Interior Integral Estimates on Weak Solutions of Nonlinear Parabolic Systems

Interior Integral Estimates on Weak Solutions of Nonlinear Parabolic Systems

J. Naumann^and M. Wolff

Abstract

This paper concerns various types of CACCIOPPOLI and POINCARE inequalities on weak solutions ^uof nonlinear parabolic systems. The main result of the paper is the local integrability of the spatial gradient^{D u}to an exponent ^p>2.

Contents

1. Introduction

2. Regularization and localization 3. CACCIOPPOLI inequalities 4. POINCARE inequalities

5. Extended CACCIOPPOLI inequality 6. Local higher integrability of^{D u} References

1 Introduction

Let

IR

ⁿ (

n

2) be a bounded domain, and let 0

< T <

+¹ be xed. Set

Q

= (0

T

).

We consider the following system of PDE's:

@u

ⁱ

@t

^;

D a

_i(

xtuDu

) =

f

i^;

D g

_i ¹⁾ in

Q

(

i

= 1

:::N

)

(1.1)

where:

u

= ^f

u

¹

:::u

^Ng (

N

1)

Du

= ^f

D u

^ig (matrix of rst spatial derivatives of

u

)

D

=

@

@x

= 1

:::n

f

=^f

f

¹

:::f

N^g and

g

=^f

g

_i^g are given functions in

Q

. Let

a

=^f

a

_i^g be a matrix. Dene

k

a

^k= ^Xn

=1

N

X

i⁼¹(

a

_i)²¹⁼²

:

1)Throughout a repeated Greek resp. Latin index implies summation over 1 ^{::: n}resp. 1 ^{::: N}.

Throughout the whole paper, we impose on the functions

a

_i in (1.1) the following conditions:

8

>

<

>

:

a

_i is a Caratheodory function on

Q

IR

^N

IR

^nN (

= 1

:::n i

= 1

:::N

)

(1.2)

8

>

<

>

:

k

a

(

xtu

)^k

a

⁰(1 +^j

u

^j(n+2)=n+^k

^k)

8(

xtu

)²

Q

IR

^N

IR

^nN (

a

⁰= const) (1.3)

8

>

<

>

:

a

_i(

xtu

)

ⁱ

^0k

^k2

8(

xtu

)²

Q

IR

^N

IR

^nN (

⁰= const

>

0)

:

(1.4)

Let

W

_p¹( ) =^f

v

²

L

^p( ) :

D v

²

L

^p( ) (

= 1

:::n

)^g denote the usual Sobolev space over . Next, dene

W

²¹¹(

Q

) =ⁿ

v

²

L

²(

Q

) :

D v @v @t

²

L

²(

Q

) (

= 1

:::n

)^o

and

W

²¹⁰(

Q

) = ^f

v

²

L

²(

Q

) :

D v

²

L

²(

Q

) (

= 1

:::n

)^g

V

²¹⁰(

Q

) = ⁿ

v

²

W

²¹⁰(

Q

) : esssup

(0T⁾

Z

v

²(

xt

)d

x <

+^1o

:

The following imbedding theorem is well-known (cf. e.g. 5]):

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

Let ! IR

ⁿ be a bounded domain with Lipschitz boundary

@!: Let

@!

be relatively open. Then:

k

v

^k_L^2(n+2)=n(_! ⁽⁰_T⁾⁾

c

⁰ esssup

(0T⁾

Z

!

v

²(

xt

)d

x

+^ZT

0 Z

!

j

Dv

^j2d

x

d

t

¹⁼² for all

v

²

V

²¹⁰(

!

(0

T

))

v

= 0 a.e. on (0

T

) (

c

⁰= const

>

0)

:

(1.5)

By

L

^p(

Q IR

^N),

W

²¹¹(

Q IR

^N) etc. we denote the space of vector valued functions

v

=^f

v

¹

:::v

^Ng the components of which belong to

L

^p(

Q

) resp.

W

²¹¹(

Q

) etc.

We introduce the notion of weak solution of (1.1) regardless of whether or not this solution is subject to any boundary or initial condition.

Let (1.2) and (1.3) be satised. Without any further reference, throughout the whole paper, we assume

g

²

L

²(

Q IR

^nN)

:

DEFINITION 1.1 Assume

1

f

²

L

^2(n+2)=(n+4)(

Q IR

^N) resp.

2

f

²

L

¹(

Q IR

^N).

