Correctors for some nonlinear monotone operators

Correctors for Some Nonlinear Monotone Operators

Johan BYSTR ¨ OM

Department of Mathematics, Lule˚ a University of Technology, SE-97187 Lule˚ a, Sweden E-mail: johanb@sm.luth.se

Received March 3, 2000; Revised June 5, 2000; Accepted August 28, 2000 Abstract

In this paper we study homogenization of quasi-linear partial differential equations of the form −div (a (x, x/ε

_h

, Du

_h

)) = f

_h

on Ω with Dirichlet boundary conditions.

Here the sequence (ε

_h

) tends to 0 as h → ∞ and the map a (x, y, ξ) is periodic in y, monotone in ξ and satisfies suitable continuity conditions.We prove that u

_h

→ u weakly in W

₀^1,p

(Ω) as h → ∞, where u is the solution of a homogenized problem of the form −div (b (x, Du)) = f on Ω. We also derive an explicit expression for the homogenized operator b and prove some corrector results, i.e. we find (P

_h

) such that Du

_h

− P

h

(Du) → 0 in L

^p

(Ω, R

ⁿ

).

1 Introduction

In mathematical models of microscopically non-homogeneous media, various local charac- teristics are usually described by functions of the form a (x/ε

_h

) where ε

_h

> 0 is a small parameter.The function a (x) can be periodic or belong to some other specific class.To compute the properties of a micro non-homogeneous medium is an extremely difficult task, since the coefficients are rapidly oscillating functions.One way to attack the problem is to apply asymptotic analysis to the problems of microlevel non-homogeneous media, which immediately leads to the concept of homogenization.When the parameter ε

_h

is very small, the heterogeneous medium will act as a homogeneous medium.To characterize this homogeneous medium is one of the main tasks in the homogenization theory.For more information concerning the homogenization theory, the reader is referred to [2, 10]

and [13].

In this paper we consider the homogenization problem for monotone operators and the local behavior of the solutions.Monotone operators are very important in the study of nonlinear partial differential equations.The problem we study here can be used to model different nonlinear stationary conservation laws, e.g. stationary temperature distribution.

For a more detailed discussion concerning different applications, see [16].

We will study the limit behavior of the sequence of solutions (u

_h

) of the Dirichlet boundary value problem

 



 

−div

 a

 x, x

ε

_h

, Du

_h



= f

_h

on Ω, u

_h

∈ W

₀^1,p

(Ω) ,

Copyright c  2001 by J Bystr¨om

where f

_h

→ f in W

^−1,q

(Ω) as ε

_h

→ 0.Moreover, the map a (x, y, ξ) is defined on Ω × R

ⁿ

× R

ⁿ

and is assumed to be periodic in y, continuous in ξ and monotone in ξ.We also need some continuity restriction in the first variable of a (x, y, ξ).We will consider two different cases, namely when a (x, y, ξ) is of the form

a (x, y, ξ) =



N i=1

χ

_Ωi

(x) a

_i

(y, ξ) ,

and when a (x, y, ξ) satisfies that

|a (x

1

, y, ξ) − a (x

2

, y, ξ) |

^q

≤ ω (|x

1

− x

2

|) (1 + |ξ|)

^p

,

where ω : R → R is continuous, increasing and ω (0) = 0. In both cases we will prove that u

_h

→ u weakly in W

^1,p

(Ω) and that u is the solution of the homogenized problem

 −div (b (x, Du)) = f on Ω, u ∈ W

₀^1,p

(Ω) .

We will prove that the operator b has the same structure properties as a and is given by b (x, ξ) =

Y

a

x, y, ξ + Dv

^ξ,x

(y) 

dy, where v

^ξ,x

is the solution of the cell-problem

 −div  a 

x, y, ξ + Dv

^ξ,x

(y)

= 0 on Y,

v

^ξ,x

∈ W

^1,p

(Y ) . (1)

Here Y is a periodic cell and W

^1,p

(Y ) is the subset of W

^1,p

(Y ) such that u has mean value 0 and u is Y -periodic.The corresponding weak formulation of the cell problem is

 



 

Y

 a

x, y, ξ + Dv

^ξ,x

(y) 

, Dw



dy = 0 for every w ∈ W

^1,p

(Y ) , v

^ξ,x

∈ W

^1,p

(Y ) .

(2)

The homogenization problem described above with p = 2 was studied in [15].Others have investigated the case where we have no dependence in x, that is, when a is of the form a (x, y, ξ) = a (y, ξ).Here we mention [9] and [6] where the problems corresponding to single valued and multi valued operators were studied.Moreover, the almost periodic case was treated in [4].

