Boundary controllability in problems of transmission for a class of second order hyperbolic systems

BOUNDARY CONTROLLABILITY IN PROBLEMS

OF TRANSMISSION FOR A CLASS OF

SECOND ORDER HYPERBOLIC SYSTEMS

JOHN E. LAGNESE

Abstract. We consider transmission problems for general second or- der linear hyperbolic systems having piecewise constant coecients in a bounded, open connected set with smooth boundary and controlled through the Dirichlet boundary condition. It is proved that such a sys- tem is exactly controllable in an appropriate function space provided the interfaces where the coecients have a jump discontinuity are all star-shaped with respect to one and the same point and the coecients satisfy a certain monotonicity condition.

1. Introduction and main results

Let , ¹be bounded, open, connected sets in^Rnwith smooth boundaries

; and ;¹, respectively, such that ¹ , and set ² = ⁿ¹, whose boundary is ;² = ;;¹. Let

m

and

n

be positive integers. For

ik

= 1

::: m

, for

j`

= 1

::: n

and for

= 1

2, let

a

^ijk` be real numbers having the symmetry

a

^ijk` =

a

^k`ij = 1

2

ik

= 1

::: m j`

= 1

::: n:

(1.1) When

m

=

n

we further assume that

a

^ijk` =

a

^jik` =

a

^ij`k = 1

2

ijk`

= 1

::: n:

(1.2) The

a

^ijk` are also assumed to satisfy the following ellipticity condition:

if

m

⁶=

n

then

a

^ijk`

ij

k`

c

⁰

ij

ij

c

⁰

>

0

⁸(

ij)^{2Rm n} (1.3) if

m

=

n

then

a

^ijk`

ij

k`

c

⁰

ij

ij

c

⁰

>

0

⁸(

ij)²

S

ⁿ

(1.4) where

S

ⁿ denotes the set of all real symmetric

n

n

matrices. In (1.3), (1.4) and in what follows, a repeated Roman index indicates summation with respect to that index over an appropriate range (either 1 to

m

or 1 to

n

, as the context suggests), unless otherwise noted.

Fix

T >

0 and let

W

1 =

0

B

@

w

¹¹

w

...¹m

1

C

A

^W2 =

0

B

@

w

²¹

w

...²m

1

C

A

Department of Mathematics, Georgetown University, Washington, DC 20057 USA.

Received by the journal April 15, 1997. Accepted for publication October 13, 1997.

c Societe de Mathematiques Appliquees et Industrielles. Typeset by L^ATEX.

be vector functions dened on the cylinders

Q

¹ = ¹(0

T

) and

Q

² = ² (0

T

), respectively. We set = ;(0

T

) and ¹ = ;¹(0

T

) and consider the following problem of transmission:

(

w

¹i^;

a

^ijk`1

w

¹k`j= 0 in

Q

¹

w

²i^;

a

^ijk`2

w

²k`j= 0 in

Q

²

i

= 1

::: m

^(1.5) where _=

@=@t

and where a subscript following a comma indicates dieren- tiation with respect to the corresponding spatial variable

W

2 =^U on

(1.6)

w

¹i=

w

²i

a

^ijk`1

w

¹k`

j =

a

^ijk`2

w

²k`

j on ¹

i

= 1

::: m

(1.7) where

= (

¹

:::

n) is the unit normal to ;¹ pointing into ¹ (and towards the exterior of ²),

W^jt⁼⁰= _^Wjt⁼⁰= 0 in ,

= 1

2

:

(1.8) The function ^U in (1.6) is viewed as a control function which is chosen to eect the evolution of the dynamics. More precisely, we consider below the reachability problem, which is to determine the set

RT =^f(^W1(

T

)

^W2(

T

)

^{W_ 1}(

T

)

^{W_ 2}(

T

)) : ^U2Ug where ^U is an appropriate space of controls.

