Lipschitz Domains

As f = 0 on ∂Q, we obtain that g(0) =g(1) = 0. Moreover, it holds ´1

0 g(y)dy = 0. Furthermore, one can check, using Jensen’s inequality, that

kgk_L^∞_(0,1)≤ kfk_L^∞_(Q),kgk_H1(0,1)≤ kfk_H1(Q)andkgk_W1,q(0,1)≤ kfk_W1,q(Q). In addition, let us define

Z(y) = ˆ y

0

g(t)dt.

Clearly,Z(0) =Z(1) = 0andZ satisfies the estimates

kZk_L∞(0,1)≤ kgk_L∞(0,1)≤ kfk_L∞(Q), kZk_H1(0,1)≤Ckfk_H1(Q), andkZk_W1,q(0,1)≤ kfk_W1,q(Q). Furthermore, let ψ∈C_c^∞((0,1))such that´1

0 ψ dx= 1. This function can be chosen independently from f. We set

h(x, y) = ˆ x

0

f(t, y)−ψ(t)g(y)dt.

Thenf(x, y) =∂xh(x, y) +ψ(x)g(y)andh= 0on∂Q. In addition, one varifies that khk_L^∞_(Q)≤Ckfk_L^∞_(Q),khk_H1

0(Q)≤Ckfk_H1

0(Q), andkhk_W1,q

0 (Q)≤Ckfk_W1,q 0 (Q). Finally, set

Y(x, y) = (h(x, y), ψ(x)Z(y)).

ThendivY =f. The desired estimates follow from those for handZ.

U Γ

_U

δ

Figure 2.6: Sketch of the situation in Lemma 2.4.2 and Lemma 2.4.3.

where δ is the length of I. Then for every f ∈L²(U)∩L^q(U) there exists Y such that divY =f. Moreover,Y satisfies

kYk_L^∞_(U;R2)+kYk_H1(U;R²)≤Ckfk_L2(U;R²), kYk_W1,q(U;R²)≤Ckfk_Lq(U;R²), andY = 0on ΓU where

ΓU ={(x, y)∈R²:x∈I, y=ψ(x)} ∪ {(x, y)∈R²:x∈∂I, ψ(x)≤y≤ψ(x) +δ}.

For a sketch ofU andΓU, see Figure 2.6.

Proof. Let I be an interval and ψ ∈ Lip(I) such that Lip(ψ) ≤ ε₀, where ε₀ will be fixed later.

Moreover, letf ∈L²(U)∩L^q(U)whereU is defined as in the statement of the lemma.

For(x, y)∈I×(0, δ) =:Qwe define

fˆ(x, y) =f(x, y+ψ(x)).

Clearly,kfˆkL²(Q)=kfkL²(U)andkfˆkL^q(Q)=kfkL^q(U). Next, considerQ˜=I×(0,2δ)and

f˜(x, y) =







fˆ(x, y) in I×(0, δ),

−fˆ(x, y−δ) in I×(δ,2δ).

By Proposition 2.3.1 (scale the second variable ofI×(0,2δ)by a factor ¹₂ to receive a function defined on a cube) there is a solution Y˜ ∈L^∞( ˜Q;R²)∩H₀¹( ˜Q;R²)∩W₀^1,q( ˜Q;R²) to div ˜Y = ˜f in Q˜ which satisfies

kY˜k_L∞( ˜Q;R²)+kY˜k_H1( ˜Q;R²)≤Ckf˜k_L2( ˜Q)≤CkfkL²(U)andkY˜k_W1,q( ˜Q;R²)≤CkfkL^q(U). Set for(x, y)∈U

Z(x, y) = ˜Y(x, y−ψ(x)).

Notice that Z= 0onΓU. Moreover, one computes for (x, y)∈U

divZ(x, y) =∂₁Y˜₁(x, y−ψ(x))−∂₂Y˜₁(x, y−ψ(x))ψ⁰(x) +∂₂Y˜₂(x, y−ψ(x))

= ˜f(x, y−ψ(x))−∂2Y˜1(x, y−ψ(x))ψ⁰(x)

=f(x, y)−∂2Y˜1(x, y−ψ(x))ψ⁰(x).

Consequently, it holds

kdivZ−fk_L2(U)≤ε₀kY˜k_H1( ˜Q;R²)≤Cε₀kfk_L2(U)andkdivZ−fk_Lq(U)≤Cε₀kfk_Lq(U). Note that by scaling one can see that the constant does not depend on δ.

In a similar way one verifies that

• kZkL^∞(U;R²)≤CkfkL²(U),

• kZk_H1(U;R²)≤C(1 +ε0)kfk_L2(U),

• kZk_W1,q(U;R²)≤C(1 +ε0)kfk_Lq(U).

We can use this construction inductively to approach the desired solution to divY =f.

Set f¹ = f and Z¹ = Z. Inductively, one finds for k ≥ 2 and f^k = divZ^k−1−f^k−1 a function Z^k∈L^∞(U;R²)∩H¹(U;R²)∩W^1,q(U;R²)such that

• kdivZ^k−f^kk_L2(U)≤Cε0kf^kk_L2(U)=Cε0kdivZ^k−1−f^k−1k_L2(U)≤(ε0C)^kkfk_L2(U),

• kdivZ^k−f^kk_Lq(U)≤(ε0C)^kkfk_Lq(U),

• kZ^kk_L^∞_(U;R2)≤CkdivZ^k−1−f^k−1k_L2(U)≤C(ε0C)^k−1kfk_L2(U),

• kZ^kk_H1(U;R²)≤(1 +ε₀)CkdivZ^k−1−f^k−1k_L2(U)≤C(1 +ε₀)(ε₀C)^k−1kfk_L2(U),

• kZk_W1,q(U;R²)≤(1 +ε₀)CkdivZ^k−1−f^k−1k_Lq(U)≤C(1 +ε₀)(ε₀C)^k−1kfk_Lq(U). Eventually, choose ε₀ >0 so small such that ε₀C < ¹₂. ThenY =P∞

k=1Z^k fulfills the claim of the lemma.

In the following lemma we show how one can get rid of the smallness conditionLip(ψ)≤ε₀ by a scaling argument.