The function

u

²

V

²¹⁰(

Q IR

^N) is called a weak solution of (1.1) if

Z

u

ⁱ(

xt

)

v

ⁱ(

xt

)d

x

^;Zt

0 Z

u

ⁱ

@v

ⁱ

@s

^d

x

d

s

+^Zt

0 Z

a

_i(

xsuDu

)

D v

ⁱd

x

d

s

=

=^Zt

0 Z

(

f

i

v

ⁱ+

g

_i

D v

ⁱ)d

x

d

s

(1.6)

for a.a.

t

²(0

T

) and all test functions

1

v

²

W

²¹¹(

Q IR

^N), supp(

v

) (0

T

] resp.

2

v

²

W

²¹¹(

Q IR

^N)^\

L

¹(

Q IR

^N), supp(

v

) (0

T

].

The aim of the present paper is to prove various interior integral estimates on any weak solution

u

of (1.1). Following 5], we rst regularize

u

and localize then (1.6) with respect to

t

. Then CACCIOPPOLI and POINCARE inequalities are readily obtained (Sections 3 and 4).

Next, in Section 5 we prove an extended version of the preceeding CACCIOPPOLI inequality which is appearently new in the theory of parabolic systems. Our main result is the interior higher integrability of

Du

(i.e.

Du

²

L

^{plo c}for a

p >

2) which is presented in Section 6. It is based on the parabolic analogue of the well-known higher integrability by reverse Holder inequality due to GEHRING-GIAQUINTA-MODICA.

This method has been used in 3], 6] to prove the higher integrability of the spatial gradient of bounded weak solutions to parabolic systems with quadratic growth nonlinearities. Under more restrictive assumptions on the coecients

a

_i, similar results have been obtained in 1], 2]

by an entirely dierent technique.

The higher integrability of the spatial gradient of weak solution to a nonlinear parabolic system is of interest in itself. On the other hand, it is also a basic tool in the proof of partial regularity of weak solutions of nonlinear parabolic systems.

2 Regularization and localization

Let

f

²

L

¹(

Q

). Given any

t

^{0 2}(0

T

) and

k > T

^;1

t

⁰ we introduce the Steklov mean of

f

:

f

k(

xt

) =

k

^t

+ 1

k

Z

t

f

(

xs

)d

s

for a.a. (

xt

)² (0

t

⁰)

:

We note some properties of

f

k which will be used in what follows. Let

t

⁰²(0

T

).

PROPOSITION 2.1 Let

f

²

L

^p(

Q

) (1

p <

+¹) and

g

²

L

^p0(

Q

). Then

t⁰⁺Z ^1k 0

Z

f

(

xt

) ^Zt

t^;k1

g

(

xs

)d

s

d

x

d

t

= ^t

0

Z

0 Z

tZ^+1k

t

f

(

xs

)d

s

g

(

xt

)d

x

d

t

for all

k > T

^;1

t

^0.

PROPOSITION 2.2 1. Let

f

²

L

^p(

Q

) (1

p

+¹). Then

t⁰

Z

0 Z

j

f

k^jpd

x

d

t

^Zt0

0 Z

j

f

^jpd

x

d

t

(1

p <

+¹)

esssup

(0t^0)j

f

k^jesssup

(0t^0)j

f

^j for all

k > T

^;1

t

^0.

2. Let

f

²

L

^p(

Q

) (1

p <

+¹). Then (i)

f

k ^!

f

in

L

^p( (0

t

⁰)) as

k

^!+¹

(ii)

f

k(

t

)^!

f

(

t

) in

L

^p( ) as

k

^!+¹ for a.a.

t

²(0

t

⁰).

3. Let

f

²

L

¹(

Q

),

g

²

L

¹(

Q

). Then there exists a subsequence^f

k

j^g such that

t⁰

Z

0 Z

f

k^j

g

k^jd

x

d

t

^;!Zt0

0 Z

fg

d

x

d

t as j

^!1

:

The following result shows the eect of regularization with respect to

t

of the Steklov mean.

PROPOSITION 2.3 Let

f

²

W

²¹⁰(

Q

). Then

f

k ²

W

²¹¹( (0

t

⁰)) and there holds

D f

k(

xt

) = (

D f

)k(

xt

) (

= 1

:::n

)

@f

k

@t

⁽

xt

) =

k

^h

f xt

+ 1

k

;

f

(

xt

)ⁱ for a.a. (

xt

)² (0

t

⁰) and all

k > T

^;1

t

^0.

We are now going to localize (1.6) with respect to

t

. To this end, let

t

^{0 2}(0

T

) be arbitrary.

