The Case of a Torus

In this section, we prove a primal version of the Bourgain-Brézis type estimate on the 2-torus Π², which we simply denote by Πin the following. To be precise, we show the following statement.

Theorem 2.2.1. Let Π be the 2-torus and 2 < q < ∞. Then there exists a constant C > 0 such that for all functions f ∈L²(Π)∩L^q(Π)satisfying ´

Πf = 0there exists a functionF ∈L^∞(Π;R²)∩ H¹(Π;R²)∩W^1,q(Π;R²)such that

(i) divF =f,

(ii) kFkL^∞ ≤Ckfk_L2,

(iii) kFkH¹≤CkfkL², (iv) kFk_W1,q ≤CkfkL^q.

Remark 2.2.1. The result by Bourgain and Brézis in [11, Theorem 1] is the same without the assumptionf ∈L^q and the resulting estimate forF in L^q. In two dimensions, the same result holds true for the curl-operator as the operators div andcurl are linked by a rotation of the vector fields.

Hence, the result can be understood as a characterization of the failure of the embeddingH¹toL^∞in two dimensions. A function inH¹-function can be decomposed such that the part of the decomposition which is not controlled inL^∞is a gradient.

Remark 2.2.2. Statements of this type in the pureL²-case hold for a more general class of operators.

In [12, Theorem 10] it is shown that it is sufficient that for an operatorS :W^1,n(Πⁿ,R^r)→X with closed range, where X is a Banach space, there exists for each1 ≤s≤r an index1 ≤is≤d such that for all functions f ∈W^1,n(Πⁿ,R^r)it holds that

kSfk ≤C max

1≤s≤rmax

i6=is

k∂ifsk_Ln

n1

n2

2j−12j 2j+1 2j+2 2j

2j+1 2j+2

Λ1j

Λ1j Λ1j+1

Λ1j+1 Λ1j+2

Λ1j+2

Figure 2.3: Sketch of Λ¹.

to guarantee that for anyf ∈W^1,n(Πⁿ,R^r) there exists g ∈W^1,n(Πⁿ,R^r)∩L^∞(Πⁿ,R^r) satisfying S(f) =S(g) and corresponding bounds. Clearly, this condition holds true for thediv-operator. The fact that the operator is blind for the derivatives∂i_sfs allows to insert oscillations in this particular direction.

Remark 2.2.3. Moreover, Bourgain and Brézis show that the correspondance of f to a solution F ∈ H¹∩L^∞ of divF = f which satisfies the bounds of the theorem cannot be linear, see [11, Proposition 2].

We prove this result following the ideas presented in the proof of Theorem 1 in [11].

The main ingredient to prove Theorem 2.2.1 is the following lemma which gives a first approximation to Theorem 2.2.1. It shows that the equationdivF =f can be almost solved by a functionF which satisfies estimates with a good linear term and a bad nonlinear term.

Lemma 2.2.2(Nonlinear approximation). Let Πbe the 2-torus and2 < q <∞. There existsc >0 such that for allf ∈L²(Π)∩L^q(Π)satisfyingkfkL² ≤c and´

Πf = 0the following holds:

For everyδ >0 there existCδ>0 andF ∈L^∞(Π;R²)∩H¹(Π;R²)∩W^1,q(Π;R²)such that (i) kFk_L^∞ ≤C_δ,

(ii) kFk_H1≤C_δkfk_L2,

(iii) kdivF−fk_L2 ≤δkfk_L2+Cδkfk²_L2, (iv) kFk_W1,q ≤CδkfkL^q,

(v) kdivF−fkL^q ≤δkfk_L2+Cδkfk_L2kfkL^q. Proof. Letf ∈L²(Π)∩L^q(Π)such that´

Πf = 0 andkfkL² ≤c wherec >0 will be fixed later.

Consider the following decomposition ofZ²\ {0}, see Figure 2.3, Λ¹_j=

2^j−1<|n1| ≤2^j;|n2| ≤2^j andΛ²_j =

2^j−1<|n2| ≤2^j;|n1| ≤2^j−1 forj∈N. (2.5)

n1

n2

Λ1j−2

Λ1j−1

Λ1j

I2 j aj,2

2j−3 2j−2 2j−1 2j

2j−2 2j−1 2j

Λ1j,1 Λ1j,2 Λ1j,3 Λ1j,4

Figure 2.4: Sketch of the subdivision ofΛ¹into the stripesΛ¹_j,r for positiven₁.