The question of boundary controllability in problems of transmission has been considered by several authors. In particular, J.-L. Lions considered the system (1.5) - (1.8) in the special case of two wave equations, that is, he considered the problem

(

w

^1;

a

¹

w

¹ = 0 in

Q

¹

w

^2;

a

²

w

² = 0 in

Q

²

^(1.9) where

a

>

0 and is the ordinary Laplacian in ^Rn,

w

² =

u

on

(1.10)

w

¹ =

w

²

a

¹

@w

¹

@

⁼

a

²

@w

²

@

^{on 1}

(1.11)

w

^jt⁼⁰= _

w

^jt⁼⁰= 0 in ,

= 1

2

:

(1.12) For this problem the following result is proved in 5, Chapter VI].

Theorem 1.1. Assume that ;¹ is star-shaped with respect to some point

x

^{0 21} and set ;(

x

⁰) =^f

x

²;^j(

x

^;

x

⁰)

>

0^g, (

x

⁰) = ;(

x

⁰)(0

T

), where

is the unit outer normal to;. Let

RT =^f(

w

¹(

T

)

w

²(

T

)

w

_¹(

T

)

w

_²(

T

)) :

u

²

L

²()

u

= 0 on ⁿ(

x

⁰)^g

:

If

a

¹

> a

² and

T > T

(

x

⁰) := 2

R

(

x

⁰)

=

^p

a

², where

R

(

x

⁰) = max_x²2

j

x

^;

x

^0j, then ^RT =

H

V

⁰ where

H

=

L

²(¹)

L

²(²)

V

=^f(

'

¹

'

²)²

H

¹(¹)

H

¹(²) :

'

^2j;= 0

'

^1j;1 =

'

^2j;1g

:

ESAIM: Cocv, , ,

The main purpose of this paper is prove the following generalization of Theorem 1.1 to the system (1.5) - (1.8). Let us dene

j

A

^2j=

8

>

>

>

>

<

>

>

>

>

:

inf

06=(ij^)2Rm n

a

^ijk`2

kl

ij

ij

ij

m

⁶=

n

inf

06=(ij⁾²Sⁿ

a

^ijk`2

kl

ij

ij

ij

m

=

n:

^(1.13)

Theorem 1.2. Assume that ;¹ is star-shaped with respect to some point

x

⁰²¹ and let;(

x

⁰), (

x

⁰) and

R

(

x

⁰) be as in Theorem 1.1. Let

RT=^f(^W1(

T

)

^W2(

T

)

^{W_ 1}(

T

)

^{W_ 2}(

T

)):^U2

L

²()]^m

^U=0 on ⁿ(

x

⁰)^g

:

If

8

<

:

a

^ijk`1

ij

k`

a

^ijk`2

ij

k`

⁸(

ij)^{2Rm n}

m

⁶=

n

a

^ijk`1

ij

k`

a

^ijk`2

ij

k`

⁸(

ij)²

S

ⁿ

m

=

n

^(1.14) and if

T > T

(

x

⁰) =

8

>

>

>

>

<

>

>

>

>

:

2

R

(

x

⁰)

p

j

A

^2j

m

⁶=

n

2^p2

R

(

x

⁰)

p

j

A

^2j

m

=

n:

^(1.15) then ^RT =

H

V

⁰ where

H

=

L

²(¹)]^m

L

²(²)]^m

V

=^f(¹

²)²

H

¹(¹)]^m

H

¹(²)]^m : ^2j;= 0

^1j;1 = ^2j;1g

:

Remark 1.3. Whether or not the monotonicity condition (1.14)is necessary for exact controllability in dimension

n

2 is unknown, even in the case of ordinary wave equations considered in Theorem 1.1. It is known that no such condition is necessary for exact boundary controllability of transmission problems in dimension one or, more generally, for exact controllability of 1- dimensional networks in ^R3 3, Chapter IV].

Remark 1.4. Set

H

^;1(²) =^f

'

²

H

¹(²) :

'

^j;= 0^g

:

The space

V

is a closed subspace of

H

¹(¹)]^m

H

^;1(²)]^m and

V

⁰ may be identied with the restriction to

V

of elements in the product space (

H

¹(¹))⁰]^m(

H

^;1(²))⁰]^m (cf. 5, Remark 5.1, p. 376].

Remark 1.5. With only minor modications in the proof, Theorem 1.2 can be extended to the situation involving and

p

2 nested open sets

!

¹

::: !

p with

!

i

!

i⁺¹

i

= 1

::: p

^;1

!

p=

:

Set ¹ =

!