We consider the Steklov mean with integers

k > T

^;1

t

^0.

THEOREM 2.1 Assume

1

f

²

L

^2(n+2)=(n+4)(

Q IR

^N) resp.

2

f

²

L

¹(

Q IR

^N).

Let

u

²

V

²¹⁰(

Q IR

^N) be a weak solution of (1.1). Then

Z

@u

_ik

@t '

^id

x

+^Z

(

a

_i)k

D '

ⁱd

x

=^Z

((

f

i)k

'

ⁱ+ (

g

_i )k

D '

ⁱ)d

x

(2.1)

for a.a.

t

²(0

t

⁰), all integers

k > T

^;1

t

⁰ and all test functions 1

'

²

W

²¹(

IR

^N), supp(

'

) resp.

2

'

²

W

²¹(

IR

^N)^\

L

¹(

IR

^N), supp(

'

) .

Proof. Fix an integer

m > n

_{2. Let} j (

j

= 1

2

:::

) be open sets such that:

j j j⁺¹

::: @

j smooth

¹

j⁼¹ j =

:

The Sobolev imbedding theorem implies

W

^2m( j)

C

( j) (

j

= 1

2

:::

). Dene

W

^2m( j) =ⁿ

'

²

W

^2m( j) :

'

=

@'

@

⁼

:::

=

@

^m;1

'

@

^m;1 = 0 a.e. on

@

j

o

:

Let

'

²

W

^2m( j

IR

^N) be arbitrary. We extend

'

by zero onto ⁿ j and denote this function in again by

'

. Let

²

C

(

IR

) have its support in (0

t

⁰). Then the function

v

(

xt

) =

k'

(

x

) ^Zt

t^;1k

(

s

)d

s

(

xt

)²

Q k

integer

> T

^;1

t

⁰ is admissible in (1.6).

Let

t

^{1 2}

t

⁰+ 1

kT

. Clearly,

v

(

t

¹) = 0 and

t¹

Z

0 Z

u

ⁱ(

xt

)

@v

ⁱ

@t

⁽

xt

)d

x

d

t

= ^;

k

^Zt0

0 Z

h

u

ⁱ

xt

+ 1

k

;

u

ⁱ(

xt

)ⁱ

'

ⁱ(

x

)

(

t

)d

x

d

t

= ^;Zt0

0 Z

@u

_ik

@t

⁽

xt

)

'

ⁱ(

x

)

(

t

)d

x

d

t

(cf. Prop. 2.3). Now (1.6) with

t

^{1 2}

t

⁰+ 1

kT

gives

t⁰

Z

0 Z

@u

_ik

@t '

ⁱ

d

x

d

t

+ ^t

0

Z

0 Z

(

a

_i)k

D '

ⁱ

d

x

d

t

= ^t

0

Z

0 Z

(

f

i)k

'

ⁱ+ (

g

_i)k

D '

ⁱ]

d

x

d

t:

Thus, by a standard argument,

Z

j

@u

_ik

@t

⁽

xt

)

'

ⁱ(

x

)d

x

+^Z

(

a

_i)k(

xt

)

D '

ⁱ(

x

)d

x

=

=^Z

j

(

f

i)k(

xt

)

'

ⁱ(

x

) + (

g

_i)k(

xt

)

D '

ⁱ(

x

)]d

x

(2.2)

for all

t

²(0

t

⁰)ⁿ

E

jk where meas

E

jk = 0 (notice that by virtue of the separability of

W

^2m( j) the set

E

jk does not depend on

'

). Dene

E

=_j^1S

=1 S

k>⁽T^;t^0);1

E

jk. Then meas

E

= 0, and (2.2) is true for a.a.

t

²(0

t

⁰), for

j

= 1

2

:::

and all integers

k > T

^;1

t

^0.

Let

'

satisfy 1 resp. 2 above. Fix

j

such that supp(

'

) j. Let

'

be the standard mollication of

'

. Then

'

²

W

^2m( j

IR

^N) for all 0

< <

dist(supp(

'

)

@

j) and

'

^!

'

in

W

²¹( j

IR

^N) as

^! 0. If, in addition,

'

²

L

¹(

IR

^N) then there exists a subsequence of

f

'

^g(not relabelled) such that

'

^!

'

a.e. in j. Hence, inserting

'

in (2.2) and letting tend

^!0 gives the claim.