Lemma 2.4.3. Let 2< q <∞, I an interval,ψ∈Lip(I), andU defined as in Lemma 2.4.2. Then there exists a constantC >0such that for everyf ∈L²(U)∩L^q(U)there exists a functionY satisfying Y = 0 onΓU (ΓU defined as in Lemma 2.4.2), divY =f,

kYk_L^∞_(U;R2)+kYk_H1(U;R²)≤Ckfk_L2(U), andkYk_W1,q(U;R²)≤Ckfk_Lq(U). The constantC may depend onLip(ψ)but not onI.

Proof. LetI be an interval,δthe length ofI,ψ∈Lip(I), andf ∈L²(U)∩L^q(U).

Define N = d^Lip(ψ)_ε

0 e where ε₀ is the constant from Lemma 2.4.2. Moreover, set I˜= N·I. Next, define for x ∈ I˜ the function ψ(x) =˜ ψ(_N^x). Then Lip( ˜ψ) ≤ ^Lip(ψ)_N ≤ ε0. Define the function f˜: ˜U :={(x, y)∈I˜×R: ˜ψ < y <ψ(x) +˜ δ} →Rby

f˜(x, y) =fx N, y

.

SubdivideI˜intoN subintervalsI˜k of lengthδ. Now, the application of Lemma 2.4.2 to each function f˜_|U˜_k on the set U˜k = {(x, y) ∈ I˜k ×R : ˜ψ(x) < y < ψ(x) +˜ δ} provides a function Z^k such that divZ^k= ˜fk onU˜k,

kZ^kk_L∞( ˜Uk;R²)+kYk_H1( ˜Uk;R²)≤Ckf˜k_L2( ˜Uk), andkZ^kk_W1,q( ˜Uk;R²)≤Ckf˜k_Lq( ˜Uk).

Moreover, it holds thatZ^k = 0 onΓU˜k. We defineZ : ˜U →R²byZ(x, y) =Z^k(x, y)for(x, y)∈U˜_k. By the boundary values ofZ^k on ΓU_k, it holds that divZ = ˜f. Moreover, we obtain that Z is a Sobolev function satisfying the estimates

kZk_L∞( ˜U;R²)+kZk_H1( ˜U;R²)≤Ckf˜k_L2( ˜U;R²)≤C(N)kfk_L2(U;R²)

andkZk_W1,q( ˜U;R²)≤C(N)kfk_Lq(U;R²).

Finally, for(x, y)∈U we setY(x, y) = (_N¹Z1(N x, y), Z2(N x, y)). From this definition one can check the desired properties forY directly.

Using the two previous lemmas discussing the local situation at the boundary, we are now able to prove Theorem 2.4.1.

Proof of Theorem 2.4.1. Letf ∈L²(Ω)∩L^q(Ω).

By the definition of a Lipschitz boundary, we can find a finite cover of∂Ωby sets(Ui)i=1,...,kwhich have up to rotation the form as in Lemma 2.4.3 such that up to rotation∂U_i∩∂Ω ={(x, ψi(x)) :x∈I_i}.

Let(θi)i=0,...,k be a partition of unity in the following sense. For i = 1, . . . , k letθi ∈C^∞(Ω) such thatθ_i= 0onΩ\U_i and letθ₀∈C_c^∞(Ω)such thatPk

i=0θ_i = 1onΩ. By applying Lemma 2.4.3, we find fori= 1, . . . , k solutions Zⁱ ∈L^∞(Ui;R²)∩H¹(Ui;R²)∩W^1,q(Ui;R²)todivZⁱ=f inUi such that Zⁱ = 0on ∂Ui∩∂Ωsatisfying the bounds from Lemma 2.4.3. Note that the constant for the bounds depends only onψ_i and hence onΩ.

Furthermore, let Q be a cube containing Ω. Extend f by 0 to Q and call this extension g. By Proposition 2.3.1, there existsZ⁰∈L^∞(Q;R²)∩H₀¹(Q;R²)∩W₀^1,q(Q;R²)such thatdivZ⁰=g,

kZ⁰kL^∞(Q;R²)+kZ⁰kH¹(Q;R²)≤CkfkL²(Ω;R²), andkZ⁰kW^1,q(Q;R²)≤CkfkL^q(Ω;R²). Define Z = Pk

i=0Zⁱθi. Then Z = 0 on∂Ω. By construction, it holds also that Z ∈ L^∞(Ω;R²)∩ H₀¹(Ω;R²)∩W₀^1,q(Ω;R²)satisfies the estimates

kZk_L^∞_(Ω;R2)+kZk_H1(Ω;R²)≤Ckfk_L2(Ω;R²), andkZk_W1,q(Ω;R²)≤Ckfk_Lq(Ω;R²). Moreover,

divZ =f+

k

X

i=0

Zⁱ· ∇θi

| {z }

=:h

.

Note thath= 0on∂Ω. In addition, straightforward estimates show that

khk_H1(Ω)≤CkZk_H1(Ω;R²)≤Ckfk_L2(Ω;) andkhk_W1,q(Ω)≤Ckfk_Lq(Ω). (2.38)

Now, fix q < r <∞. By the Sobolev embedding theorem, we derive from (2.38) that

khk_Lr(Ω)≤Ckhk_H1(Ω)≤Ckfk_L2(Ω) andkhk_Lr(Ω)≤Ckfk_Lq(Ω). (2.39) By [11, Theorem 2’], there exists a functionR∈W₀^1,r(Ω,R²)such that

divR=handkRkW^1,r(Ω;R²)≤CkhkL^r(Ω). Asr > q >2it follows from the Sobolev embedding theorem and (2.39) that

kRkH¹(Ω;R²)+kRkL^∞(Ω;R²)≤CkRk_W1,r(Ω;R²)≤CkhkL^r(Ω)≤ kfkL²(Ω)

andkRkW^1,q(Ω;R²)≤ kfkL^q(Ω).

Eventually,Y =Z−Rhas the demanded properties.