¹

i =

!

iⁿ

!

i^;1

i

= 1

::: p:

Setting ;i:=

@!

i, one should assume that ;iis star-shaped with respect to a point

x

^{0 21},

i

= 1

::: p

^;1, and that the parameters

a

^ijk` ,

= 1

::: p

,

ESAIM: Cocv, , ,

associated with satisfy appropriate monotonicity conditions with respect to

.

Example 1.6. (Wave equations) Suppose that

m

= 1. In this case the system (1.5) may be written

(

w

^1;

a

^j`1

w

¹j` in

Q

¹

w

^2;

a

^j`2

w

²j` in

Q

²

where

a

j`=

a

`j. If we set

A

= (

a

j`), the condition (1.14) with

m

= 1

< n

becomes the requirement that the matrix

A

^1;

A

² be positive semidenite.

The control time

T

(

x

⁰) is given by 2

R

(

x

⁰)

=

^pj

A

^2j, where

j

A

^2j= inf

06=^2Rn

A

²

j

^j2

:

In particular, if

A

=

a

I

, where

I

is the identity matrix in ^Rn, we recover Theorem 1.1.

Example 1.7. (Linear isotropic elastodynamics) Here

n

=

m

= 3 and the system of equations is (we do not use summation convention in this example)

8

>

>

>

>

>

<

>

>

>

>

>

:

¹

w

¹i^;

¹

w

¹i^;(

¹+

¹)^X3

j⁼¹

w

¹jji= 0 in

Q

¹

²

w

²i^;

²

w

²i^;(

²+

²)^X3

j⁼¹

w

²jji= 0 in

Q

²

i

= 1

2

3

where

is the (constant) mass density per unit of reference volume and

,

are the Lam e parameters for the elastic material occupying the region . We assume that

>

0 and

0

:

(1.16)

By a change of notation we may assume that

¹ =

²= 1. Then the nonzero coe!cients

a

^ijk` are given by

a

ⁱⁱⁱⁱ =

+ 2

i

= 1

2

3

a

^ijij =

a

^ijji =

a

^iijj =

i

⁶=

j

= 1

2

3

and therefore for (

ij)²

S

³ we have

X

a

^ijk`

ij

k`= (

+ 2

)^X3

i⁼¹

²_ii+

3

X

ij_i⁼¹

6=j

ii

jj+

2

3

X

ij_i⁼¹

6=j

(

ij +

ji)²

=

3

X

i⁼¹

ii

!

2

+ 2

3

X

ij⁼¹

²_ij 2

3

X

ij⁼¹

_ij²

:

In the same way, the monotonicity condition (1.14) (with strict inequality) will be satised if

¹

² and

¹

²

:

The control time

T

(

x

⁰) is given by 2^p2

R

(

x

⁰)

=

^pj

A

^2j= 2

R

(

x

⁰)

=

^p

², which is what is expected since the two wave speeds associated with the region ² are ^p

² and ^p

²+ 2

².

ESAIM: Cocv, , ,

Let us brie"y recount other works on boundary controllability in problems of transmission or, more generally, in interface problems. These works are for the most part disjoint from the present paper. For one-dimensional interface problems (for example, string or beam equations on one-dimensional graphs in ^R3), one may refer to 3] for a comprehensive analysis of the reachabil- ity problem utilizing controls supported at the nodes of the graph. These systems, however, do not generally fall within the framework of the prob- lem (1.5) - (1.8). For higher dimensional systems, one may cite 2], where boundary controllability in transmissions problems for the Reissner-Mindlin system of thin plate theory is considered. This system consists of two cou- pled second order equations with constant coe!cients but with lower order terms as well, so they too fall outside of the scope of the present work. There are also several works dealing with boundary controllability in problems of transmission or in interface problems for the wave equation in the singular case. If, for example, ;^1\; ⁶= in the problem (1.9) - (1.12) above, the solution will generally suer a loss of regularity in a neighborhood of points belonging to this set. In this case the proof technique for Theorem 1.1 given in 5] fails that is the reason for the assumption ¹ . In the singular case, we can refer to two recent papers of S. Nicaise 6], 7], where the question of exact boundary controllability for the wave equation on two-dimensional networks in ^Rnis treated. In these papers it is shown, among other things, that a result analogous to Theorem 1.1 may be obtained provided that, in addition to boundary controls, controls are also imposed near the singular vertices. The reader is referred to the cited papers for the precise statements of Nicaise's results. References to other works related to the topic of the present paper may be found in the bibliographies of the books and papers cited above.