3 CACCIOPPOLI inequalities

Dene

B

r =

B

r(

x

⁰) =^f

x

²

IR

ⁿ:^j

x

^;

x

^0j

< r

^g

Q

r =

Q

r(

x

⁰

t

⁰) =

B

r(

x

⁰)(

t

^0;

r

²

t

⁰)

:

Let (

x

⁰

t

⁰)²

Q

. Fix any 0

< r <

¹₂^p

t

⁰ such that

B

²r . Let

²

C

¹(

IR

ⁿ)²⁾and

²

C

¹(

IR

) be cut-o functions as follows:

8

>

<

>

:

1 on

B

r 0 in

IR

ⁿⁿ

B

²r

0

1

^j

D

^j

c

⁰

r

ⁱⁿ

IR

ⁿ

8

>

<

>

:

1 on (

t

^0;

r

²

+¹) 0 on (^;1

t

^0;4

r

²)

0

1

0

⁰

c

⁰

r

^{2 on}

IR

ⁿ (

c

⁰= const

>

0 independent of

r

).

Following 3] we dene for any

v

²

L

¹(

Q

²r)

v

~²r(

t

) = ^Z

B^2r

²d

x

^{;1 Z}

B^2r

v

(

yt

)

²(

y

)d

y

for a.a.

t

²(

t

^0;4

r

²

t

⁰)

:

Let

v

²

W

²¹¹(

Q

²r). Then the function

t

^7! ~

v

²r(

t

) possesses a weak derivative d~

v

²r

d

t

²

L

²(

t

^{0 ;} 4

r

²

t

⁰) and there holds

d~

v

²r

d

t

⁽

t

) = ^gd

v

d

t

2r(

t

) for a.a.

t

²(

t

^0;4

r

²

t

⁰)

:

2)To emphasize the dependence of on^B2r, below we shall write^2r in place of,^r etc.

Let (1.2) { (1.4) be satised. In what follows, we consider the two cases³⁾

8

>

<

>

:

f

²

L

^2(n+2)=(n+4)(

Q IR

^N)

u

²

V

²¹⁰(

Q IR

^N) is a weak solution of (1.1)

(3.1)

or

8

>

<

>

:

f

²

L

¹(

Q IR

^N)

u

²

V

²¹⁰(

Q IR

^N)^\

L

¹(

Q IR

^N) is a weak solution of (1.1)

:

(3.2)

We begin by proving the following

THEOREM 3.1 Assume (3.1) or (3.2). Then, for every

" >

0, 12

Z

B^2r

j

u

(

xt

)^;j2

²(

x

)d

x

²(

t

) + (

^0;

"

) ^Zt

t^0;4r²

Z

B^2r

k

Du

^k2

²

²d

x

d

s

c

¹ 1 + 1

"

1

r

²

Z

Q^2r

j

u

^;j2d

x

d

s

+

c

^{1 Z}

Q^2r(1 +^j

u

^j2(n+2)=n

²

²)d

x

d

s

(3.3)

+ ^Z

Q^2r

j

f

^jj

u

^;j

²

²d

x

d

s

+

c

¹ 1 + 1

"

Z

Q^2r

k

g

^k2d

x

d

s

for a.a.

t

²(

t

^0;4

r

²

t

⁰) and all ²

IR

^N

12

Z

B^2r

j

u

(

xt

)^;

u

~²r(

t

)^j2

²(

x

)d

x

²(

t

) + (

^0;

"

) ^Zt

t^0;4r²

Z

B^2r

k

Du

^k2

²

²d

x

d

s

c

² 1 + 1

"

1

r

²

Z

Q^2r

j

u

^;

u

~²r^j2d

x

d

s

+

c

^{2 Z}

Q^2r (1 +^j

u

^j2(n+2)=n

²

²)d

x

d

s

(3.4)

+ ^Z

Q^2r

j

f

^jj

u

^;

u

~²r^j

²

²d

x

d

s

+

c

² 1 + 1

"

Z

Q^2r

k

g

^k2d

x

d

s

for a.a.

t

²(

t

^0;4

r

²

t

⁰) where the constants

c

¹,

c

² depend neither on

r

nor on

"

. Proof. Let

k

be any integer

> T

^;1

t

⁰. The function

'

= (

u

k(

t

)^;)

²

²(

t

)

t

²(

t

^0;4

r

²

t

⁰) is admissible in (2.1). We obtain

12

Z

B^2r

j

u

k(

t

)^;j2

²d

x

²(

t

) + ^Zt

t^0;4r²

Z

B^2r(

a

_i)k(

D u

_ik)

²

²d

x

d

s

=

=^;2 ^Zt

t^0;4r²

Z

B^2r(

a

_i)k(

u

_ik^;i)

(

D

)

²d

x

d

s

3)Recall that^g2L2(^QIRnN) throughout.