For α= 1,2 set Λ^α =S

jΛ^α_j. Correspondingly, let f^α =P_Λαf =P

n∈Λ^αfˆ(n)e^in·x and decompose f =f¹+f². In the following, we construct functionsYα: Π→Rwhich satisfy

1. kYαkL^∞ ≤Cδ, 2. kYαkH¹ ≤C_δkfkL²,

3. k∂αYα−f^αk_L2 ≤δkfk_L2+Cδkfk²_L2, 4. kYαk_W1,q ≤CδkfkL^q,

5. k∂αY_α−f^αkL^q ≤δkfkL²+C_δkfkL²kfkL^q.

Without loss of generality we may assume thatf =f¹ and construct onlyY1. Let us define

f_j =P_Λ1

jf = X

n∈Λ¹_j

fˆ(n)e^in·xandF_j=X

n

1 n1

fˆ_j(n)e^in·x= X

n∈Λ¹_j

1 n1

fˆ(n)e^in·x.

Moreover, fix a smallε >0and subdivide Λ¹_j in stripes of length∼ε2^j−1by setting Λ¹_j= [

0≤r≤2bε⁻¹c+1

Λ¹_j,r,

where for 0≤r≤ bε⁻¹cwe setΛ¹_j,r =I_j^r×[−2^j,2^j]whereas forbε⁻¹c+ 1≤r≤2bε⁻¹c+ 1we set Λ¹_j,r =I_j^{r−bε−1c−1}×[−2^j,2^j] where I_k^r and J_k^r are defined as in (2.1) in the previous section. For a sketch of the situation, see Figure 2.4.

Next, define

F˜_j(x) =X

r

X

n∈Λ¹_j,r

1 n1

fˆ_j(n)e^in·x .

The main property ofF˜j is the smallness of its partial derivative inx1-direction.

In fact, we can rewriteF˜j(x) =P

r

P

n∈Λ¹_j,r 1 n1

fˆj(n)e^in·xe^−iaj,rx1

where aj,r is the left endpoint of I_j^r, respectively the right endpoint ofJ_j^{r−bε−1c−1}. Differentiation leads to

|∂1F˜j|=X

r

X

n∈Λ¹_j,r

n1−aj,r

n₁

fˆj(n)e^in·x

=X

r

X

n∈Λ¹_j,r

mε(n1) ˆfj(n)e^in·x

, (2.6)

wheremεis the the special function defined in (2.2) in the previous section. As0≤mε≤εand using Plancharel’s identity and Hölder’s inequality for the sum overr, we derive that

k∂1F˜jk_L2 ≤Cε⁻¹² X

r

X

n∈Λ¹_j,r

mε(n1) ˆf(n)e^in·x L²

≤Cε¹²kfjk_L2.

For anL^q-version of this estimate, we will later use Proposition 2.1.4 and Lemma 2.1.5.

As we also need an appropiate localization in Fourier space ofF˜_j, let us recall that the n-th one-dimensional Féjer-kernel is given by

K_n(t) = X

|k|<n

n− |k|

n e^ikt= 1 n

1−cos(nt) 1−cos(t) ≥0.

If we define

Gj = 9 ˜Fj∗(K₂j+1⊗K₂j+1),

we obtain by the properties of the Fejéjer kernel discussed in the beginning of the previous section that

supp ˆG_j⊂[−2^j+1,2^j+1]×[−2^j+1,2^j+1]⊂ {|n| ≤C2^j} and|F_j| ≤ |F˜_j| ≤G_j. (2.7) Moreover, in the proof of [11, Theorem 1] it is shown that

kGjkL^∞≤9kF˜jk_∞≤Ckfjk_L2, (2.8) kGjk_L2≤Cε⁻¹²2^−jkfjk_L2, (2.9) k∂1Gjk_L2≤Cε¹²kfjk_L2, (2.10)

k∇GjkL²≤Cε⁻¹²kfjkL². (2.11)

Let us only prove (2.8), the rest can be proved similarly:

|F˜_j(x)| ≤X

r

X

n∈Λ¹_j,r

| 1 n1

fˆ(n)| ≤2^−j+1 X

n∈Λ¹_j

|fˆ(n)| ≤C



 X

n∈Λ¹_j

|fˆ(n)|²





1 2

=Ckfjk_L2.

As in [11], we define

Y1=X

j

Fj

Y

k>j

(1−Gk).