2. A priori estimates Let

H

and

V

be as in Theorem 1.2. For

k

0 set

H

k() =

H

^k(¹)]^m

H

^k(²)]^m

:

Let := (¹

²), $ := ($¹

$²) be elements of

V

and consider the bilinear form

(

$) =^X2

⁼¹

Z

a

^ijk`

'

k`

ij

dx

$²

V:

This form is continuous on

V

V

in the topology induced by^H1() and is symmetric by virtue of (1.1). If

m

⁶=

n

,

is also coercive on

V

because of the ellipticity assumption (1.3) and ^2j; = 0, that is, for some

k >

0,

(

)

k

^kk2H1()

⁸²

V:

On the other hand, if

m

=

n

, because of the symmetry assumptions (1.2) we may write

(

$) =^X2

⁼¹

Z

a

^ijk`

"

k`()

"

ij($)

dx

ESAIM: Cocv, , ,

where

"

k`() = 12(

'

k`+

'

`k)

:

If ²

V

then the functions

'

i(

x

) =

(

'

¹i(

x

)

x

²¹

'

²i(

x

)

x

²²

i

= 1

::: m

are in

H

⁰¹() and therefore the following Korn's inequality holds 8, Theorem 2.1]:

2

X

⁼¹

Z

"

ij()

"

ij()

dx

¹₂^X2

⁼¹

Z

'

ij

'

ij

dx:

(2.1) It follows from (2.1) and (1.4) that

(

)

c

o

2

X

⁼¹

Z

"

ij()

"

ij()

dx

k

^kk2H1()

⁸²

V

Therefore in all cases

(

)]¹⁼²denes a norm on

V

equivalent to the one induced by^H1(). We retopologize

V

with this norm and let

V

⁰denote the dual space of

V

. With

H

identied with its dual space we have the dense and continuous embeddings

V H V

⁰.

Let us consider the homogeneous boundary value problem

(

'

¹i^;

a

^ijk`1

'

¹k`j= 0 in

Q

¹

'

²i^;

a

^ijk`2

'

²k`j= 0 in

Q

²

i

= 1

::: m

^(2.2)

²= 0 on

(2.3)

'

¹i =

'

²i

a

^ijk`1

'

¹k`

j =

a

^ijk`2

'

²k`

j on ¹

i

= 1

::: m

(2.4) ^jt⁼⁰= ⁰

_^jt⁼⁰= ¹ in ,

= 1

2

:

(2.5) Assume that ⁰ := (⁰¹

⁰²) ²

V

, ¹ := (¹¹

¹²) ²

H

. According to standard variational theory (2.2) - (2.5) has a unique solution in the fol- lowing sense: there is a unique function := (¹

²) ²

C

(0

T

]

V

) ^\

C

¹(0

T

]

H

)^\

C

²(0

T

]

V

⁰) such that

h

$ⁱV +

(

$) = 0

⁸$²

V

0

< t

T

and which satises (2.5), where ^h

$ⁱV denotes the pairing between ele- ments ²

V

⁰ and $²

V

in the

V

^0;

V

duality. Furthermore, the mapping

S

(

t

) : (⁰

¹)^7!((

t

)

_(

t

)) is a denes a unitary group on

V

H

, that is

k(

t

)^k2_V +^k_(

t

)^k2_H =^k0k2_V +^k1k2_H

t

0

:

(2.6) Let

A

denote the canonical isomorphism of

V

onto

V

⁰, so that

(

$) =

h

A

$ⁱV,⁸

$²

V

, and set

D

(

A

) =^f$²

V

:

A

$²

H

^g

^k$^kD⁽A⁾=^k

A

$^kH

:

If (⁰

¹)²

D

(

A

)

V

, then ²

C

(0

T

]

D

(

A

))^\

C

¹(0

T

]

V

)^\

C

²(0

T

]

H

) and

S

(

t

) is a unitary group on

D

(

A

)

V

. Because of our assumptions on the regions and ¹, an element ²

D

(

A

) has the regularity ²

ESAIM: Cocv, , ,

H ()^\

V

and

A

=^;(

a

^ijk`1

'

¹kj`

a

^ijk`2

'

²kj`). Moreover, satises both of the transmission conditions in (2.4) in the trace sense and the norm

k kD⁽A⁾ is equivalent to that induced on

D

(

A

) by^H2(). These results are standard and may be proved by the usual methods.

