Weak approximation of multipliers

In order to show that a pair of multipliers (ν, w) belongs to the limiting normal cone we need to find certain sequences{ν_n}_n∈N⊂H⁻¹(Ω), {w_n}_n∈N⊂H₀¹(Ω) that converge weakly to ν and w, see Definition 2.4.1. For the construction of {w_n}_n∈N we use Theorem 5.3.1, which means that we need to show all the conditions of that theorem.

A certain type of pointwise orthogonality of wn and νn is required for all n ∈N that are large enough. Therefore, we will construct {ν_n}_n∈N as a sequence of functions in L^∞(Ω) such that ν_n = 0 a.e. on Ωn and w_n = 0 q.e. on Ω\Ωn, where {Ωn}_n∈N is a sequence of open subsets of Ω such that the assumptions ofTheorem 5.3.1can be satisfied.

Although the functions ν_n, w_n are pointwise orthogonal for all n ∈ N that are large enough, it will turn out that the weak limits ν∈H⁻¹(Ω),w∈H₀¹(Ω) are not necessarily pointwise orthogonal and that our construction works for arbitraryν ∈L^p(Ω)⊂H⁻¹(Ω), w∈H₀¹(Ω), where p∈(1,2) is such that L^p(Ω)↪→H⁻¹(Ω).

Throughout this section, letp∈(1,2)∩[2d/(d+ 2),2), ν ∈L^p(Ω), and w∈H₀¹(Ω) be chosen arbitrarily but fixed. Additionally, we make the assumption that d≥2 holds.

Note that according to the Sobolev embedding theorem, these possible values for pare

precisely thosep∈(1,2) such that the embeddingL^p(Ω)↪→H⁻¹(Ω) holds, see [Adams, Fournier, 2003, Theorem 4.12].

Let us start with defining the sequence{Ωn}_n∈N of open subsets of Ω and some related objects that will be important in this section. In order to avoid some case distinctions betweend= 2 and d≥3, we introduce the auxiliary functionL^dvia

L^d(a) :=

{︄−log(a)⁻¹ fora∈(0,1) andd= 2, a^d−2 fora∈(0,∞) andd≥3. In any case,L^d is monotonically increasing and the range is (0,∞).

We are going to cover Ω by closed cubes. Therefore, fix a number n ∈ N and let {ω_iⁿ}_i∈N= _n²Z^dbe a regular grid, and we define the cubes P_iⁿ:=ω_iⁿ+ [−_n¹,_n¹]^d. These cubes have edge length ²_n and their interiors are pairwise disjoint. We also define O_iⁿ:= intB_1/n(ω_iⁿ)⊂P_iⁿ to be the (open) ball with radius 1/n that is contained inP_iⁿ andInas the set of indicesiwithP_iⁿ∩Ω̸=∅. Note thatInis finite for alln∈Nbecause Ω is bounded. Furthermore, each cube contains a closed hole T_iⁿ :=Bai,n(ωⁿ_i) at the center, which is a closed ball with radiusa_i,n≥0. For technical reasons, we introduce another index setJn, defined as

J_n:={i∈I_n|P_iⁿ⊂Ω, 0<∥ν∥^p_Lp(Pn

i )< n^1−d}.

Contrary to the approach in [Harder, G. Wachsmuth, 2018c, Section 3], the radii of the holesT_iⁿ are not uniform and depend on the local values of ν. Fori∈Jn, we define the radiusa_i,n>0 of the holeT_iⁿ implicitly via

L^d(ai,n) = meas(P_iⁿ) avg(P_iⁿ,|ν|^p)^2p−1, (5.16) where

avg(P_iⁿ,|ν|^p) := 1 meas(P_iⁿ)

∫︂

P_iⁿ

|ν|^pdω

is the average of the function |ν|^p over the set P_iⁿ. In the case that i∈I_n\J_n we set ai,n:= 0. The radiiai,nare well-defined because L^dis injective and has range (0,∞) and because avg(P_iⁿ,|ν|^p) is positive for i∈J_n. Finally, we define the perforated domain

Ωn:= Ω\ ^⋃︂

i∈J_n

T_iⁿ.

The next lemma shows that we have a_i,n ≤ _n¹ for n large enough, i.e., T_iⁿ ⊂ P_iⁿ holds. Afterwards, we will only consider these parameters n ∈ N which guarantee ai,n≤1/n.