The weak convergence of u

_h

to u in W

^1,p

(Ω) implies that u

_h

− u → 0 in L

^p

(Ω) , but in general, we only have that Du

_h

− Du → 0 weakly in L

^p

(Ω, R

ⁿ

).However, we will prove that it is possible to express Du

_h

in terms of Du up to a remainder which converges strongly in L

^p

(Ω, R

ⁿ

).This is done by constructing a family of correctors P

_h

(x, ξ, t), defined by

P

_h

(x, ξ, t) = P

 x ε

_h

, ξ, t



= ξ + Dv

^ξ,t

 x ε

_h



. (3)

Let (M

_h

) be a family of linear operators converging to the identity map on L

^p

(Ω, R

ⁿ

) such that M

_h

f is a step function for every f ∈ L

^p

(Ω, R

ⁿ

).Moreover, let γ

_h

be a step function approximating the identity map on Ω.We will show that

Du

_h

− P

_h

(x, M

_h

Du, γ

_h

) → 0 in L

^p

(Ω, R

ⁿ

) .

The problem of finding correctors has been studied by many authors, see e.g. [7] where single valued monotone operators of the form −div

a

x

εh

, Du

_h



were considered and [3]

where the corresponding almost periodic case was considered.

2 Preliminaries and notation

Let Ω be a open bounded subset of R

ⁿ

, |E| denote the Lebesgue measure in R

ⁿ

and ·, · denote the Euclidean scalar product on R

ⁿ

.Moreover, if X is a Banach space, we let X

^∗

denote its dual space and · | · denote the canonical pairing over X

^∗

× X.

Let {Ω

i

⊂ Ω : i = 1, . . . , N} be a family of disjoint open sets such that  Ω \ ∪

^N_i=1

Ω

_i

 = 0 and |∂Ω

_i

| = 0. Let (ε

_h

) be a decreasing sequence of real numbers such that ε

_h

→ 0 as h → ∞.Furthermore, Y = (0, 1)

ⁿ

is the unit cube in R

ⁿ

and we put Y

_h^j

= ε

_h

(j + Y ), where j ∈ Z

ⁿ

, i.e. the translated image of ε

_h

Y by the vector ε

_h

j.We also define the following index sets:

J

_h

=



j ∈ Z

ⁿ

: Y

^j_h

⊂ Ω 

, J

_hⁱ

=



j ∈ Z

ⁿ

: Y

^j_h

⊂ Ω

_i

 , B

_hⁱ

=



j ∈ Z

ⁿ

: Y

^j_h

∩ Ω

_i

= ∅, Y

^j_h

\Ω

_i

= ∅  . Moreover, we define Ω

^h_i

= ∪

_j∈Ji

h

Y

^j_h

and F

_i^h

= ∪

_j∈Bi h

Y

_h^j

. In a corresponding way let 

Ω

^k_i

⊂ Ω : i ∈ I

_k



denote a family of disjoint open sets with diameter less than

¹_k

such that  Ω \ ∪

_i∈Ik

Ω

^k_i

 = 0 and  ∂Ω

^k_i

 = 0.We also define the following index sets:

J

_h^i,k

=



j ∈ Z

ⁿ

: Y

^j_h

⊂ Ω

^k_i

 , B

_h^i,k

=



j ∈ Z

ⁿ

: Y

^j_h

∩ Ω

^k_i

= ∅, Y

^j_h

\Ω

^k_i

= ∅  . Let Ω

^k,h_i

= ∪

_j∈Ji,k

h

Y

^j_h

and F

_i^k,h

= ∪

_j∈Bi,k h

Y

_h^j

.

Corresponding to f ∈ L

^p

(Ω, R

ⁿ

) we define the function M

_h

f : R

ⁿ

→ R

ⁿ

by (M

_h

f ) (x) = 

j∈Jh

χ

_Yj h

(x) ξ

_h^j

, where ξ

_h^j

=

¹

|

^Yh^j

|



Y_h^j

f dx and χ

_E

is the characteristic function of the set E (in order to define ξ

_h^j

for all j ∈ Z

ⁿ

, we treat f as f = 0 outside Ω).It is well known that

M

_h

f → f in L

^p

(Ω, R

ⁿ

), (4)

see [14, p.129]. We also define the step function γ

_h

: Ω → Ω by γ

_h

(x) = 

j∈Jh

χ

_Yj

h

(x) x

^j_h

, (5)

where x

^j_h

∈ Y

_h^j

. Finally, C will denote a positive constant that may differ from one place to an other.