Lemma 2.1. Suppose that (⁰

¹) ²

D

(

A

)

V

, and let

h

²

W

¹¹()]ⁿ. Then the solution of (2.2) - (2.5) satises the following identity:

2

X

⁼¹

Z

'

_i(

h

^r

'

i)

dx

T

0

+ 12

Z

Q(div

h

) _

'

i

'

_i^;

a

^ijk`

'

k`

'

ij]

dxdt

+

Z

Q

a

^ijk`

'

k`

'

ip

h

pj

dxdt

= 12

Z

(

h

)

a

^ijk`2

@'

²k

@

@'

²i

@

j

`

d

+ 12

Z

1

(

h

)^h(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹ij

'

¹k`

+

a

^ijk`2

@'

¹i

@

^;

@'

²i

@

@'

¹k

@

^;

@'

²k

@

j

`

d :

(2.7) Proof. The proof utilizes classical multipliers. Form

0 =

Z

Q(

'

i^;

a

^ijk`

'

k`j)

h

^r

'

i

dxdt

(2.8) and integrate by parts. One has

Z

Q

'

i(

h

^r

'

i)

dxdt

=

Z

'

_i(

h

^r

'

i)

dx

T

0

+ 12

Z

Q(div

h

) _

'

i

'

_i

dxdt

^;1₂^Z

(

h

) _

'

i

'

_i

d

(2.9) where =

@

(0

T

) and

is the unit normal vector pointing into the exterior of . In addition,

Z

Q(

a

^ijk`

'

k`j)

h

^r

'

i

dxdt

=

Z

a

^ijk`

'

k`

'

ip

h

p

j

d

; Z

Q

a

^ijk`

'

k`

'

ip

h

pj

dxdt

^;Z

Q

a

^ijk`

'

k`

'

ipj

h

p

dxdt:

From the symmetry property (1.1) we obtain

a

^ijk`

'

k`

'

ipj = 12(

a

^ijk`

'

k`

'

ij)p

ESAIM: Cocv, , ,

and therefore

Z

Q(

a

^ijk`

'

k`j)

h

^r

'

i

dxdt

=

Z

a

^ijk`

'

k`

'

ip

h

p

j

d

^;1 2

Z

(

h

)

a

^ijk`

'

k`

'

ij

d

; Z

Q

a

^ijk`

'

k`

'

ip

h

pj

dxdt

_{+ 12}^Z_Q

(div

h

)

a

^ijk`

'

k`

'

ij

dxdt:

(2.10) We then obtain from (2.8) - (2.10) the identity

Z

'

_i(

h

^r

'

i)

dx

T

0

+

Z

Q

a

^ijk`

'

k`

'

ip

h

pj

dxdt

+ 12

Z

Q(div

h

) _

'

i

'

_i^;

a

^ijk`

'

k`

'

ij]

dxdt

= 12

Z

(

h

)( _

'

i

'

_i^;

a

^ijk`

'

k`

'

ij)

d

+

Z

a

^ijk`

'

k`

'

ip

h

p

j

d :

(2.11) We sum (2.11) over

= 1

2, noting the relations

'

¹i^j1 =

'

²i^j1

'

²ij^j =

@'

²i

@

^j

i

= 1

::: m j

= 1

::: n

and observing that along the boundary region ;¹ that is common to ¹ and ² the unit normal vector pointing out of ² points into ¹. We obtain

2

X

⁼¹

(

Z

'

_i(

h

^r

'

i)

dx

T

0

+^Z

Q

a

^ijk`

'

k`

'

ip

h

pj

dxdt

+ 12

Z

Q(div

h

)( _

'

i

'

_i^;

a

^ijk`

'

k`

'

ij)

dxdt

= 12

Z

(

h

)

a

^ijk`2

@'

²k

@

@'

²i

@

j

`

d

+ 12

Z

1

(

h

)(

a

^ijk`1

'

¹k`

'

¹ij^;

a

^ijk`2

'

²k`

'

²ij)

d

+

Z

1

(

a

^ijk`2

'

²k`

'

²ip^;

a

^ijk`1

'

¹k`

'

¹ip)

h

p

j

d :

(2.12) To complete the proof we need to show that the last two integrals in (2.12) agree with the last integral in (2.7).