As discussed at the beginning of the chapter we use Theorem 2.4.1 to prove a decomposition statement for functions in H₀¹∩W₀^1,q. This statement is the content of the next theorem. This theorem can be understood as the primal version to the Bourgain-Brézis type estimate (Theorem 2.0.1) as the the Bourgain-Brézis type estimate can be derived from Theorem 2.4.4 by dualization.

Theorem 2.4.4 (The primal result). Let 2 < q <∞and Ω⊂R² open, simply connected, bounded with Lipschitz boundary. Then there exists a constant C > 0 such that for every ϕ ∈H₀¹(Ω;R²)∩ W₀^1,q(Ω;R²)there existh∈H₀²(Ω)∩W₀^2,q(Ω)andg∈L^∞(Ω;R²)∩H₀¹(Ω;R²)∩W₀^1,q(Ω;R²)satisfying

(i) ϕ=g+∇h,

(ii) kgkL^∞(Ω;R²)+kgk_H¹

0(Ω;R²)+khk_H²

0(Ω)≤Ckϕk_H¹

0(Ω;R²), (iii) kgk_W1,q

0 (Ω;R²)+khk_W2,q

0 (Ω)≤Ckϕk_W1,q 0 (Ω;R²). Proof. Letϕ∈H₀¹(Ω;R²)∩W₀^1,q(Ω).

By the boundary values of ϕ it holds that ´

Ωcurlϕ dx = 0. The application of Theorem 2.4.1 to curlϕ ∈ L²(Ω)∩L^q(Ω) provides a function Y ∈ L^∞(Ω;R²)∩H₀¹(Ω;R²)∩W₀^1,q(Ω;R²) such that divY = curlϕand

kYk_L∞(Ω;R²)+kYk_H1

0(Ω;R²)≤Ckcurlϕk_L2(Ω)≤Ckϕk_H1(Ω;R²)andkYk_W1,q

0 (Ω;R²)≤Ckϕk_W1,q(Ω;R²). Setg=Y^⊥= (−Y₂, Y₁). Theng satisfies the same bounds asY andcurlg= divY = curlϕ. AsΩis simply-connected, by the Hodge decomposition there exists a vector fieldh∈H²(Ω)∩W^2,q(Ω) such that ϕ−g=∇h,

khk_H2(Ω)≤Ckg−ϕk_H1

0(Ω;R²)≤Ckϕk_H1(Ω;R²), andkhk_W2,q(Ω)≤Ckϕk_W1,q 0 (Ω;R²).

Moreover,∇h=ϕ−g= 0on∂Ω. Therefore,his constant on the boundary ofΩand we may assume it is zero. Hence,h∈H₀²(Ω)∩W₀^2,q(Ω).

Remark 2.4.1. From Theorem 2.4.4 we can derive the corresponding dual statement i.e., a function f ∈ L¹(Ω;R²) satisfying divf = a+b ∈ H⁻²(Ω) +W^−2,p(Ω), p < 2, is an element of the space H⁻¹(Ω;R²) +W^−1,p(Ω;R²)and

kfk_H⁻¹_(Ω;R2)+W^−1,p(Ω;R²)≤C

kfk_L1(Ω;R²)+kak_H−2(Ω)+kbk_W−2,p(Ω)

.

In fact, letϕ∈H₀¹(Ω;R²)∩W₀^1,p0(Ω;R²). We use the decompositionϕ=g+∇hfrom Theorem 2.4.4 and estimate

ˆ

Ω

f ϕ dx= ˆ

Ω

f(g+∇h)dx

≤C

kfk_L1(Ω;R²)kgk_L∞(Ω;R²)+kak_H−2(Ω)khk_H2

0(Ω)+kbk_W−2,p(Ω)khk

W₀^2,p0(Ω)

≤C

kfk_L1(Ω;R²)+kak_H−2(Ω)+kbk_W−2,p0

(Ω)

max

kϕk_H1

0(Ω;R²),kϕk_W1,p 0 (Ω;R²)

. In particular,f ∈(H₀¹(Ω;R²)∩W₀^1,p0(Ω;R²))⁰=H⁻¹(Ω;R²)+W^−1,p(Ω;R²). Hence, it can be written as f =A+B ∈H⁻¹(Ω;R²) +W^−1,p(Ω;R²). The difference to the Bourgain-Brézis type estimate, which is our final goal in this chapter, is thatA andB only satisfy a combined estimate, precisely

kAk_H−1(Ω;R²)+kBk_W−1,p(Ω;R²)≤C(kfk_L1(Ω;R²)+kak_H−2(Ω)+kbk_W−2,p(Ω)).

The point is to use a scaling argument to obtain separate estimates forAandB.

The classicalW^k,p-norm and the homogeneousW₀^k,p-norm are equivalent norms on the spaceW₀^k,p. So far, it has not been important which of these norms we use onW₀^k,p. Next, we are interested in the scaling of the optimal constant in Theorem 2.4.4. We show the scaling invariance of the optimal con-stant in Theorem 2.4.4 for the homogeneousW₀^k,p-norms i.e.,kfk_Wk,p

0 (Ω,R^m)=P

|α|=kkD^αfk_Lp(Ω;R^m). Proposition 2.4.5. Let 2 < q < ∞ and Ω ⊂ R² open, simply connected, bounded with Lipschitz boundary. LetR >0 andΩ_R=R·Ω. If we denote byC(Ω), respectivelyC(Ω_R), the optimal constant of Theorem 2.4.4 for the domainΩ, respectivelyΩR, thenC(Ω) =C(ΩR).

Proof. Let ϕ ∈ W₀^1,q Ω_R;R²

∩H₀¹ Ω_R;R²

. Define the function ϕ_R−1 : Ω → R² for x ∈ Ω by ϕ_R⁻¹(x) =ϕ(Rx). Thenϕ_R⁻¹ ∈W₀^1,q(Ω;R²)∩H₀¹(Ω;R²)and ϕ_R⁻¹ fulfills

kϕ_R⁻¹k_W1,q

0 (Ω;R²)=

2

X

i=1

k∂iϕ_R⁻¹k_Lq(Ω;R²)=R^1−2qkϕk_W1,q 0 (ΩR;R²)

and kϕR⁻¹k_H1

0(Ω;R²)=kϕk_H1

0(Ω_R;R²).