Let a : Ω × R

ⁿ

× R

ⁿ

→ R

ⁿ

be a function such that a (x, ·, ξ) is Lebesgue measurable and Y -periodic for x ∈ Ω and ξ ∈ R

ⁿ

. Let p be a real constant 1 < p < ∞ and let q be its dual exponent,

¹_p

+

¹_q

= 1. We also assume that a satisfies the following continuity and monotonicity conditions: There exists two constants c

₁

, c

₂

> 0, and two constants α and β, with 0 ≤ α ≤ min (1, p − 1) and max (p, 2) ≤ β < ∞ such that

|a (x, y, ξ

₁

) − a (x, y, ξ

₂

) | ≤ c

₁

(1 + |ξ

₁

| + |ξ

₂

|)

^p−1−α

|ξ

₁

− ξ

₂

|

^α

, (6) a (x, y, ξ

1

) − a (x, y, ξ

2

) , ξ

₁

− ξ

2

 ≥ c

2

(1 + |ξ

1

| + |ξ

2

|)

^p−β

|ξ

1

− ξ

2

|

^β

, (7) for x ∈ Ω, a.e. y ∈ R

ⁿ

and every ξ ∈ R

ⁿ

.Moreover, we assume that

a (x, y, 0) = 0, (8)

for x ∈ Ω, a.e. y ∈ R

ⁿ

.Let (f

_h

) be a sequence in W

^−1,q

(Ω) that converges to f .

Remark 1. We will use these continuity and monotonicity conditions to show theorems and properties.However, we concentrate on showing the non-trivial cases, for instance when β = p, and omit the simple ones, in this case when β = p.

The solution v

^ξ,x

of the cell-problem (1) can be extended by periodicity to an element in W

_loc^1,p

(R

ⁿ

), still denoted by v

^ξ,x

, and

Rⁿ

 a

x, y, ξ + Dv

^ξ,x

(y) 

, Dφ (y)



dy = 0 for every φ ∈ C

₀^∞

(R

ⁿ

) . (9)

3 Some useful lemmas

The following lemma, see e.g. [12], is fundamental to the homogenization theory.

Lemma 1 (Compensated compactness). Let 1 < p < ∞. Moreover, let (v

h

) be a sequence in L

^q

(Ω, R

ⁿ

) which converges weakly to v, ( −div v

_h

) converges to −div v in W

^−1,q

(Ω) and let (u

_h

) be a sequence which converges weakly to u in W

^1,p

(Ω) . Then

Ω

v

_h

, Du

_h

 φ dx →

Ω

v, Du φ dx, for every φ ∈ C

₀^∞

(Ω) .

We will also use the following estimates, which are proved in [5].

Lemma 2. Let a satisfy (6), (7) and (8). Then the followinginequalities hold:

(a) |a (x, y, ξ)| ≤ c

a

1 + |ξ|

^p−1



, (10)

(b) |ξ|

^p

≤ c

_b

(1 + a (x, y, ξ) , ξ) , (11)

(c)

Y

 ξ + Dv

^ξ,xi

 

^p

dy ≤ c

_c

(1 + |ξ|

^p

) . (12) Lemma 3. For every ξ

₁

, ξ

₂

∈ R

ⁿ

we have

Y

 ξ

1

+ Dv

^ξ1,x

− ξ

₂

− Dv

^ξ2,x

 

^p

dy ≤ C (1 + |ξ

₁

|

^p

+ |ξ

₂

|

^p

)

^{β−α−1β−α}

|ξ

₁

− ξ

₂

|

^β−αp

. (13)

4 Some homogenization results

Let a (x, y, ξ) satisfy one of the conditions 1. a is of the form

a (x, y, ξ) =



N i=1

χ

_Ωi

(x) a

_i

(y, ξ) . (14)

2.There exists a function ω : R → R that is continuous, increasing and ω (0) = 0 such that

|a (x

₁

, y, ξ) − a (x

₂

, y, ξ) |

^q

≤ ω (|x

₁

− x

₂

|) (1 + |ξ|)

^p

(15) for x

₁

, x

₂

∈ Ω, a.e. y ∈ R and every ξ ∈ R

ⁿ

.