Lemma 5.3.2. (a) For largen∈Nwe have a_i,n<

(︃ 1 ^)︃1+ε ∀i∈J_n,

whereε depends on the dimension, but not oni andn. In the case of d= 2 we even have

a_i,n<exp(−n/8)∀i∈J_n for largen∈N.

(b) For every large n∈Nthere exists a constant Cn>1 such that L^d(a_i,n)≤ 1

L^d(ai,n)⁻¹−L^d(1/n)⁻¹ ≤C_nL^d(a_i,n) holds for all i∈Jn. Moreover,Cn→1 as n→ ∞.

(c) The convergence

n→∞lim meas^(︂{ν ̸= 0} \ ^⋃︂

i∈J_n

P_i^n)︂= 0 holds.

Proof. We start with part(a). Using the definition of the index setJ_nand 2/p−1∈(0,1), we have

L^d(a_i,n) = meas(P_iⁿ) avg(P_iⁿ,|ν|^p)^2/p−1≤meas(P_iⁿ)(1 + avg(P_iⁿ,|ν|^p))

=^(︃2 n

)︃d

+∥ν∥^p_Lp(Pn i)<

(︃2 n

)︃d

+n^1−d<2^d+1n^1−d. (5.17) For d≥3 and largen∈N this implies

a_i,n<2^d+1n(1−d)/(d−2) <

(︃ 1 2n

)︃1+ε

,

where we setε:= 1/(2(d−2)). For d= 2, the inequality(5.17)yields a_i,n<exp(−n/8).

For part(b), we note that by part (a) it follows thatL^d(ai,n)/L^d(1/n)→0 asn→ ∞, uniformly in i∈J_n. This implies the claim.

It remains to prove part(c). Ifi∈In does not belong toJn then there are three possible reasons: ∥ν∥^p_Lp(P_iⁿ)= 0,∥ν∥^p_Lp(P_iⁿ)≥n^1−d, orP_iⁿ̸⊂Ω. Therefore, we have

meas^(︂{ν ̸= 0} \ ^⋃︂

i∈Jn

P_i^n)︂= ^∑︂

i∈I_n\J_n

meas({ν ̸= 0} ∩P_iⁿ)

≤ ^∑︂

i:∥ν∥^p

Lp(P n i)≥n^1−d

meas(P_iⁿ) + ^∑︂

i:P_iⁿ̸⊂Ω

meas(Ω∩P_iⁿ).

For the first term we have

∑︂

i:∥ν∥^p

Lp(P n i)≥n^1−d

meas(P_iⁿ) = ^∑︂

i:∥ν∥^p

Lp(P n i)≥n^1−d

(2n)^−d

≤2^−dn^{−1 ∑︂}

i:∥ν∥^p

Lp(P n i)≥n^1−d

∥ν∥^p_Lp(P_iⁿ)

≤2^−dn⁻¹∥ν∥^p_Lp(Ω) →0

asn→ ∞. The convergence of the second term follows from Ω =^⋃︁n∈N

⋃︁

i:P_iⁿ⊂ΩP_iⁿ. This proves the claim.

For the next lemma, the adaptive choice of the radii a_i,n in(5.16)is crucial.

Lemma 5.3.3. We define the measurable functionν˜n via ν˜n:= ^∑︂

i∈Jn

χ_Tⁿ

i β_i,n, where the real-valued coefficients βi,n satisfy

|β_i,n| ≤ 1 meas(T_iⁿ)

∫︂

P_iⁿ

|ν|dω.

Then there is a constant C >0 (depending only on the domain Ω and p) such that

∥ν˜n∥_H−1(Ω) ≤C∥ν∥^p/2_Lp(Ω)+C∥ν∥_Lp(Ω)

holds for alln∈N.