By Theorem 2.4.4, there exist h_R⁻¹ ∈ H₀²(Ω)∩W₀^2,q(Ω) and g_R⁻¹ ∈ L^∞(Ω;R²)∩H₀¹(Ω;R²)∩ W₀^1,q(Ω;R²)such thatϕ_R−1 =g_R−1+∇h_R−1 and

kg_R−1k_L∞(Ω;R²)+kg_R−1k_H1

0(Ω;R²)+kh_R−1k_H2

0(Ω)≤C(Ω)kϕ_R−1k_H1 0(Ω;R²)

=C(Ω)kϕk_H1

0(ΩR;R²), (2.40) kgR⁻¹k_W^1,q

0 (Ω;R²)+khR⁻¹k_W^2,q

0 (Ω)≤C(Ω)kϕR⁻¹k_W^1,q

0 (Ω;R²)=C(Ω)R^1−2qkϕk_W^1,q

0 (Ω_R;R²). (2.41) Next, define the functions g : Ω_R → R² and h : Ω_R → R for x ∈ Ω_R by g(x) = g_R−1 x

R

and h(x) =R h_R−1 x

R

. Then it holdsϕ=g+∇h. Moreover, by (2.40) and (2.41), it follows that kgk_L^∞_(ΩR;R2)+kgk_H1

0(Ω_R;R²)+khk_H2

0(Ω_R)=kg_R−1k_L^∞_(Ω;R2)+kg_R−1k_H1

0(Ω;R²)+kh_R−1k_H2 0(Ω)

≤C(Ω)kϕk_H1 0(Ω_R)

andkgk_W1,q

0 (Ω_R;R²)+khk_W2,q

0 (Ω_R)=R^−1+2q(kg_R−1k_W1,q

0 (Ω;R²)+kh_R−1k_W2,q(Ω))

≤C(Ω)kϕk_W1,q 0 (ΩR).

This proves C(Ω)≤C(ΩR). The reverse inequality follows byΩ = (ΩR)_R−1.

Using Proposition 2.4.5, we can finally prove the main result of this chapter by a scaling argument.

Theorem (Bourgain-Brézis type estimate). Let 1< p <2 andΩ⊂R² open, simply connected, and bounded with Lipschitz boundary. Then there exists a constantC >0such that for everyf ∈L¹(Ω;R²) satisfying divf =a+b ∈H⁻²(Ω) +W^−2,p(Ω) there exist A ∈H⁻¹(Ω;R²) and B ∈ W^−1,p(Ω;R²) such that the following holds:

(i) f =A+B,

(ii) kAk_H−1(Ω;R²)≤C(kfk_L1(Ω;R²)+kak_H−2(Ω)), (iii) kBkW^−1,p(Ω;R²)≤CkbkW^−2,p(Ω).

Proof. Letf ∈L¹(Ω;R²),R >0, and ΩR=R·Ω.

Define the function fR : ΩR → R² byfR(x) = f _R^x

for x∈ ΩR . Now, consider a test function ϕ∈H₀¹ Ω_R;R²

∩W₀^1,p0 Ω_R;R²

. By Theorem 2.4.4, there exist functionsh∈H₀²(Ω_R)∩W₀^2,p0(Ω_R) andg∈L^∞ ΩR;R²

∩H₀¹ ΩR;R²

∩W₀^1,p0 ΩR;R²

such thatϕ=g+∇h, kgk_L^∞_(ΩR;R2)+kgk_H1

0(Ω_R;R²)+khk_H2

0(Ω_R)≤Ckϕk_H1 0(Ω_R;R²), andkgk_W1,p0

0 (Ω_R;R²)+khk_W2,p0

0 (Ω_R)≤Ckϕk_W1,p0 0 (Ω_R;R²). Note that by Proposition 2.4.5 the constant Cdoes not depend onR. Next, notice that

ˆ

Ω_R

f_R·ϕ dx= ˆ

Ω_R

f_R·(g+∇h)dx

=< fR, g >_L1(Ω_R;R²),L^∞(Ω_R;R²)−< aR, h >_H−2(ΩR),H²₀(ΩR)−< bR, h >

W^−2,p(ΩR),W₀^2,p0(ΩR), (2.42) where a_R is defined by< a_R, h >_H−2(ΩR),H₀²(ΩR)=R < a, h_R−1 >_H−2(Ω),H₀²(Ω) and b_R is defined by

< bR, h >_W−2,p(Ω

R),W₀^2,p0(Ω_R)=R < b, h_R⁻¹ >_{W−2,p(Ω),W}2,p0

0 (Ω) forh_R⁻¹(x) =h(Rx). By scaling one sees that

kaRk_H−2(Ω_R)=R²kak_H−2(Ω) and kbRk_W−2,p(Ω_R)=R^1+p2kbk_W−2,p(Ω). (2.43) Moreover, from (2.42) we derive that

ˆ

Ω_R

fR·ϕ dx

≤C

kfRk_L1(ΩR;R²)+kaRk_H−2(ΩR)+kbRk_W−2,p(ΩR)

max

kϕk_H1(ΩR;R²),kϕk_W1,p0

(ΩR;R²)

.

The dual space ofH₀¹(ΩR;R²)∩W₀^1,p0(ΩR;R²)equipped with the norm kϕk_H1

0(Ω_R;R²)∩W₀^1,p0(Ω_R;R²)= max kϕk_H1

0(Ω_R;R²),kϕk_W1,p0 0 (Ω_R;R²)

is isomorphic to the spaceH⁻¹(ΩR;R²) +W^−1,p(ΩR;R²)endowed with the norm

kFk_H−1(ΩR;R²)+W^−1,p(ΩR;R²)= inf{kF1k_H−1(ΩR;R²)+kF2k_W−1,p(ΩR;R²):F1+F2=F}.

Hence,fR∈H⁻¹ ΩR;R²

+W^−1,p ΩR;R² and kfRk_H−1(Ω_R;R²)+W^−1,p(Ω_R;R²)≤C

kfRk_L1(Ω_R;R²)+kaRk_H−2(Ω_R)+kbRk_W−2,p(Ω_R)

.