Now we consider the weak Dirichlet boundary value problems (one for each choice of h):

 



 

Ω

 a

 x, x

ε

_h

, Du

_h

 , Dφ



dx = f

_h

| φ for every φ ∈ W

₀^1,p

(Ω) , u

_h

∈ W

₀^1,p

(Ω) .

(16)

By a standard result in the existence theory for boundary value problems defined by monotone operators, these problems have a unique solution for each h, see e.g. [17].

We let φ = u

_h

in (16) and use H¨ older’s reversed inequality, (7), (8) and Poincare’s inequality.This implies that

C



Ω

(1 + |Du

_h

|)

^p

dx



^p−β

p



Ω

|Du

_h

|

^p

dx



^β

p

≤ c

₂

Ω

(1 + |Du

_h

|)

^p−β

|Du

_h

|

^β

dx

≤

Ω

 a

 x, x

ε

_h

, Du

_h

 , Du

_h



dx = f

_h

| u

_h



≤ f

_h



_W−1,q

u

_h



_W^1,p

0

≤ C Du

_h



_Lp(Ω,Rⁿ)

,

(17)

where C does not depend on h. Now if Du

_h



^p_Lp(Ω,Rⁿ)

< |Ω| , then clearly u

_h



_W^1,p

0 (Ω)

≤ C by Poincare’s inequality.Hence assume that Du

_h



^p_Lp(Ω,Rⁿ)

≥ |Ω| . But then we have by (17) that

2

^p−β

Ω

|Du

_h

|

^p

dx =

 2

^p−1

Ω

2 |Du

_h

|

^p

dx



^p−β

p



Ω

|Du

_h

|

^p

dx



^β

p

≤



Ω

2

^p−1

(1 + |Du

_h

|

^p

) dx



^p−β

p



Ω

|Du

_h

|

^p

dx



^β

p

≤ C



Ω

|Du

_h

|

^p

dx



¹

p

,

that is, Du

_h



_Lp(Ω,Rⁿ)

≤ C. According to Poincare’s inequality we thus have that

u

_h



_W1,p

0 (Ω)

≤ C. Summing up, we have that

u

_h



_W^1,p

0 (Ω)

≤ C. (18)

Since u

_h

is bounded in W

₀^1,p

(Ω), there exists a subsequence (h

^

) such that

u

_h

→ u

_∗

weakly in W

₀^1,p

(Ω) . (19)

The following theorems will show that u

_∗

satisfy an equation of the same type as those which are satisfied by u

_h

.

Theorem 4. Let a satisfy (6), (7), (8) and (14). Let (u

h

) be solutions of (16). Then we have that

u

_h

→ u weakly in W

₀^1,p

(Ω) , a

 x, x

ε

_h

, Du

_h



→ b (x, Du) weakly in L

^q

(Ω, R

ⁿ

) ,

(20)

where u is the unique solution of the homogenized problem

 



 

Ω

b (x, Du) , Dφ dx = f | φ for every φ ∈ W

₀^1,p

(Ω) , u ∈ W

₀^1,p

(Ω) .

(21)

The operator b : Ω × R

ⁿ

→ R

ⁿ

is defined a.e. as b (x, ξ) =



N i=1

χ

_Ωi

(x)

Y

a

_i

y, ξ + Dv

^ξ,xi

(y) 

dy =



N i=1

χ

_Ωi

(x) b

_i

(ξ) , where x

_i

∈ Ω

_i

, b

_i

(ξ) = 

Y

a

_i



y, ξ + Dv

^ξ,xi

(y)

dy and v

^ξ,xi

is the unique solution of the cell problem

 



 

Y

 a

_i

y, ξ + Dv

^ξ,xi

(y) 

, Dφ (y)



dy = 0 for every φ ∈ W

_✷^1,p

(Y ) , v

^ξ,xi

∈ W

_✷^1,p

(Y ) .

(22)

Remark 2. An equivalent formulation of the equations (21) and (22) above can be given in the following unified manner

 



 

Ω

Y

a (x, y, Du (x) + Dv (x, y)) , Du (x) + Dv (x, y) dxdy =

Ω

f u dx, (u, v) ∈ W

₀^1,p

(Ω) × L

^q



Ω; W

_✷^1,p

(Y ) , for all (u, v) ∈ W

₀^1,p

(Ω) × L

^q

Ω; W

_✷^1,p

(Y ) 

. This formulation often occurs in the notion of two-scale limit introduced in e.g. [1].

Proof. We have shown that u

_h^

→ u

_∗

weakly in W

₀^1,p

(Ω) for a subsequence (h

^

) since u

_h

is bounded in W

₀^1,p

(Ω) . We define

ψ

ⁱ_h

= a

_i

 x ε

_h

, Du

_h

 .