+ ^Zt

t^0;4r²

Z

B^2r

j

u

k ^;j2

²

⁰d

x

d

s

+ ^Zt

t^0;4r²

Z

B^2r(

f

i)k(

u

_ik^;i)

²

²d

x

d

s

+ ^Zt

t^0;4r²

Z

B^2r(

g

_i)k(

D u

_ik)

²+ 2(

u

_ik^;i)

(

D

)]

²d

x

d

s

for all

t

²(

t

^0;4

r

²

t

⁰). Using Prop. 2.2 we nd by letting tend

k

^!+¹

12

Z

B^2r

j

u

(

t

)^;j2

²d

x

²(

t

) +

^{0 Zt}

t^0;4r²

Z

B^2r

k

Du

^k2

²

²d

x

d

s

2

c

⁰

r

t

Z

t^0;4r²

Z

B^2r(1 +^j

u

^j(n+2)=n+^k

Du

^k)^j

u

^;j

²d

x

d

s

+

c

⁰

r

²

t

Z

t^0;4r²

Z

B^2r

j

u

^;j2d

x

d

s

+ ^Zt

t^0;4r²

Z

B^2r

j

f

^jj

u

^;j

²

²d

x

d

s

+ ^Zt

t^0;4r²

Z

B^2r

k

g

^{k k}

Du

^k

²

²+ 2

c

⁰

r

^j

u

^;jd

x

d

s:

Then (3.3) is readily seen by employing Young's inequality.

To prove (3.4), we rst note that

Z

B^2r

u

_ik(

xt

)^;(^g

u

_ik)²_r(

t

)]

²(

x

)d

x

= 0 (

i

= 1

:::N

) for all

t

²(

t

^0;4

r

²

t

⁰). Hence

Z

B^2r

@u

_ik

@t

⁽

t

)(

u

_ik(

t

)^;(^g

u

_ik)²_r(

t

))

²d

x

_{= 12} ^Z

B^2r

dtd ^j

u

k(

t

)^;(^g

u

k)²_r(

t

)^j2

²d

x

and therefore

t

Z

t^0;4r²

Z

B^2r

@u

_ik

@s

⁽

u

_ik^;(^g

u

_ik)²_r)

²

²d

x

d

s

=

= 12_B^Z

2r

j

u

k(

t

)^;(^g

u

k)²_r(

t

)^j2

²d

x

²(

t

)^{; Z}

t^0;4r²

Z

B^2r

j

u

k^;(^g

u

k)²_r^j2

²

⁰d

x

d

s

for all

t

²(

t

^0;4

r

²

t

⁰).

Now we insert

'

= (

u

k(

t

)^;(^g

u

k)²_r(

t

))

²

²(

t

)

t

²(

t

^0;4

r

²

t

⁰) into (2.1) and nd

12

Z

B^2r

j

u

k(

xt

)^;(^g

u

k)²_r(

t

)^j2

²(

x

)d

x

²(

t

) + ^Zt

t^0;4r²

Z

B^2r(

a

_i)k(

D u

_ik)

²

²d

x

d

s

=

=^;2 ^Zt

t^0;4r²

Z

B^2r(

a

_i)k(

u

_ik^;(^g

u

_ik)²_r)

(

D

)

²d

x

d

s

+ ^Z

t^0;4r²

Z

B^2r

j

u

k ^;(^g

u

k)²_r^j2

²

⁰d

x

d

s

+ ^Z

t^0;4r²

Z

B^2r(

f

i)k(

u

_ik ^;(^g

u

_ik)²_r)

²

²d

x

d

s

+ ^Zt

t^0;4r²

Z

B^2r(

g

_i)k(

D u

_ik)

²+ 2(

u

_ik^;(^g

u

_ik)²_r)

(

D

)]

²d

x

d

s

for all

t

² (

t

^0;4

r

²

t

⁰). Letting

k

^! +¹ and using an analogous reasoning as above we nd (3.4).