In particular, there existAR∈H⁻¹ ΩR;R²

, BR∈W^−1,p ΩR;R²

such thatfR=AR+BR and kARk_H−1(Ω_R)+kBRk_W−1,p(Ω_R)≤C

kfRk_L1(Ω_R)+kaRk_H−2(Ω_R)+kbRk_W−2,p(Ω_R)

. (2.44) We defineA∈H⁻¹(Ω;R²)andB∈W^−1,p(Ω;R²)by

< A, ϕ >_H−1(Ω;R²),H₀¹(Ω;R²)=R⁻²< AR, ϕR>_H−1(Ω_R;R²),H₀¹(Ω_R;R²)

and < B, ϕ >_W−1,p(Ω;

R²),W₀^1,p0(Ω;R²)=R⁻²< BR, ϕR>_W−1,p(Ω

R;R²),W₀^1,p0(Ω_R;R²)

for everyϕ∈C_c^∞(Ω)andϕR∈C_c^∞(ΩR)given byϕR(x) =ϕ _R^x

. Then it holds for everyϕ∈C_c^∞(Ω) ˆ

Ω

f·ϕ dx=R⁻² ˆ

Ω_R

fR·ϕRdx=< A, ϕ >_H−1(Ω;R²),H₀¹(Ω;R²)+< B, ϕ >_W−1,p(Ω;

R²),W₀^1,p0(Ω;R²). Consequently,f =A+B. Moreover, by (2.43) and (2.44) we see that

kAk_H−1(Ω;R²)=R⁻²kA_Rk_H−1(ΩR;R²)≤C

kfk_L1(Ω;R²)+kak_H−2(Ω)+R^2p−1kbk_W−2,p(Ω)

kBk_W−1,p(Ω;R²)=R^−1−2pkBRk_W−1,p(Ω_R;R²)

≤C R^1−p2

kfk_L1(Ω;R²)+kak_H−2(Ω)

+kbk_W−2,p(Ω)

. ChoosingRsuch thatR^1−2p = _kfk ^{kbkW−2,p(Ω)}

L1(Ω;R2)+kak_H−2 (Ω) finishes the proof.

Remark 2.4.2. Let us remark here that by a similar argumentation this result also holds for Radon measures.

3 A Generalized Rigidity Estimate with Mixed Growth

The goal of this section is to prove a rigidity estimate for fields with non-vanishingcurlin the case of a nonlinear energy density with mixed growth. Precisely, we show the following theorem.

Theorem 3.0.1. Let1< p <2andΩ⊂R²open, simply connected, bounded with Lipschitz boundary.

There exists a constantC >0such that for everyβ ∈L^p(Ω;R^2×2)such thatcurlβ∈ M(Ω;R²)there exists a rotationR∈SO(2)such that

ˆ

Ω

|β−R|²∧ |β−R|^pdx≤C ˆ

Ω

dist(β, SO(2))²∧dist(β, SO(2))^pdx+|curlβ|(Ω)²

. One of the simplest versions of a rigidity statement is the following. For a given a connected set Ω⊂R² and a function u∈C²(Ω;R²)such that∇u(x)∈SO(2)for allx∈Ω, there exists a rotation R∈SO(2)and vectorsa, b∈R² such thatu(x) =R(x−a) +b, see [45].

Intuitively, this means that a deformation that is locally a rotation, is already a global rotation. The same statement is also true for infinitesimal rotations i.e., for the set of skew-symmetric matrices.

First qualitative versions of this statement are the different versions of Korn’s inequality, see [36,52,53].

An estimate in the nonlinear case was developed by Friesecke, James and Müller in [37]. Extensions to the case of energy densities with mixed growth include [19, 58].

First results for fields with non-vanishingcurlare anL²-version of our result in [59, Theorem 3.3] and the generalized Korn’s inequality in [38, Theorem 11].

The proofs of the statements in [38, 59] make use of the Bourgain-Brézis inequality as stated at the beginning of the previous chapter, see also [14, Lemma 3.3 and Remark 3.3]. The proof in our case is based on its counterpart in the case of mixed growth i.e., Theorem 2.0.1.

Before we are able to prove Theorem 3.0.1, we need to show two simple lemmas and a version of the classical rigidity estimates for gradients in the case of mixed growth which involves the weak L²-norm, Proposition 3.0.5.

We start proving an easy triangle-inequality for a quantity with mixed growth.

Lemma 3.0.2. Let m∈Nand1< p <2. There exists a constantC >0 such that for alla, b∈R^m it holds

|a+b|²∧ |a+b|^p≤C |a|²∧ |a|^p+|b|²∧ |b|^p . Proof. We can restrict ourselves to the following cases:

1. |a+b| ≤1

a) |a|,|b| ≤1. Here, the statement follows by the usual triangle inequality.

b) |b|>1. Then|a+b|²≤ |b|^p≤ |a|²∧ |a|^p+|b|^p=|a|²∧ |a|^p+|b|²∧ |b|^p.

2. |a+b|>1.

a) |a|,|b| ≤1. Then|a+b| ≤2 and|a| ≥ ¹₂ or|b| ≥ ¹₂. Wlog |b| ≥ ¹₂. Then |a+b|^p ≤2^p ≤ 2^p+2|b|²≤C |a|²+|b|²

=C |a|²∧ |a|^p+|b|²∧ |b|^p . b) |a|<1,|b| ≥1. Then|a+b|^p≤2^p|b|^p ≤2^p |a|²+|b|^p

= 2^p |a|²∧ |a|^p+|b|²∧ |b|^p . c) |a|,|b| ≥1. This follows again from the usual triangle-inequality.

Next, we prove a simple decomposition result for the sum of two functionsf ∈L^2,∞, g∈L^p, where 1< p <2, which we need later in the proof of the rigidity statement involving weakL²-norms.

Lemma 3.0.3. LetU ⊂Rⁿand1< p <2. Then for everyk >0there exists a constantC(k)>0such that for every two nonnegative functions f ∈L^2,∞(U), g ∈L^p(U)there exist functions f˜∈L^2,∞(U) andg˜∈L^p(U)such that

(i) f+g= ˜f+ ˜g, (ii)

f˜

2

L^2,∞(U)+k˜gk^p_Lp(U)≤C(k)

kfk²_L2,∞(U)+kgk^p_Lp(U)

, (iii) g˜∈ {0} ∪(k,∞]andf˜≤k.

Proof. Letk >0. Definef˜= (f+g)1_{f+g≤k} and˜g= (f+g)1_{f+g>k}. Then (i) and (iii) are clearly satisfied. Moreover, we can estimate

f˜

2

L^2,∞(U)≤4kfk²_L2,∞(U)+ 4

g1_{g≤k}

2 L²(U)

≤4kfk²_L2,∞(U)+ 4k^2−p

g1_{g≤k}

p L^p(U)

≤C(k)

kfk²_L2,∞(U)+kgk^p_Lp(U)

. Forg, notice that˜ f1_{f+g>k}=f1_{f+g>k}1_{f≤k

2}+f1_{f+g>k}1_{{f >}k

2}≤g+f1_{{f >}k

2}. Thus, we can conclude thatk˜gk^p_Lp(U)≤C

f1_{{f >}k

2}

p

L^p(U)+kgk^p_Lp(U)

. We estimate

1_{{f >}k

2}f

p L^p(U)

= ˆ ∞

0

pt^p−1Lⁿ({1_{{f >}k

2}f > t})dt (3.1)

= ˆ ^k₂

0

pt^p−1Lⁿn 1_{{f >}k

2}f > to dt+

ˆ ∞

k 2

pt^p−1L²n 1_{{f >}k

2}f > to dt

≤C(k)Lⁿ

1_{{f >}k 2}f > k

2

+ ˆ ∞

k 2

pt^p−3kfk²_L2,∞(U)dt

≤C(k)Lⁿ

f > k 2

+C(k)kfk²_L2,∞(U)

≤C(k)kfk²_L2,∞(U). Hence,k˜gk^p_Lp(U)≤C(k)

kfk²_L2,∞(U)+kgk^p_Lp(U)

. This finishes the proof.

As a second ingredient for the proof of the preliminary mixed-growth rigidity result we need the following truncation argument from [37, Proposition A.1].

c1=c1(U)such that for allu∈W^1,1(U,R^m)and allλ >0 there exists a measurable setE⊂U such that

(i) uisc₁λ-Lipschitz onE, (ii) Lⁿ(U\E)≤ ^c_λ¹´

{|∇u|>λ}|∇u|dx .

With the use of this result, we are now able to prove the mixed-growth rigidity estimate involving weak norms. This result will be the last ingredient to prove the main result of this chapter, Theorem 3.0.1. In [19], the authors prove rigidity estimates for fields whose distance toSO(n)is either the sum of anL^p- and anL^q-function, or in a weak spaceL^p,∞. Our result is a combination of these results.

Proposition 3.0.5. Let 1 < p < 2, n ≥2, andU ⊂Rⁿ open, simply connected and bounded with Lipschitz boundary. Letu∈W^1,1(U;R^n×n)such that there exist f ∈L^2,∞(U) andg ∈L^p(U) such that

dist(∇u, SO(n)) =f +g.

Then there exist matrix fields F ∈ L^2,∞(U;R^n×n) and G ∈ L^p(U;R^n×n) and a proper rotation R∈SO(n)such that

∇u=R+G+F and

kFk²_L2,∞(U;R^n×n)+kGk^p_Lp(U;R^n×n)≤C(kfk²_L2,∞(U)+kgk^p_Lp(U)).

The constantC does not depend onu, f, g.

Proof. Without loss of generality we may assume thatf andgare nonnegative. According to Lemma 3.0.3, we may also assume thatf ≤kandg∈ {0} ∪(k,∞)wherek will be fixed later.

First, we apply Proposition 3.0.4 forλ= 2nto obtain a measurable setE⊂U such thatuis Lipschitz continuous onE with Lipschitz constant M = 2c1n. Let uM be a Lipschitz continuous extension of u_|EtoU with the same Lipschitz constant. In particular,uM =uonE. Setk= 2M. Then we obtain

dist(∇u_M, SO(2))≤f + 2M1_U\E. (3.2)

Indeed, notice that

dist(∇uM, SO(2))≤2c1n+√

n≤2M. (3.3)

Hence, we derivedist(∇u_M, SO(2))≤2M onU\E. OnE, we obtain that dist(∇uM, SO(2)) = dist(∇u, SO(2)) =f+g.

As we may assume thatg∈ {0} ∪(2M,∞], in view of equation (3.3), it holdsdist(∇uM, SO(2)) =f onE. This shows (3.2).

By applying the weak-type rigidity estimate forL^2,∞from [19, Corollary 4.1], we find a proper rotation R∈SO(2)such that

k∇uM−Rk²_L2,∞(U;R^n×n)≤Ckdist(∇uM, SO(2))k²_L2,∞(U)

≤4Ckfk²_L2,∞(U)+ 16CM² 1_U\E

2

L^2,∞(U). (3.4)

Next, note that if|∇u|>2n, then

|∇u| ≤√

n+ dist(∇u, SO(n))≤2 dist(∇u, SO(n)) = 2(f+g)≤4 max{f, g}. (3.5) Using Proposition 3.0.4 (ii) and (3.5), we can estimate

Lⁿ(U\E)≤ c₁ 2n

ˆ

{|∇u|>2n}

|∇u|dx

≤ c1

2n ˆ

{4f≥n}

4f dx+ c1

2n ˆ

{4g≥n}

4g dx

≤ c1

2n^p ˆ

{4f≥n}

4^pf^pdx+ c1

2n^p ˆ

{4g≥n}

4^pg^pdx

≤C

f1_{4f≥n}

p

L^p(U)+kgk^p_Lp(U)

≤C

kfk²_L2,∞(U)+kgk^p_Lp(U)

,

where we used a similar estimate as in (3.1) for the last inequality. In particular, it follows from (3.4) that

k∇uM −Rk²_L2,∞(U;R^n×n)≤C(kfk²_L2,∞(U)+kgk^p_Lp(U)).

Hence, we can write∇u−R=∇u− ∇uM+∇uM−Rand it remains to control∇u− ∇uM. Clearly, we only have to consider∇u− ∇uM onU\E. OnU\E, it holds the pointwise estimate

|∇u− ∇uM| ≤ |∇u|+ 2c1n≤dist(∇u, SO(2)) + 2M1_U\E=f+g+ 2M1_U\E. As before, we know that

1_U\E

2

L^2,∞(U)≤C(kfk²_L2,∞(U)+kgk^p_Lp(U)). Therefore, we are able to write

∇u− ∇uM = h1+h2 where kh1k²_L2,∞(U;R^n×n),kh2k^p_Lp(U;R^n×n) ≤ C(kfk²_L2,∞(U)+kgk^p_Lp(U)). This finishes the proof.

Armed with this weak-type rigidity estimate for mixed growth we are now able to prove the gener-alized rigidity estimate for fields with non-vanishingcurlin our setting, Theorem 3.0.1. The proof is similar to the one of the corresponding statement in [59, Theorem 3.3] but uses quantities with mixed growth instead of quantities inL², in particular the Bourgain-Brézis type estimate for mixed growth, Theorem 2.0.1.

Proof of Theorem 3.0.1. Defineδ= ´

Ωdist(β, SO(2))²∧dist(β, SO(2))^pdx+|curlβ|(Ω)² .

As 1 < p <2, the embeddingM(Ω;R²),→W^−1,p(Ω;R²)is bounded. Hence, there exists a unique solutionv to the problem







∆v= curlβ, v∈W₀^1,p(Ω;R²).

(3.6)

Define β˜=∇vJ where

J = 0 −1

1 0

! .

Optimal regularity for elliptic equations with measure valued right hand side yields (see e.g. [31])

β˜

_L2,∞(U;R^2×2)

≤C|curlβ|(Ω). (3.7)

∇u=β−β. Clearly,˜

|dist(∇u, SO(2))| ≤ |β|˜ +|dist(β, SO(2))|. (3.8) Notice that

dist(β, SO(2)) = dist(β, SO(2))1_{|dist(β,SO(2))|≤1}

| {z }

=:f1

+ dist(β, SO(2))1{|dist(β,SO(2))|>1}

| {z }

=:f2

,

wherekf1k²_L2(Ω) ≤δ and kf2k^p_Lp(Ω) ≤δ. Together with (3.7) and (3.8), this proves the existence of functionsg1∈L^2,∞(Ω) andg2∈L^p(Ω)such that

dist(∇u, SO(2)) =g1+g2 where kg1k²_L2,∞(Ω)≤4

β˜

2

L^2,∞(Ω)+ 4kf1k²_L2,∞(Ω)≤Cδ and kg2k^p_Lp(Ω)≤ kf2k^p_Lp(Ω)≤Cδ.

By Proposition 3.0.5, we derive the existence of a rotation Q ∈ SO(2) and G₁ ∈ L^2,∞(Ω;R^2×2), G2∈L^p(Ω;R^2×2)such that

∇u−Q=G1+G2,kG1k²_L2,∞ ≤Cδ, and kG2k^p_Lp≤Cδ.

Without loss of generality we may assume thatQ=Id(otherwise replaceβ byQ^Tβ).

Next, letϑ: Ω→[−π, π)be a measurable function such that the corresponding rotation

R(ϑ) = cos(ϑ) −sin(ϑ) sin(ϑ) cos(ϑ)

!

satisfies

|β(x)−R(ϑ(x))|= dist(β, SO(2))for almost everyx∈Ω. (3.9) Now, let us decompose

R(ϑ(x))−Id=R(ϑ(x))−β+β− ∇u+∇u−Id

=R(ϑ(x))−β+ ˜β+G1+G2. (3.10) As SO(2) is a bounded set, it is true that |Id−R(ϑ(x))|² ≤ C|Id−R(ϑ(x))|^p ∧ |Id−R(ϑ(x))|². In addition, one can check that |R(ϑ(x))−Id| ≥ ^|ϑ(x)|₂ . Hence, by (3.9), (3.10), and the triangle inequality in Lemma 3.0.2, we can estimate

|ϑ(x)|²

4 ≤ |R(ϑ(x))−Id|²≤C

dist(β, SO(2))²∧dist(β, SO(2))^p+|β|˜²+|G1|²+|G2|^p . Taking theL^1,∞-quasinorm on both sides of the inequality we obtain

kϑk²_L2,∞(Ω)≤Cδ. (3.11)

Following [59, Theorem 3.3], we define the linearized rotation by

R_lin(ϑ) = 1 −ϑ ϑ 1

! .

Using [59, Lemma 3.2], we derive from (3.11) that

kR(ϑ)−R_lin(ϑ)k²_L2 ≤Cδ.

Thus, we can find functionsh₁∈L²(Ω;R^2×2)andh₂∈L^p(Ω;R^2×2)such that β−R_lin(ϑ) = β−R(ϑ)

| {z }

∈L^p(Ω;R^2×2)+L²(Ω;R^2×2)

+R(ϑ)−R_lin(ϑ)

| {z }

∈L²(Ω;R^2×2)

=h₁+h₂ (3.12)

andkh1k^p_Lp(Ω;R^2×2),kh2k²_L2(Ω;R^2×2)≤Cδ. By definition, we see thatcurlRlin(ϑ) =−∇ϑ. Hence, curlβ =−∇ϑ+ curlh1+ curlh2,

which implies

div (curlβ)^⊥

= div (curlh1)^⊥

| {z }

∈W^−2,p(Ω)

+ div (curlh2)^⊥

| {z }

∈H⁻²(Ω)

.

Therefore, we can apply Theorem 2.0.1 to obtainA∈H⁻¹(Ω;R²)andB∈W^−1,p(Ω;R²)such that (curlβ)^⊥=A+B,kAk²_H−1(Ω;R²)≤C(|curlβ|(Ω)²+

div(curlh1)^⊥

2 H⁻²(Ω)), and kBk^p_W−1,p(Ω;R²)≤C

div(curlh₂)^⊥

p

W^−2,p(Ω). (3.13)

In particular, one derives from (3.12) and (3.13) that kAk²_H−1(Ω;R²)≤C(|curlβ|(Ω)²+

div(curlh₁)^⊥

2

H⁻²(Ω))≤C(δ+kh1k²_L2(Ω;R^2×2))≤Cδ (3.14) and similarly

kBk^p_W−1,p(Ω;R²)≤Cδ. (3.15)

Clearly, the same holds for curlβ,−A^⊥ and−B^⊥. According to this decomposition ofcurlβ we can also decompose the solution vto (3.6).

In fact, as v is the unique solution to the linear problem (3.6), in view of (3.13), (3.14) and (3.15) there exists a decompositionv=v1+v2wherekv1k²_H1(Ω;R²)≤Cδandkv2k^p_W1,p(Ω;R²)≤Cδ. Following the notation from the beginning of the proof, we define

β˜₁=∇v1J andβ˜₂=∇v2J.

Then ∇u=β−β˜=β−β˜₁−β˜₂. Now, using the classical mixed growth rigidity estimate from [58, Proposition 2.3], there exists a proper rotationR∈SO(2)such that

ˆ

Ω

|∇u−R|²∧ |∇u−R|^pdx≤C ˆ

Ω

dist(∇u, SO(2))²∧dist(∇u, SO(2))^p dx.

Eventually, we obtain with the use of Lemma 3.0.2 the following chain of inequalities ˆ

Ω

|β−R|²∧ |β−R|^p dx

≤C ˆ

Ω

|∇u−R|²∧ |∇u−R|^p dx+

β˜1

2 L²+

β˜2

p L^p

≤C

Ω

dist(∇u, SO(2))²∧dist(∇u, SO(2))^p dx+δ

≤C ˆ

Ω

dist(β, SO(2))²∧dist(β, SO(2))^p dx+

β˜1

2 L²+

β˜2

p L^p+δ

≤Cδ, which finishes the proof.

4 Plasticity as the Γ-Limit of a Nonlinear Dislocation Energy with Mixed Growth and the Assumption of Diluteness

In section 1.2, we discussed how to model the behavior of an infinite cylindrical body in which only straight, parallel edge dislocations appear. In this chapter, we investigate the behavior of the stored energy as the interatomic distance goes to zero under the assumption of well-separateness of dislo-cations. We focus on the situation of a nonlinear energy with subquadratic growth for large strains.

This allows us to compute the stored energy without introducing an ad-hoc cut-off radius, see Section 1.3.

We characterize theΓ-limit of the suitably rescaled energy in all existing scaling regimes, Theorem 4.3.2, Theorem 4.4.2 and Theorem 4.5.1. In particular, in the so-called critical scaling regime we derive the same strain-gradient plasticity model as the authors in [38, 59] who started from models involving an ad-hoc cut-off radius around the dislocations. Hence, our result justifies a-posteriori their cut-off approach in this regime. Moreover, we discuss compactness properties in the different regimes, Theorem 4.3.1 and Theorem 4.4.1.

4.1 Setting of the Problem

In this section, we introduce the mathematical setting of the problem. For a discussion of the physical situation, see Section 1.2 and in particular Figure 1.9b.

We considerΩ⊂R²to be a simply-connected, bounded domain with Lipschitz boundary representing the cross section of an infinite cylindrical crystal. The set of (normalized) minimal Burgers vectors for the given crystal is denoted byS={b1, b₂}for two linearly independent vectorsb₁, b₂∈R². Moreover, we write

S= span_ZS={λ₁b₁+λ₂b₂:λ₁, λ₂∈Z} for the set of (renormalized) admissible Burgers vectors.

Letε >0the interatomic distance for the given crystal. The set of admissible dislocation densities is defined as

X_ε= (

µ∈ M(Ω;R²) :µ=

M

X

i=1

εξ_iδ_xi, M ∈N, B_ρε(x_i)⊂Ω,|x_j−x_k| ≥2ρ_εforj6=k,06=ξ_i∈S )

,

where we assume thatρε satisfies

1) lim_ε→0ρ_ε/ε^s=∞for all fixeds∈(0,1)and 2) limε→0|logε|ρ²_ε= 0.

This means that we assume the dislocations to be separated on an intermediate scaleερε→0.

Furthermore, we define the set of admissible strains generatingµ∈Xεby ASε(µ) =

β ∈L^p(Ω,R^2×2) : curlβ=µin the sense of distributions . (4.1) The energy densityW :R^2×2→[0,∞)satisfies the usual assumptions of nonlinear elasticity:

(i) W ∈C⁰(R^2×2)andW ∈C² in a neighbourhood ofSO(2);

(ii) stress-free reference configuration: W(Id) = 0;

(iii) frame indifference: W(RF) =W(F)for allF∈R^2×2andR∈SO(2).

In addition, we assume thatW satisfies the following growth condition:

(iv) there exists1<p<2and0< c≤C such that for every F∈R^2×2it holds c dist(F, SO(2))²∧dist(F, SO(2))^p

≤W(F)≤C dist(F, SO(2))²∧dist(F, SO(2))^p . (4.2)

According to the scaling heuristics discussed in Section 1.3, we define the rescaled energy forε >0by

Eε(µ, β) =







1 ε²N_ε|logε|

´

ΩW(β)dx if(µ, β)∈Xε× ASε(µ),

+∞ else inM(Ω;R²)×L^p(Ω;R^2×2),

(4.3)

where we used the usual trick of extending the energy by+∞to non-admissible strains and dislocation densities.

Our goal is to determine theΓ-limit ofE_ε asε→0.

The behavior of the energy depends highly on the scaling of Nε with respect to ε. As discussed in Section 1.3, the different scaling regimes are: the subcritical regime N_ε |logε|, the critical regime Nε∼ |logε|, and the supercritical regime Nε |logε|.

Before we can state the different Γ-convergence results, we introduce the self-energy of a disloca-tion, which corresponds to the minimal energy that a single dislocation induces. As discussed in Section 1.3, the self-energy of the dislocations is expected to contribute to the limit in the subcritical and the critical regime.

Im Dokument Plasticity as the Γ-limit of a Dislocation Energy Dissertation (Seite 44-62)