Then according to (6), (8), H¨ older’s inequality, Poincare’s inequality and (18), we find that

Ωi

 ψ

ⁱ_h



^q

dx =

Ωi

 a

ⁱ

 x

ε

_h

, Du

_h





^q

dx ≤ c

^q₁

Ωi

(1 + |Du

_h^

|)

^{q(p−1−α)}

|Du

_h^

|

^αq

dx

≤ C



Ωi

(1 + |Du

_h^

|)

^p

dx



^p−1−α_p−1



Ωi

|Du

_h^

|

^p

dx



_p−1^α

≤ C

Ωi

(1 + |Du

_h^

|)

^p

dx ≤ C,

that is, ψ

_hⁱ_

is bounded in L

^q

(Ω

_i

, R

ⁿ

) . Hence there is a subsequence (h

^

) of (h

^

) such that ψ

ⁱ_h

→ ψ

_∗ⁱ

weakly in L

^q

(Ω

_i

, R

ⁿ

) .

From our original problem (16) we conclude that

 

 



 

 



N i=1

Ωi

 a

_i

 x

ε

_h

, Du

_h

 , Dφ



dx = f

_h^

| φ for every φ ∈ W

₀^1,p

(Ω) ,

u

_h

∈ W

₀^1,p

(Ω) . In the limit we have



N i=1

Ωi

 ψ

_∗ⁱ

, Dφ 

dx = f | φ for every φ ∈ W

₀^1,p

(Ω) . If we could show that

ψ

ⁱ_∗

= b

_i

(Du

_∗

) for a.e. x ∈ Ω

_i

, (23)

then it follows by uniqueness of the homogenized problem (21) that u

_∗

= u.Let w

^ξ,x_h ⁱ

(x) = ξ, x + ε

_h

v

^ξ,xi

 x ε

_h



for a.e. x ∈ R

ⁿ

. By the monotonicity of a

_i

we have for a fix ξ that

Ωi

 a

_i

 x

ε

_h

, Du

_h

(x)



− a

_i

 x

ε

_h

, Dw

_h^ξ,x_ⁱ

(x)



, Du

_h

(x) − Dw

^ξ,x_hⁱ

(x)



φ (x) dx ≥ 0 for every φ ∈ C

₀^∞

(Ω

_i

), φ ≥ 0. We now note that by (16) it also holds that

Ωi

 a

_i

 x

ε

_h

, Du

_h

(x)

 , Dφ



dx = f

_h^

| φ for every φ ∈ W

₀^1,p

(Ω

_i

) .

Since f

_h

→ f in W

^−1,q

(Ω) , this implies that

−div

 a

_i

 x

ε

_h

, Du

_h

(x)



= −div ψ

ⁱ_h

→ −div ψ

_∗ⁱ

in W

^−1,q

(Ω

_i

) . (24) We also have that

−div

 a

_i

 x

ε

_h

, Dw

^ξ,x_hⁱ

(x)



= 0 on Ω

_i

. (25)

By the compensated compactness lemma (Lemma 1) we then get in the limit

Ωi

 ψ

ⁱ_∗

− b

_i

(ξ) , Du

_∗

(x) − ξ 

φ (x) dx ≥ 0

for every φ ∈ C

₀^∞

(Ω

_i

), φ ≥ 0. Hence for our fixed ξ ∈ R

ⁿ

we have that

 ψ

_∗ⁱ

− b

_i

(ξ) , Du

_∗

(x) − ξ 

≥ 0 for a.e. x ∈ Ω

_i

. (26)

In particular, if (ξ

_m

) is a countable dense set in R

ⁿ

, then (26) implies that

 ψ

_∗ⁱ

− b

_i

(ξ

_m

) , Du

_∗

(x) − ξ

_m



≥ 0 for a.e. x ∈ Ω

_i

. (27)

By the continuity of b (see Remark 3) it follows that

 ψ

_∗ⁱ

− b

_i

(ξ) , Du

_∗

(x) − ξ 

≥ 0 for a.e. x ∈ Ω

_i

and for every ξ ∈ R

ⁿ

. (28) Since b is monotone and continuous (see Remark 3) we have that b is maximal monotone and hence (23) follows.We have proved the theorem up to a subsequence (u

_h

) of (u

_h

) . By uniqueness of the solution to the homogenized equation it follows that this holds for the whole sequence.