By (2.7) and (2.8), it holds|Fj| ≤Ckfjk_L2 ≤Ckfk_L2. We assume thatkfk_L2, respectively cin the

formulation of the theorem, is so small that Ckfk_L2 <1. Then, one can show that

|Y₁| ≤X

j

|F_j|Y

k>j

(1− |F_k|)≤1. (2.12)

Another calculation, see [11, equation (5.19)], shows that Y₁=X

j

F_j−X

j

G_jH_j,

where

H_j=X

k<j

F_k Y

k<l<j

(1−G_l).

Thus,

∂₁Y₁=X

j

f_j−X

j

∂₁(G_jH_j) =f−X

j

∂₁(G_jH_j). (2.13)

Moreover, by definition ofH_j andF_j and (2.7) it can be seen that

|H_j| ≤1, supp ˆH_j⊂ {|n| ≤C2^j}, P_k(G_jH_j) = 0 for allk > j+m, respectivelyGjHj= X

k≤j+m

Pk(GjHj), (2.14)

where the Pk are smooth Littlewood-Paley-projections on{|n| ∼2^k} as discussed at the end of the previous section and mis independent of j. In [11, proof of Theorem 1], Bourgain and Brézis show, using (2.7) - (2.14), that

k∂1Y1−fk_L2 ≤Clog(ε⁻¹)

ε¹²kfk_L2+ε⁻¹²kfk²_L2

andkY1k_H1 ≤Cεkfk_L2. (2.15) Hence, properties 1.–3. for Y1 are already shown. In what follows, we adopt the ideas of their proof to show the corresponding estimates in L^q i.e., properties 4. and 5., forY1.

First, we estimate

k∇Y1kL^q≤ k∇X

j

F_jkL^q+k∇X

j

G_jH_jkL^q. (2.16)

For the first term on the right hand side, we observe that

X

j

∇F_{j Lq}

=

X

j

X

n∈Λ¹_j

n n₁

fˆ(n)e^in·x L^q

≤C

X

j

X

n∈Λ¹_j

fˆ(n)e^in·x L^q

=Ckfk_Lq. (2.17)

Note that we used for the first inequality that _nⁿ

11^S

jΛ¹_j is anL^p-multiplier. This can be shown by multiplier transference and the Marcinkiewicz multiplier theorem (note that inΛ_j the second variable n2 is controlled by 2n1). Next, we estimate the second term of the right hand side of (2.16). Using (2.14) and classical Littlewood-Paley estimates as discussed at the end of the previous section, we

obtain

X

j

∇(GjHj) _Lq

≤C





 X

k

Pk

X

j

∇(GjHj)

2





1 2

L^q

. (2.18)

Note that the operator∇can also be seen as a Fourier multiplication operator. Hence, it commutes with the Littlewood-Paley-projectionsPk. In particular, the localization in Fourier space for GjHj

in (2.14) also holds for ∇G_jH_j. The triangle inequality and rewriting with the change of variables j→k+syield

≤C X

s≥−m

X

k

|Pk∇(Gk+sH_k+s)|²

!¹₂ L^q

. (2.19)

The changek→k−sleads to

=C X

s≥−m



 X

k≥s

|P_k−s∇(GkHk)|²





1 2

_Lq

. (2.20)

The Littlewood-Paley inequality for gradients yields

≤C X

s≥−m

2^−s





 X

k

|2^kGk Hk

|{z}

|·|≤1

|²







1 2

_Lq

(2.21)

≤C X

s≥−m

2^−s



 X

j

|2^kGk|²





1 2

_Lq

. (2.22)

By definition,G_k is the convolution ofF˜_k with a Fejér kernel. Applying Proposition 2.1.2 leads to

≤C X

s≥−m

2^−s

X

k

|2^kF˜k|²

!¹_{2 Lq}

. (2.23)

=C X

s≥−m

2^−s





 X

k



 X

r≤2bε⁻¹c−1

X

n∈Λ¹_k,r

2^k n₁

fˆ(n)e^in·x





2





1 2

_Lq

. (2.24)

Using Hölder’s inequality for the sum overr yields

≤Cε⁻¹² X

s≥−m

2^−s





 X

k

X

r≤2bε⁻¹c−1

X

n∈Λ¹_k,r

2^k n1

fˆ(n)e^in·x

2





1 2

_Lq

. (2.25)

Now, we use a one-sided Littlewood-Paley-type inequality for non-dyadic decompositions which goes back to Rubio de Francia, [69, Theorem 8.1]. For the case of a torus, see [41, Theorem 2.5] or [10] for the dual statement. The statement is the following: Forq >2there exists a constantC >0such that for all partitions ofZinto intervals(Ik)k it holds that