Proof. Letn∈Nbe fixed. We define ui,n as the solution of

−∆u_i,n=χ_Tⁿ

i −a^d_i,nn^dχ_Oⁿ

i in Ω, u_i,n∈H₀¹(Ω). It follows that

ν˜n=−∆^{(︂ ∑︂}

i∈J_n

βi,nui,n

)︂+ ^∑︂

i∈J_n

a^d_i,nn^dβi,nχOⁿ_i. (5.18) FromLemma 2.2.14 (a)we know that{u_i,n̸= 0} ⊂clOⁿ_i ⊂P_iⁿ and

∥u_i,n∥²_H1

0(Ω)≤C a^2d_i,nL^d(a_i,n)⁻¹,

where the constantC >0 does not depend onnandi. We continue with the boundedness

of ∥ν˜n∥_H−1(Ω). Using the isometry of−∆, (5.18)yields

∥ν˜n∥_H−1(Ω)≤

⃦

⃦

⃦

⃦

∑︂

i∈Jn

βi,nui,n

⃦

⃦

⃦

⃦H₀¹(Ω)

+^⃦⃦⃦

⃦

∑︂

i∈Jn

a^d_i,nn^dβi,nχ_Oⁿ

i

⃦

⃦

⃦

⃦H⁻¹(Ω)

.

Since the functions u_i,n are orthogonal with respect to the inner product in H₀¹(Ω), we have

⃦

⃦

⃦

⃦

∑︂

i∈Jn

βi,nui,n

⃦

⃦

⃦

⃦

2 H₀¹(Ω)

= ^∑︂

i∈Jn

|β_i,n|²∥u_i,n∥²_H1 0(Ω)

≤C ^∑︂

i∈Jn

a²_i,n^dL^d(a_i,n)⁻¹ 1 meas(T_iⁿ)²

(︃∫︂

P_iⁿ

|ν|dω )︃²

≤C ^∑︂

i∈J_n

L^d(a_i,n)^{−1(︃∫︂}

P_iⁿ

|ν|dω )︃²

≤C ^∑︂

i∈Jn

L^d(ai,n)⁻¹ meas(P_iⁿ)^2−2p∥ν∥²_Lp(P_iⁿ)

=C ^∑︂

i∈Jn

L^d(a_i,n)⁻¹ meas(P_iⁿ) avg(P_iⁿ,|ν|^p)^2p−1∥ν∥^p_Lp(Pn i)

=C ^∑︂

i∈Jn

∥ν∥^p_Lp(P_iⁿ)≤C∥ν∥^p_Lp(Ω).

For the other term we have

⃦

⃦

⃦

⃦

∑︂

i∈J_n

a^d_i,nn^dβ_i,nχ_Oⁿ

i

⃦

⃦

⃦

⃦

p H⁻¹(Ω)

≤C

⃦

⃦

⃦

⃦

∑︂

i∈J_n

a^d_i,nn^dβ_i,nχ_Oⁿ

i

⃦

⃦

⃦

⃦

p L^p(Ω)

=C ^∑︂

i∈Jn

(ai,nn)^dp|β_i,n|^p∥χ_Oⁿ

i∥^p_Lp(Ω)

≤C ^∑︂

i∈J_n

n^dp−d (︃∫︂

P_iⁿ

|ν|dω^)︃p

≤C ^∑︂

i∈Jn

n^dp−dmeas(P_iⁿ)^{(1−1p)p∫︂}

P_iⁿ

|ν|^pdω

≤C ^∑︂

i∈Jn

∫︂

P_iⁿ

|ν|^pdω≤C∥ν∥^p_Lp(Ω). This completes the proof.

After we have established this boundedness, we are in a position to prove that every functionν˜ which is pointwise bounded by|ν|can be approximated weakly by functions living on the holes T_iⁿ.

Lemma 5.3.4. Let ν˜ be a function inL^p(Ω)⊂H⁻¹(Ω) such that|ν˜| ≤ |ν|. If we define ν˜n:= ^∑︂

i∈Jn

χ_Tⁿ

i

1 meas(T_iⁿ)

∫︂

P_iⁿ

ν˜ dω, thenν˜n⇀ ν˜ in H⁻¹(Ω).

Proof. Becauseν˜nsatisfies the requirements forLemma 5.3.3, we know thatν˜nis bounded in H⁻¹(Ω). Hence, it suffices to show the convergence on the dense linear subspace C_c^∞(Ω)⊂H₀¹(Ω). Let f ∈C_c^∞(Ω) be given. We have

⃓

⃓

⃓⟨ν˜n−ν˜, f⟩_H−1(Ω)×H¹₀(Ω)

⃓

⃓

⃓=^⃓⃓_⃓^∫︂

Ω(ν˜n−ν˜)fdω^⃓⃓_⃓

≤^⃓⃓_⃓^∑︂

i∈J_n

∫︂

P_iⁿ(ν˜n−ν˜)fdω^⃓⃓_⃓+^⃓⃓_⃓^∫︂

{ν˜̸=0}\⋃︁

i∈JnP_iⁿ

f ν˜ dω^⃓⃓_⃓. The second term converges to 0 because ofLemma 5.3.2 (c). For the first term we can use that f is uniformly continuous. This means that for each ε >0 we can find arbitrarily largen∈Nsuch that|f(ω)−f(ωˆ)|< ε for allω, ωˆ ∈P_iⁿ, i∈Jn. Thus,

⃓

⃓

⃓

⃓

∑︂

i∈Jn

∫︂

P_iⁿ(ν˜n−ν˜)fdω

⃓

⃓

⃓

⃓

≤ ^∑︂

i∈Jn

⃓

⃓

⃓

⃓ 1 meas(T_iⁿ)

∫︂

P_iⁿ

ν˜(ω) dω

∫︂

T_iⁿ

f(ωˆ) dωˆ−

∫︂

P_iⁿ

f(ω)ν˜(ω) dω

⃓

⃓

⃓

⃓

≤ ^∑︂

i∈Jn

∫︂

P_iⁿ

|ν˜(ω)|

⃓

⃓

⃓

⃓ 1 meas(T_iⁿ)

∫︂

T_iⁿ

|f(ωˆ)−f(ω)|dωˆ^⃓⃓⃓

⃓dω

≤ε ^∑︂

i∈Jn

∫︂

P_iⁿ

|ν˜|dω ≤ε∥ν˜∥_L1(Ω).

Sinceεcan be arbitrarily small, this proves that ⟨ν˜n−ν˜, f⟩_H−1(Ω)×H₀¹(Ω) →0 asn→ ∞. If we use the above lemma withν˜ = ν we get an explicit description of the sequence {ν_n}_n∈N that converges weakly toν.

The next lemma provides some technical estimates, which will be used to verify the assumptions of Theorem 5.3.1 in our setting with differently sized holes. Again, the specific choice(5.16)is crucial for these estimates.

Lemma 5.3.5. Let the size of the holesT_iⁿ be chosen according to (5.16). Then, there exists a constantC >0, such that

∑︂

i∈Jn

L^d(a_i,n)≤C∥ν∥^2−p_Lp(Ω), (5.19a) n^{d−2 ∑︂}L^d(ai,n)² →0, (5.19b)

n^{d q−d ∑︂}

i∈Jn

L^d(ai,n)^q≤C∥ν∥^p_Lp(Ω), (5.19c) whereq=p/(2−p).

Proof. We start with (5.19a). Using the definition (5.16) and meas(P_iⁿ) = (2/n)^d, we find

∑︂

i∈Jn

L^d(ai,n) = ^∑︂

i∈Jn

meas(P_iⁿ) avg(P_iⁿ,|ν|^p)^p2−1= ^∑︂

i∈Jn

meas(P_iⁿ)^{2−p2(︂∫︂}

P_iⁿ

|ν|^pdω^)︂

2 p−1

= 2^2d(1−p1)n^{d(2p−2) ∑︂}

i∈Jn

(︂∫︂

P_iⁿ

|ν|^pdω^)︂

2 p−1

.

Since ²_p −1∈(0,1), we can use Holder’s inequality to obtain

∑︂

i∈Jn

L^d(a_i,n)≤2^2d(1−1p)n^d(2p−2) (︃

∑︂

i∈Jn

∫︂

P_iⁿ

|ν|^pdω )︃²_p⁻¹

(︂ ∑︂

i∈Jn

1^)︂2−

2 p

≤C n^{d(2p−2)+d(2−2p)}∥ν∥^p(

2 p−1)

L^p(Ω) ≤C∥ν∥^2−p_Lp(Ω). This shows(5.19a).

Next, we verify(5.19b). We use (5.17)and (5.19a)and obtain n^{d−2 ∑︂}

i∈Jn

L^d(ai,n)²≤n^{d−2 ∑︂}

i∈Jn

L^d(ai,n) 2^d+1n^1−d= 2^d+1n^{−1 ∑︂}

i∈Jn

L^d(ai,n)→0. Finally, we address (5.19c). Using (5.16)and q(2/p−1) = 1 we get

n^{d q−d ∑︂}

i∈Jn

L^d(a_i,n)^q=n^{d q−d ∑︂}

i∈Jn

meas(P_iⁿ)^{q−1 ∫︂}

P_iⁿ

|ν|^pdω ≤2^d(q−1)∥ν∥^p_Lp(Ω).

As a next step, we verify that the conditions(H.1)to(H.5’)ofTheorem 5.3.1are satisfied for the above choice of the perforated domain Ωn. We are following the strategy of [Cioranescu, Murat, 1997, Theorem 2.2]. However, due to the variable size of the holes, the analysis is more involved.

We start by defining an appropriate v_n∈H¹(Ω). Fori∈J_n letv_i,n∈H₀¹(Ω) be defined as the solution to

v_i,n= 1 inT_iⁿ,

−∆v_i,n= 0 inOⁿ_i \T_iⁿ, v_i,n= 0 in Ω\O_iⁿ.

Functions of this type are discussed inLemma 2.2.14 (b). Note that the requirements on ai,nin this lemma are satisfied byLemma 5.3.2for n∈Nlarge enough. We then define

v_n:= 1− ^∑︂

i∈J_n

v_i,n.

The next two lemmas show that (H.1)to(H.5’) are satisfied.

Lemma 5.3.6. The conditions(H.1)to(H.3)are satisfied by the above choice of{v_n}_n∈N. Proof. The conditions (H.1)and (H.2)follows directly from our choice of v_n.

Let us show that {v_n}_n∈N is a bounded sequence in H¹(Ω). Because of 0 ≤vn ≤1 it suffices to calculate^⃦⃦|∇v_n|^⃦⃦

2

H₀¹(Ω). Due toLemma 2.2.14 (b)and(5.19a) we have

⃦

⃦|∇v_n|^⃦⃦

2≤ ^∑︂

i∈Jn

∥v_i,n∥²_H1

0(Ω)≤C ^∑︂

i∈Jn

L^d(ai,n)≤C∥ν∥^2−p_Lp(Ω), (5.20) which shows that the sequence {v_n}_n∈N is bounded in H¹(Ω). We check that the convergencev_n→1 holds in L¹(Ω). Indeed,

∥v_n−1∥_L1(Ω)= ^∑︂

i∈J_n

∫︂

Oⁿ_i

|v_i,n|dω ≤C ^∑︂

i∈J_n

1

nL^d(a_i,n)→0,

where we used(2.22)and(5.19a). Together with the boundedness of{v_n}_n∈NinH¹(Ω)↪→ L¹(Ω) and the reflexivity ofH¹(Ω), this impliesvn⇀1 in H¹(Ω). This shows(H.3).

We remark that(5.20)shows that the capacity of the holes ^⋃︁i∈JnT_iⁿ remains bounded.

Indeed, the function 1−vn can be used inLemma 2.6.16 and we obtain cap^{(︂ ⋃︂}

i∈Jn

T_i^n)︂≤ ∥1−vn∥²_H1

0(Ω)≤C∥ν∥^2−p_Lp(Ω)

Lemma 5.3.7. The conditions (H.4’) and (H.5’) are satisfied by the above choice of {v_n}_n∈Nand some sequences{µ_n}_n∈Nand{γ_n}_n∈N. In particular, we haveµ=C_d|ν|^2−p, whereCd= max(1, d−2)Sd andSd denotes the surface measure of the boundary of the d-dimensional unit ball B1(0)⊂R^d.

Proof. First, we prove(H.5’). We note that ∆v_n only acts on the boundaries ∂T_iⁿ and

∂Oⁿ_i. We set γn, µn ∈ H⁻¹(Ω) such that −∆vn = µn−γn and µn only acts on ∂Oⁿ_i whereas γ_n only acts on ∂T_iⁿ. Then it can be seen that the condition⟨γ_n, z_n⟩= 0 is true for allz_n∈H₀¹(Ωn). It is possible to explicitly calculateµ_n. We denote byδ_i,n∈H⁻¹(Ω) the surface measure on∂Oⁿ_i, i.e.,

⟨δ_i,n, f⟩_H−1(Ω)×H¹(Ω)=^∫︂ f(s) ds ∀f ∈C_c^∞(Ω).