Theorem 5. Let a satisfy (6), (7), (8) and (15). Let (u

_h

) be solutions of (16). Then we have that

u

_h

→ u weakly in W

₀^1,p

(Ω) , a

 x, x

ε

_h

, Du

_h



→ b (x, Du) weakly in L

^q

(Ω, R

ⁿ

) , where u is the unique solution of the homogenized problem

 



 

Ω

b (x, Du) , Dφ dx = f | φ for every φ ∈ W

₀^1,p

(Ω) , u ∈ W

₀^1,p

(Ω) .

The operator b : Ω × R

ⁿ

→ R

ⁿ

is defined a.e. as b (x, ξ) =

Y

a

x, y, ξ + Dv

^ξ,x

(y) 

dy,

where v

^ξ,x

is the unique solution of the cell problem

 



 

Y

 a

x, y, ξ + Dv

^ξ,x

(y) 

, Dφ (y)



dy = 0 for every φ ∈ W

_✷^1,p

(Y ) , v

^ξ,x

∈ W

_✷^1,p

(Y ) .

(29)

Before we prove this theorem, we make some definitions that will be useful in the proof.

Define the function a

^k

(x, y, ξ) = 

i∈Ik

χ

_Ωk i

(x) a

x

^k_i

, y, ξ

 ,

where x

^k_i

∈ Ω

^k_i

. Consider the boundary value problems

 



 

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h

 , Dφ



dx = f

_h

| φ for every φ ∈ W

₀^1,p

(Ω) , u

^k_h

∈ W

₀^1,p

(Ω) .

(30)

The conditions for Theorem 4 are satisfied and the theorem implies that there exists a u

^k_∗

such that

u

^k_h

→ u

^k_∗

weakly in W

₀^1,p

(Ω) as h → ∞,

where u

^k_∗

is the unique solution of

 



 

Ω

 b

^k

x, Du

^k_∗

 , Dφ



dx = f | φ for every φ ∈ W

₀^1,p

(Ω) , u

^k_∗

∈ W

₀^1,p

(Ω) .

(31)

Here

b

^k

(x, ξ) = 

i∈Ik

χ

_Ωk i

(x)

Y

a

x

^k_i

, y, ξ + Dv

^ξ,xki

(y) 

dy = 

i∈Ik

χ

_Ωk i

(x) b

x

^k_i

, ξ

 ,

and v

^ξ,xki

is the solution of

 



 

Y

 a

x

^k_i

, y, ξ + Dv

^ξ,xki

(y) 

, Dφ (y)



dy = 0 for every φ ∈ W

_✷^1,p

(Y ) , v

^ξ,xki

∈ W

_✷^1,p

(Y ) .

Proof. First we prove that u

_h

→ u weakly in W

₀^1,p

(Ω) . If g ∈ W

^−1,q

(Ω) , we have that

h→∞

lim g | u

_h

− u = lim

k→∞

lim

h→∞

g | u

_h

− u

= lim

k→∞

lim

h→∞



g | u

_h

− u

^k_h

+ u

^k_h

− u

^k_∗

+ u

^k_∗

− u 

≤ lim

k→∞

lim

h→∞

g

_W−1,q(Ω)

 u

_h

− u

^k_h

 

W₀^1,p(Ω)

+ lim

k→∞

lim

h→∞



g | u

^k_h

− u

^k_∗

 + lim

k→∞

g

_W−1,q(Ω)

 u

^k_∗

− u  

W₀^1,p(Ω)

.

We need to show that all terms on the right hand side are zero.

Term 1. Let us prove that

k→∞

lim lim

h→∞

 u

h

− u

^k_h

 

W₀^1,p(Ω)

= 0. (32)

By the definition we have that

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h

 , Dφ



dx = f

_h

| φ for every φ ∈ W

₀^1,p

(Ω) ,

Ω

 a

 x, x

ε

_h

, Du

_h

 , Dφ



dx = f

_h

| φ for every φ ∈ W

₀^1,p

(Ω) . This implies that we with φ = u

^k_h

− u

_h

have

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h



− a

^k

 x, x

ε

_h

, Du

_h



, Du

^k_h

− Du

h

 dx

=

Ω

 a

 x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

_h



, Du

^k_h

− Du

h



dx.