(X

k

|Skf|²)¹² L^q(Π¹)

≤Ckfk_Lq(Π¹),

where S_kf =P

l∈I_kfˆ(l)e^il·x. We use this inequality in the first variable for the decomposition of the n1-axis given byΛ¹_k,r, respectivelyI_k^randJ_k^r:

(2.25)≤Cε⁻¹² X

s≥−m

2^−s

X

k,r≤2bε⁻¹c−1

X

n∈Λ¹_k,r

2^k n1

fˆ(n)e^in·x L^q

. (2.26)

Finally, we use that we know from Proposition 2.1.4 and Lemma 2.1.5 that P

k 2^k n11_Λ1

k is an L^p -multiplier to obtain

≤Cε⁻¹² X

s≥−m

2^−s

X

k,r≤2bε⁻¹c−1

X

n∈Λ¹_k,r

| {z }

P

n∈Λ1

fˆ(n)e^in·x _Lq

(2.27)

=Cε⁻¹² X

s≥−m

2^−skfk_Lq (2.28)

≤Cε⁻¹²kfk_Lq. (2.29)

Collecting (2.16), (2.17), and (2.18) - (2.29) leads to

k∇Y1k_Lq ≤Cε⁻¹²kfk_Lq. As we may assume without loss of generality that ´

ΠY1= 0, this implies

kYk_W1,q ≤Cε⁻¹² kfk_Lq. (2.30)

Hence, it is left to prove property 5. forY1. By (2.13), it remains to control ∂1P

j(GjHj)

_Lq. As in

(2.18)–(2.20), we can estimate

∂1

X

j

GjHj

_Lq

≤ X

s≥−m



 X

j

|P_j−s∂1(GjHj)|²





1 2

_Lq

.

Now, fixs∗∈Nand estimate fors > s∗as in (2.20)–(2.28)



 X

j

|P_j−s∂₁(G_jH_j)|²





1 2

_Lq

≤Cε⁻¹²2^−skfk_Lq. (2.31)

Fors≤s_∗ we estimate, using that |Hj| ≤1,



 X

j

|P_j−s∂₁(G_jH_j)|²





1 2

_Lq

≤C



 X

j

|∂₁G_j|²





1 2

_Lq

+C



 X

j

|G_j∂₁H_j|²





1 2

_Lq

. (2.32)

AsGj is the convolution ofF˜j with a Fejér kernel, we may apply Proposition 2.1.2 to the first term on the right hand side to derive



 X

j

|∂1Gj|²





1 2

_Lq

≤C



 X

j

|∂1F˜j|²





1 2

_Lq

.

Equation (2.6) and Hölder’s inequality for the sum overrlead to

≤Cε⁻¹²





 X

j

X

r≤2bε⁻¹c−1

X

n∈Λ¹_j,r

mε(n1) ˆf(n)e^in·x

2





1 2

_Lq

.

Using the Rubio-de-Francia-inequality for arbitrary intervals in the first variable as in (2.25)–(2.26) yields

≤Cε⁻¹²

X

j

X

r≤2bε⁻¹c−1

X

n∈Λ¹_j,r

mε(n1) ˆf(n)e^in·x L^q

.

By the improvement of the Marcinkiewicz multiplier theorem due to Coifman, de Francia, and Semmes, Proposition 2.1.4 and Lemma 2.1.5, the functionm_ε defines a multiplier whose associated operator-norm fromL^q toL^q can be estimated byCrε^r−1r for anyrsuch that|¹₂−¹_q|< ¹_r. In particular, there existsr >2such that

≤Cε^{r−1r −12}

X

n∈Λ¹

fˆ(n)e^in·x L^q

=Cε^{r−1r −12}kfk_Lq. (2.33)

For the second term of the right hand side of (2.32), note that in [11] Bourgain and Brézis show that k∇H_jk_L∞ ≤2^jkfk_L2.

Hence, we can estimate



 X

j

|G_j∂₁H_j|²





1 2

_Lq

≤



 X

j

|2^jG_j|²





1 2

_Lq

kfk_L2.