From Theorem 3.1 we derive

COROLLARY 3.1 (CACCIOPPOLI inequalities) Assume (3.1). Then esssup

(t^0;r²t⁰⁾

Z

B^r

j

u

(

xt

)^;j2d

x

+^Z

Q^r

k

Du

^k2d

x

d

t

c

²

8

>

<

>

:

r

1²

Z

Q^2r

j

u

^;j2d

x

d

t

+ ^Z

Q^2r(1 +^j

u

^j2(n+2)=n+^k

g

^k2)d

x

d

t

(3.5)

+ ^Z

Q^2r

j

f

^{j2(n+2)=(n+4)}d

x

d

t

^(n+4)=(n+2)

9

>

=

>

for all ²

IR

^N, and esssup

(t^0;r²t⁰⁾

Z

B^r

j

u

(

xt

)^;

u

~r(

t

)^j2d

x

+^Z

Q^r

k

Du

^k2d

x

d

t

c

²

8

>

<

>

:

r

1²

Z

Q^2r

j

u

^;

u

~²r^j2d

x

d

t

+ ^Z

Q^2r(1 +^j

u

^j2(n+2)=n+^k

g

^k2)d

x

d

t

(3.6)

+ ^Z

Q^2r

j

f

^{j2(n+2)=(n+4)}d

x

d

t

^(n+4)=(n+2)

9

>

=

>

(

c

²= const

>

0 independent of

r

).

Proof. Fix

"

=

⁰

2 . Then (3.3) implies 12

Z

B^2r

j

u

(

xt

)^;j2

²(

x

)d

x

²(

t

) +

⁰ 2

t

Z

t^0;4r²

Z

B^2r

k

Du

^k2

²

²d

x

d

s

c

8

>

<

>

:

r

1²

Z

Q^2r

j

u

^;j2d

x

d

s

+ ^Z

Q^2r(1 +^j

u

^j2(n+2)=n+^k

g

^k2)d

x

d

s

(3.7)

+^Z

Q^2r

j

f

^jj

u

^;j

²

²d

x

d

s

9

>

=

>

for a.a.

t

²(

t

^0;4

r

²

t

⁰). Hence, the essential supremum of the function

t

^{7! Z}

B^2r

j

u

(

xt

)^;j2

²(

x

)d

x

²(

t

)

over the interval (

t

^0;4

r

²

t

⁰), as well as the integral

Z

Q^2r

k

Du

^k2

²

²d

x

d

s

are bounded by the right-hand side of (3.7). Thus esssup

(t^0;4r²t⁰⁾

Z

B^2r

j

u

(

xt

)^;j2

²(

x

)d

x

²(

t

) + ^Z

Q^2r

k

Du

^k2

²

²d

x

d

s

c

8

>

<

>

:

r

1²

Z

Q^2r

j

u

^;j2d

x

d

s

+ ^Z

Q^2r(1 +^j

u

^j2(n+2)=n+^k

g

^k2)d

x

d

s

(3.8)

+^Z

Q^2r

j

f

^jj

u

^;j

²

²d

x

d

s

9

>

=

>

:

Next, we combine the imbedding (1.5) and Holder's and Young's inequality to obtain

Z

Q^2r

j

f

^jj

u

^;j

²

²d

x

d

s

8

>

<

>

:

esssup

(t^0;4r²t⁰⁾

Z

B^2r

j

u

(

xt

)^;j2

²(

x

)d

x

²(

t

) + ^Z

Q^2r

k

D

((

u

^;)

²

²)^k2d

x

d

t

9

>

=

>

+

c

Z

Q^2r

j

f

^{j2(n+2)=(n+4)}d

x

d

t

^(n+4)=(n+2)

for all

>

0. Thus, by choosing

appropriately small, from (3.8) we obtain (3.5).

By an analogous argument, from (3.4) it follows that esssup

(t^0;4r²t⁰⁾

Z

B^2r

j

u

(

xt

)^;

u

~²r(

t

)^j2

²(

x

)d

x

²(

t

) + ^Z

Q^2r

k

Du

^k2

²

²d

x

d

t

c

8

>

<

>

:

r

1²

Z

Q^2r

j

u

^;

u

~²r^j2d

x

d

s

+ ^Z

Q^2r(1 +^j

u

^j2(n+2)=n

²

²+^k

g

^k2)d

x

d

s

(3.9)

+ ^Z

Q^2r

j

f

^{j2(n+2)=(n+4)}d

x

d

s

^(n+4)=(n+2)

9

>

=

>

:

We estimate the rst term on the left of (3.9) from below. To this end, let

r denote the cut-o function with respect to

B

r (analogously as

²r with respect to

B

²r cf. footnote 2). We have

Z

B^r

_r²d

x

^j

B

_r=^2j_{= 12n}^j

B

r^j

: