Let us assume (3.4) and (3.5). Let ε = ε(n) be an arbitrarysequence such that ε(n)→ 0 as n→ +∞. Then the sequence of the restric-tions to Ω(ξ) of the functions u∗n (defined in (7.18)) converges to the functionu (defined in (7.2)) weakly in H1(Ω(ξ)).

8 APPENDIX

8.1. Scale irregular Koch curves.We recall the definition of scale irregular Koch curves built on two fam-

ilies of contractive similitudes (for general irregular scale fractals and theirmain properties, see [4], [31], and [32]).

Let A = 1, 2 : let 2 < `1 < `2 < 4, and, for each a ∈ A, let

Ψ(a) = ψ(a)1 , . . . , ψ

(a)4 (8.1)

be the family of contractive similitudes ψ(a)i : C → C, i = 1, . . . , 4, with

contraction factor `a−1 :

ψ(a)1 (z) =

z

`a, ψ

(a)2 (z) =

z

`aeıθ(`a) +

1

`a,

ψ(a)3 (z) =

z

`ae−ıθ(`a) +

1

2+ ı

√1

`a− 1

4, ψ

(a)4 (z) =

z − 1

`a+ 1,

where

θ(`a) = arcsin

(√`a(4− `a)

2

). (8.2)

Let Ξ = AN; we call ξ ∈ Ξ an environment. We define a left shift S on Ξsuch that if ξ = (ξ1, ξ2, ξ3, . . .) , then Sξ = (ξ2, ξ3, . . .) . For O ⊂ R2, we set

Φ(a)(O) =4⋃i=1

ψ(a)i (O)

and

Φ(ξ)n (O) = Φ(ξ1) · · · Φ(ξn) (O) .

Let K be the line segment of unit length with A = (0, 0) and B = (1, 0)

as endpoints. We set, for each n in N, K(ξ),n = Φ(ξ)n (K): K(ξ),n is the

so-called n-th prefractal curve.The fractal K(ξ) associated with the environment sequence ξ is defined

by

33

K(ξ) =+∞⋃n=1

Φ(ξ)n (Γ)

where Γ = A,B with A = (0, 0) and B = (1, 0). We remark that thesefractals do not have any exact self-similarity, i.e. there is no scaling factorwhich leaves the set invariant; the family K(ξ), ξ ∈ Ξ, however, satisfiesthe following relation

K(ξ) = Φ(ξ1)(K(Sξ)). (8.3)

Moreover, the spatial symmetry is preserved and the set K(ξ) is locallyspatially homogeneous, i.e. the volume measure µ(ξ) on K(ξ) satisfies thefollowing locally spatially homogeneous condition (8.4). Before describingthis measure, we introduce some notations. For ξ ∈ Ξ, we set i|n = (i1, ..., in)

and ψi|n = ψ(ξ1)i1 · · · ψ(ξn)

inand, for any set O ⊂ R2, ψi|n(O) = Oi|n.

The volume measure µ(ξ) is the only Radon measure on K(ξ) such that

µ(ξ)(ψi|n(K(Snξ))) =1

4n(8.4)

(see Section 2 in [4]) as, for each a ∈ A, the family Ψ(a) has 4 contractivesimilitudes.

8.2.Trace and Extension Theorems.We collect some trace and extension results used in the previous sections.

We recall that, for v in L1loc(D), where D is an arbitrary open set of R2, the

trace operator γ0 is defined as

γ0v(P ) := limr→0

1

m2(B(P, r) ∩D)

∫B(P,r)∩D

v(x, y) dx dy (8.5)

(m2 denotes the 2-dimensional Lebesgue measure) at every point P ∈ Dwhere the limit exists (see, for example, page 15 in [22]). We suppress γ0

in the notation, when it does not give rise to misunderstanding, by writingsimply v instead of γ0v.

The following theorem characterizes the trace to the set ∂Ω(ξ),n of afunction belonging to Sobolev spaces Hα(R2) (for the definitions and themain properties of Sobolev spaces, see [2]). We recall that

σn =`(ξ)(n)

4nwith `(ξ)(n) =

n∏i=1

`ξi (8.6)

Theorem 8.1 Let u ∈ Hα(R2) and σn as in (8.6). Then, for 12 < α 6 1,

||u||2L2(∂Ω(ξ),n)

6Cασn||u||2Hα(R2), (8.7)

where Cα is independent of n.

34

For the proof, see Theorem 3.5 in [14].

The following theorem that characterizes the trace to the set ∂Ω(ξ) ofa function belonging Sobolev spaces Hα(R2) is a consequence of Theorem3.9 in [14] (see also Chapter V in [22]). We remark that the fractal is notnecessarily a d-set.

Theorem 8.2 Let u ∈ Hα(R2). Then, for 12 < α 6 1,

||u||2L2(∂Ω(ξ))

6 C∗α||u||2Hα(R2). (8.8)

Following Theorem 8.3 provides an extension operator from H1(Ω(ξ),n)to the space H1(R2) whose norm is independent of the (increasing) num-ber of sides. This result is a consequence of the extension theorem due toJones (Theorem 1 in [21]) as the domains Ω(ξ),n are (ε,∞) domains with εindependent of n (for the proof, see Lemma 3.3 in [12]).

Theorem 8.3 There exists a bounded linear extension operator ExtJ : H1(Ω(ξ),n)→H1(R2), such that

||ExtJ v||2H1(R2) 6 CJ ||v||2H1(Ω(ξ),n)

(8.9)

with CJ independent of n.

We conclude this Section with an extension theorem for fractional Sobolevspaces Hα(Ω(ξ)) with 1

2 < α ≤ 1, (see Theorem 5.8 in [12] and Theorem 8in [36] ).

Theorem 8.4 There exists a linear extension operator ExtR such that, forany α ∈ (1

2 , 1], ExtR : Hα(Ω(ξ))→ Hα(R2),

||ExtR v||2Hα(R2) 6 CR||v||2Hα(Ω(ξ))

(8.10)

with CR depending on α.

Remark. We note that Theorem 8 in [36] is an improvement of theextension theorem by Jones as it provides a degree-independent Sobolevextension operator for (ε, δ) domains acting on all ordinary Sobolev spaces.We observe, finally, that if 1

2 < α < 1, estimate (8.10) can be also deducedfrom Theorem 1 on page 103 in [22] (see also Theorem 3 on page 155 in [22])since the domain Ω(ξ) can be seen as a 2-set.

35

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