The right hand side can now be treated as in (2.22)–(2.28) to obtain



 X

j

|Gj∂1Hj|²





1 2

_Lq

≤Cε⁻¹²kfk_Lqkfk_L2. (2.34)

Collecting (2.31), (2.33) and (2.34) yields

∂1

X

j

GjHj

_Lq

≤C2^−s∗ε⁻¹²kfk_Lq+ X

−m≤s≤s∗

ε⁻¹²kfk_Lqkfk_L2+ε^{r−1r −12} kfk_Lq

. (2.35)

Eventually, choose s_∗such that2^−s∗∼ε. Then (2.35) provides

∂₁X

j

G_jH_{j Lq}

≤Clog(ε⁻¹)

ε^{r−1r −12}kfk_Lq+ε⁻¹²kfk_Lqkfk_L2

. (2.36)

Here, we used that ε¹² ≤ε^{r−1r −12} forε < 1. Notice that r >2 and therefore log(ε⁻¹)ε^{r−1r −12} →0 as ε→0. Comparing with (2.15) and (2.30) shows that for givenδ >0the properties 1.–5. ofY can be achieved forε >0 small enough. This finishes the proof.

Remark 2.2.4. Note that the explicit construction given in the proof is nonlinear which is in accor-dance with Remark 2.2.3.

From the nonlinear estimate we can now derive a linear estimate.

Lemma 2.2.3 (Linear estimate). Let Πbe the 2-torus and2 < q <∞. Then for every δ >0 there exists a constant Cδ > 0 such that for every function f ∈ L²(Π)∩L^q(Π) satisfying ´

Πf = 0 there existsF ∈L^∞(Π;R²)∩H¹(Π;R²)∩W^1,q(Π;R²)such that

(i) kFkL^∞ ≤Cδkfk_L2, (ii) kFk_H1 ≤Cδkfk_L2, (iii) kdivF−fkL²≤δkfkL²,

(iv) kFkW^1,q ≤CδkfkL^q, (v) kdivF−fkL^q≤δkfkL^q.

Proof. As we want to prove a linear estimate, we may assume without loss of generality that it holds kfkL² =δC_δ⁻¹< c where c >0 is the constant from Lemma 2.2.2. The application of Lemma 2.2.2 provides the existence of F ∈L^∞(Π;R²)∩H¹(Π;R²)∩W^1,q(Π;R²)such that

(i) kFkL^∞≤Cδ =δ⁻¹C_δ²kfk_L2, (ii) kFk_H1 ≤C_δkfk_L2,

(iii) kdivF−fkL² ≤δkfkL²+C_δkfk²_L2 = 2δkfkL², (iv) kFkW^1,q ≤CδkfkL^q,

(v) kdivF−fkL^q ≤δkfkL^q+Cδkfk_L2kfkL^q = 2δkfkL^q. Now, takeδ˜= 2δandC˜δ=δ⁻¹C_δ².

Armed with this approximation we are now able to prove Theorem 2.2.1 by iterating this approxi-mation.

Proof of Theorem 2.2.1. Letf ∈L²(Π)∩L^q(Π)such that´

Πf = 0. We apply Lemma 2.2.3 forδ= ¹₂. Hence, there existsF1 such that

• kF1kL^∞ ≤C1 2kfk_L2,

• kF₁k_H1≤C1 2kfk_L2,

• kdivF₁−fk_L2 ≤¹₂kfk_L2,

• kF1kW^1,q ≤C1 2kfkL^q,

• kdivF1−fkL^q ≤¹₂kfkL^q.

We defineFi fori≥2inductively: letf˜i=f−divPi−1

j=1Fj. Note that by the periodicity of theFj it holds´

Πf˜i= 0. Reapplication of Lemma 2.2.3 forδ= ¹₂ andf˜iprovides the existence ofFisuch that (i) kFikL^∞≤C1

2kf−divPi−1

j=1Fjk_L2 ≤C1

2(¹₂)ⁱ⁻¹kfk_L2, (ii) kFik_H1 ≤C1

2kf−divPi−1

j=1Fjk_L2≤C1

2(¹₂)ⁱ⁻¹kfk_L2, (iii) kdivF_i+ divPi−1

j=1F_j−fk_L2 ≤¹₂kdivPi−1

j=1F_j−fk_L2≤(¹₂)ⁱkfk_L2, (iv) kFikW^1,q ≤C1

2kf−divPi−1

j=1F_jkL^q ≤C1

2(¹₂)ⁱ⁻¹kfkL^q, (v) kdivFi+ divPi−1

j=1Fj−fkL^q ≤ ¹₂kdivPi−1

j=1Fj−fkL^q ≤(¹₂)ⁱkfkL^q. Define F =P∞

j=1Fj. Then, divF =f and the claimed estimates follow by the triangle inequality withC= 2C1

2.

Im Dokument Plasticity as the Γ-limit of a Dislocation Energy Dissertation (Seite 32-41)