Then, using integration by parts and(2.23), it turns out that µ_n= ^∑︂

i∈Jn

∂v_n

∂n

⃓

⃓

⃓

⃓_∂On i

δ_i,n=−^∑︂

i∈Jn

∂v_i,n

∂n

⃓

⃓

⃓

⃓_∂On i

δ_i,n= ^∑︂

i∈Jn

1

n dα_i,nδ_i,n (5.21) holds, where _∂n^∂vˆ⃓⃓⃓_∂On

i

denotes the outer normal derivative of a function vˆ on∂O_iⁿ and the coefficients α_i,nare given by

α_i,n:= max(1, d−2)n^dd

L^d(ai,n)⁻¹−L^d(1/n)⁻¹. (5.22) For later use we note that Lemma 5.3.2 (b)implies the existence of a constant C >0 independent of iand nsuch that

0≤αi,n≤Cn^dL^d(ai,n). (5.23) Now we introduce the function z_i,n fori∈J_n as the solution of the equation

−∆z_i,n=α_i,n inOⁿ_i, z_i,n= 0 on Ω\Oⁿ_i. This function can be calculated explicitly and we find

z_i,n(ω) = α_i,n 2d

(︁n⁻²− |ω−ωⁿ_i|^2)︁ ∀ω∈Oⁿ_i and

−∆z_i,n=α_i,nχ_Oⁿ

i − 1

n dα_i,nδ_i,n∈H⁻¹(Ω). (5.24) For the H₀¹(Ω)-norm ofz_i,n we can calculate

∥z_i,n∥²_H1

0(Ω)=C α²_i,nn^−d−2 and because of the orthogonality we have

⃦

⃦

⃦

∑︂

i∈Jn

zi,n

⃦

⃦

⃦

2

H₀¹(Ω)= ^∑︂

i∈Jn

∥z_i,n∥²_H1

0(Ω)=C ^∑︂

i∈Jn

α²_i,nn^−d−2≤C ^∑︂

i∈Jn

L^d(ai,n)²n^d−2 →0 due to (5.23)and (5.19b). Hence,(5.21)and (5.24) imply

µ_n− ^∑︂

i∈J_n

α_i,nχ_Oⁿ

i = ∆^{(︂ ∑︂}

i∈J_n

z_i,n^)︂→0 in H⁻¹(Ω) (n→ ∞).

Using Lemma 5.3.8 below yields µn → µ in H⁻¹(Ω), where µ := Cd|ν|^2−p. Finally, γn⇀ µfollows from−∆vn⇀0 andµn→µ, which completes the proof of(H.5’).

Now,µ=Cd|ν|^2−p,ν ∈L^p(Ω) and the bounds onp imply µ∈L^p/(2−p)(Ω)⊂

{︄W^−1,2+ε(Ω) if d= 2, W^−1,d(Ω) ifd≥3 for someε >0. Thus, the remaining condition (H.4’)follows.

It remains to check the announced convergence ofµn towardsµ=Cd|ν|^2−p. Lemma 5.3.8. Letαi,nbe defined as in (5.22). Then we have the convergence

∑︂

i∈Jn

αi,nχOⁿ_i →Cd|ν|^2−p inH⁻¹(Ω), whereC_d is the constant defined inLemma 5.3.7.

Proof. We will prove this by showing the weak convergence inL^q(Ω), whereq=p/(2−p)∈ (1,∞). Indeed, the boundedness follows from

⃦

⃦

⃦

⃦

∑︂

i∈Jn

αi,nχOⁿ_i

⃦

⃦

⃦

⃦

q L^q(Ω)

= ^∑︂

i∈Jn

α^q_i,nmeas(O_iⁿ)≤C ^∑︂

i∈Jn

n^{q d}L^d(ai,n)^qn^−d≤C∥ν∥^p_Lp(Ω), where the last two inequalities are due to (5.23) and (5.19c), respectively. Thus, it is sufficient to show the weak convergence on the dense subsetC_c^∞(Ω)⊂L^q(Ω)^⋆. Due to the definition ofα_i,nand Lemma 5.3.2 (b) we have

⃓

⃓αi,n−max(1, d−2)dn^dL^d(ai,n)^⃓⃓

= max(1, d−2)dn^d⃓⃓_⃓ 1

L^d(ai,n)⁻¹−L^d(1/n)⁻¹ −L^d(a_i,n)^⃓⃓_⃓

≤(C_n−1) max(1, d−2)d n^dL^d(a_i,n),

where{C_n}_n∈N is a sequence of constants such thatC_n→1. It follows that

⃦

⃦

⃦

⃦

∑︂

i∈Jn

α_i,nχ_Oⁿ

i −max(1, d−2)d n^{d ∑︂}

i∈Jn

L^d(a_i,n)χ_Oⁿ

i

⃦

⃦

⃦

⃦

q L^q(Ω)