By the monotonicity of a (7), Schwarz’ inequality and H¨ older’s inequality it follows that c

₂

Ω

1 +  Du

^k_h

  + |Du

h

| 

_p−β

 Du

^k_h

− Du

_h

 

^β

dx

≤

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h



− a

^k

 x, x

ε

_h

, Du

_h



, Du

^k_h

− Du

_h

 dx

=

Ω

 a

 x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

_h



, Du

^k_h

− Du

_h

 dx

≤



Ω

 a  x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

_h





^q

dx



¹_q



Ω

 Du

^k_h

− Du

_h

 

^p

dx



¹

p

.

Now since  Du

^k_h



^p

L^p(Ω,Rⁿ)

≤ C and Du

h



^p_Lp(Ω,Rⁿ)

≤ C, we have, according to H¨older’s reversed inequality, that

C



Ω

 Du

^k_h

− Du

h

 

^p

dx



^β_p

≤ c

2



Ω

1 +  Du

^k_h

  + |Du

h

| 

_p

dx



^p−β_p



Ω

 Du

^k_h

− Du

_h

 

^p

dx



^β_p

≤ c

₂

Ω

1 +  Du

^k_h

  + |Du

h

| 

_p−β

 Du

^k_h

− Du

_h

 

^β

dx.

(33)

Hence we see that

 Du

^k_h

− Du

_h

 

^p

L^p(Ω,Rⁿ)

≤ C



Ω

 a  x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

_h





^q

dx



_β−1^p−1

. Thus in view of the continuity condition (15) it yields that

 Du

^k_h

− Du

_h

 

^p

L^p(Ω,Rⁿ)

≤ C

 ω

 1 k



Ω

(1 + |Du

h

|)

^p

dx



_β−1^p−1

≤ C

 ω

 1 k



_β−1^p−1

,

(34)

where we in the last inequality have used the fact that there exists a constant C inde- pendent of h such that Du

_h



_Lp(Ω,Rⁿ)

≤ C. Since D·

_Lp(Ω,Rⁿ)

is an equivalent norm on W

₀^1,p

(Ω), we have that

 u

_h

− u

^k_h

 

W₀^1,p(Ω)

→ 0

as k → ∞ uniformly in h. This means that we can change order in the limit process in (32) and (32) follows by taking (34) into account.

Term 2. We observe that

k→∞

lim lim

h→∞



g | u

^k_h

− u

^k_∗



= 0

as a direct consequence of Theorem 4.

Term 3. Let us prove that

k→∞

lim

 u

^k_∗

− u  

W₀^1,p(Ω)

= 0. (35)

By the definition we have that

Ω

 b

^k

x, Du

^k_∗

 , Dφ



dx = f | φ for every φ ∈ W

₀^1,p

(Ω) ,

Ω

b (x, Du) , Dφ dx = f | φ for every φ ∈ W

₀^1,p

(Ω) . Hence

Ω

 b

^k

x, Du

^k_∗

 − b

^k

(x, Du) , Dφ

 dx =

Ω



b (x, Du) − b

^k

(x, Du) , Dφ

 dx

for every φ ∈ W

₀^1,p

(Ω) . Now let φ = u

^k_∗

− u and use the strict monotonicity of b

^k

(see Remark 3) on the left hand side and use Schwarz’ and H¨ older’s inequalities on the right hand side.We get

c

₂

Ω

1 +  Du

^k_∗

  + |Du| 

_p−β

 Du

^k_∗

− Du  

^β

dx

≤

Ω

 b

^k

x, Du

^k_∗

 − b

^k

(x, Du) , Du

^k_∗

− Du  dx

=

Ω



b (x, Du) − b

^k

(x, Du) , Du

^k_∗

− Du  dx

≤



Ω

 b (x, Du) − b

^k

(x, Du)  

^q

dx



¹

q



Ω

 Du

^k_∗

− Du  

^p

dx



¹

p

.

Inequality (33) with Du

^k_∗

and Du instead of Du

^k_h

and Du

_h

, respectively, then yields that

 Du

^k_∗

− Du  

^p

L^p(Ω,Rⁿ)

≤ C

 ω

 1 k



Ω

(1 + |Du|)

^p

dx



^p−1

β−1

≤ C

 ω

 1 k



^p−1

β−1

. (36)

The right hand side tends to 0 as k → ∞. We now obtain (35) by noting that

D·

_Lp(Ω,Rⁿ)

is an equivalent norm on W

₀^1,p

(Ω).