≤C(Cn−1)^{q ∑︂}

i∈Jn

n^dq−dL^d(ai,n)^q ≤C(Cn−1)^q∥ν∥^p_Lp(Ω)→0,

(5.25)

where we used (5.19c)again. Now usingLemma 5.3.2 (c) we also have

⃦

⃦

⃦

⃦

∑︂

i∈I_n\J_n

avg(P_iⁿ,|ν|^p)^2p−1χO_iⁿ

⃦

⃦

⃦

⃦

q L^q(Ω)

= ^∑︂

i∈I_n\J_n

avg(P_iⁿ,|ν|^p)^(2p−1)qmeas(O_iⁿ)

≤

∫︂

|ν|^pdω→0

asn→ ∞. By combining this with(5.25) and the definition(5.16) ofai,n we arrive at

∑︂

i∈Jn

α_i,nχ_Oⁿ

i −max(1, d−2)d2^d∑︂

i∈In

avg(P_iⁿ,|ν|^p)^2p−1χ_Oⁿ

i →0

in L^q(Ω). Let f ∈C_c^∞(Ω) be given. Using the uniform continuity off ∈C_c^∞(Ω) (similar to the proof of Lemma 5.3.4) it is possible to replace χ_Oⁿ

i withχ_Pⁿ

i , i.e.

⟨︂ ∑︂

i∈Jn

αi,nχOⁿ_i −max(1, d−2)S_d^∑︂

i∈In

avg(P_iⁿ,|ν|^p)^2p−1χP_iⁿ, f^⟩︂

H⁻¹(Ω)×H₀¹(Ω)→0, where we used that 2^−dd⁻¹Sd= meas(O_iⁿ)/meas(P_iⁿ), andd⁻¹Sd is the volume of the d-dimensional unit ball. Now we apply Lemma 2.2.4 (b)tov=|ν|^p. As a consequence, we have

Cd

∑︂

i∈I_n

avg(P_iⁿ,|ν|^p)^2p−1χP_iⁿ →Cd|ν|^2−p

in L^q(Ω) with the constantCd= max(1, d−2)Sd. Combined with the calculations above, we have

⟨︂ ∑︂

i∈Jn

α_i,nχ_Oⁿ

i −C_d|ν|^2−p, f^⟩︂

H⁻¹(Ω)×H₀¹(Ω)

→0.

The boundedness inL^q(Ω) of^∑︁i∈Jnαi,nχO_iⁿ and the compact embedding intoH⁻¹(Ω) (which follows from q > p, see also [Adams, Fournier, 2003, Theorem 6.3]) completes the

proof.

We note that choosing the constant functions ν:=^(︁2^dC0)︁−1/(2−p)

ifd= 2, ν :=^(︁2^dC₀^{2−d)︁−1/(2−p)}ifd≥3

yields the same size of the holes as in [Cioranescu, Murat, 1997, (2.4)] and we obtain the same value of µ, cf. [Cioranescu, Murat, 1997, (2.3)].

Now, the assumptions ofTheorem 5.3.1are satisfied and therefore we can finally construct our sequence {w_n}_n∈N⊂H₀¹(Ω) that converges weakly to w.

Lemma 5.3.9. There exists a sequence{w_n}_n∈Nwithw_n∈H₀¹(Ωn)⊂H₀¹(Ω) such that w_n⇀ win H₀¹(Ω).

Proof. We choosewn∈H₀¹(Ωn)⊂H₀¹(Ω) as the solution of

−∆wn=−∆w+µw inH⁻¹(Ωn),

where µ= C_d|ν|^2−p and the constant C_d is defined as in Lemma 5.3.7. Then we can apply Theorem 5.3.1, whose assumptions are satisfied due to Lemmas 5.3.6and 5.3.7.

This yields the weak convergencew_n⇀ wˆ where wˆ ∈H₀¹(Ω) is the unique solution of the

partial differential equation

−∆wˆ +µwˆ =−∆w+µw.

Then the claim follows fromwˆ =w.

Im Dokument OPUS 4 | On bilevel optimization problems in infinite-dimensional spaces (Seite 172-183)