Next we prove that a

x,

_ε^x

h

, Du

_h

 → b (x, Du) weakly in L

^q

(Ω, R

ⁿ

) . Now if g ∈ (L

^q

(Ω, R

ⁿ

))

^∗

, then

h→∞

lim

 g | a

 x, x

ε

_h

, Du

_h



− b (x, Du)



= lim

k→∞

lim

h→∞

 g | a

 x, x

ε

_h

, Du

_h



− b (x, Du)



= lim

k→∞

lim

h→∞

 g | a

 x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

^k_h



+ lim

k→∞

lim

h→∞

 g | a

^k

 x, x

ε

_h

, Du

^k_h



− b

^k

x, Du

^k_∗



+ lim

k→∞

lim

h→∞

 g | b

^k

x, Du

^k_∗

 − b (x, Du) 

≤ lim

k→∞

lim

h→∞

g

_(Lq(Ω,Rⁿ))^∗

 a  x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

^k_h





L^q(Ω,Rⁿ)

+ lim

k→∞

lim

h→∞

 g | a

^k

 x, x

ε

_h

, Du

^k_h



− b

^k

x, Du

^k_∗



+ lim

k→∞

g

_(Lq(Ω,Rⁿ))^∗

 b

^k

x, Du

^k_∗

 − b (x, Du)  

L^q(Ω,Rⁿ)

. We now prove that the three terms on the right hand side are zero.

Term 1. Let us show that

k→∞

lim lim

h→∞

 a  x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

^k_h





L^q(Ω,Rⁿ)

= 0. (37)

By Minkowski’s inequality we have that

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h



− a

 x, x

ε

_h

, Du

_h





^q

dx

≤ C

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h



− a

^k

 x, x

ε

_h

, Du

_h





^q

dx +C

Ω

 a

^k

 x, x

ε

_h

, Du

_h



− a

 x, x

ε

_h

, Du

_h





^q

dx.

The second integral on the right hand side is bounded by the continuity condition (15) since

Ω

 a

^k

 x, x

ε

_h

, Du

_h



− a

 x, x

ε

_h

, Du

_h





^q

dx

≤ ω

 1 k



Ω

(1 + |Du

_h

|)

^p

dx ≤ Cω

 1 k

 ,

(38)

where the last inequality follows since (Du

_h

) is bounded in L

^p

(Ω, R

ⁿ

) . For the first integral

on the right hand side we use the continuity restriction (6), H¨ older’s inequality and (34)

to derive that

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h



− a

^k

 x, x

ε

_h

, Du

_h





^q

dx

≤ c

^q₁

Ω

1 +  Du

^k_h

  + |Du

h

| 

_{q(p−1−α)}

 Du

^k_h

− Du

_h

 

^αq

dx

≤ C



Ω

1 +  Du

^k_h

  + |Du

h

| 

_p

dx



^p−1−α

p−1



Ω

 Du

^k_h

− Du

_h

 

^p

dx



^α

p−1

≤ C



Ω

 Du

^k_h

− Du

_h

 

^p

dx



^α

p−1

≤ C

 ω

 1 k



^α

β−1

(39)

since (Du

_h

) and  Du

^k_h

are bounded in L

^p

(Ω, R

ⁿ

) . Hence by (38) and (39) we have that

Ω

 a

^k

 x, x

ε

_h

, Du

^k_h



− a

 x, x

ε

_h

, Du

_h





^q

dx ≤ C

 ω

 1 k



_β−1^α

+ Cω

 1 k



. (40) By the properties of ω, it follows that

 a  x, x

ε

_h

, Du

_h



− a

^k

 x, x

ε

_h

, Du

^k_h





L^q(Ω,Rⁿ)

→ 0

as k → ∞ uniformly in h.This implies that we may change the order in the limit process in (37) and (37) follows.

Term 2. We immediately see that

k→∞

lim lim

h→∞

 g | a

^k

 x, x

ε

_h

, Du

^k_h



− a

^k

 x, x

ε

_h

, Du

^k_∗



= 0 as a direct consequence of Theorem 4.

Term 3. Let us show that

k→∞

lim

 b

^k

x, Du

^k_∗

 − b (x, Du)  

L^q(Ω,Rⁿ)

= 0. (41)

We note that

Ω

 b

^k

x, Du

^k_∗

 − b (x, Du)  

^q

dx

=

Ω

 b

^k

x, Du

^k_∗

 − b

^k

(x, Du) + b

^k

(x, Du) − b (x, Du)  

^q

dx

≤ C



Ω

 b

^k

x, Du

^k_∗

 − b

^k

(x, Du)  

^q

dx +

Ω

 b

^k

(x, Du) − b (x, Du)  

^q

dx



.