Consider the last integral in (2.12). By choosing suitable extensions of ¹ and ² we may assume that

'

i²

H

²() for

= 1

2 and

i

= 1

::: n

. We have

(

a

^ijk`2

'

²k`

'

²ip^;

a

^ijk`1

'

¹k`

'

¹ip)

h

p

j

=

a

^ijk`2

'

²k`

j

h

p(

'

²ip^;

'

¹ip) = (

h

)

a

^ijk`2

'

²k`

j

@'

²i

@

^;

@'

¹i

@

(2.13)

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on ¹, since (

'

²ip^;

'

¹ip)^j;1 =

@'

²i

@

^;

@'

¹i

@

p

i

= 1

::: m p

= 1

::: n:

(2.14) It also follows from (2.14) that on ¹

a

^ijk`1

'

¹k`

'

¹ij =

a

^ijk`1

'

¹k`

'

²ij+

@'

¹i

@

^;

@'

²i

@

j

and

a

^ijk`2

'

²k`

'

²ij=

a

^ijk`2

'

²k`

'

¹ij^;

@'

¹i

@

^;

@'

²i

@

j

:

If the last expression is subtracted from the preceding one we obtain

a

^ijk`1

'

¹k`

'

¹ij^;

a

^ijk`2

'

²k`

'

²ij

= (

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

'

²ij+ 2

a

^ijk`2

'

²k`

j

@'

¹i

@

^;

@'

²i

@

on ¹

(2.15) where we have used the symmetry property (1.1) and the second transmis- sion condition in (2.4). It follows from (2.13) and (2.15) that

12

Z

1

(

h

)(

a

^ijk`1

'

¹k`

'

¹ij^;

a

^ijk`2

'

²k`

'

²ij)

d

+

Z

1

(

a

^ijk`2

'

²k`

'

²ip

h

p

j^;

a

^ijk`1

'

¹k`

'

¹ip

h

p

j)

d

= 12

Z

1

(

h

)(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

'

²ij

d :

(2.16) We use (2.14) once again to obtain on ¹

(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

'

²ij

=(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

'

¹ij+ (

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

j

@'

²i

@

^;

@'

¹i

@

=(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

'

¹ij+

a

^ijk`2

'

²k`

j

@'

²i

@

^;

@'

¹i

@

;

a

^ijk`2

'

¹k`

j

@'

²i

@

^;

@'

¹i

@

=(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

'

¹ij+

a

^ijk`2 (

'

²k`^;

'

¹k`)

j

@'

²i

@

^;

@'

¹i

@

=(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹k`

'

¹ij+

a

^ijk`2

@'

²k

@

^;

@'

¹k

@

@'

²i

@

^;

@'

¹i

@

j

`

:

(2.17) Now substitute (2.17) into (2.16).

Theorem 2.2. Let (⁰

¹) ²

V

H

. Then the solution of (2.2) - (2.5) satises

Z

a

^ijk`2

@'

²k

@

@'

²i

@

j

`

d

C

(

T

+ 1)(^k0k2_V +^k1k2_H) (2.18) for some constant

C

.

ESAIM: Cocv, , ,

Proof. By density it su!ces to prove (2.18) for (

) ²

D

(

A

)

V

. Since

; and ;¹ are smooth, there is a vector eld

h

²

C

¹()]ⁿ such that

h

=

on ; and

h

= 0 on ;¹ (see Komornik 1, Lemma 2.1]). We utilize this vector eld in (2.7) and observe that the left side of (2.7) is bounded above in absolute value by

C

o(^k_(

T

)^k2_H+^k(

T

)^k2_V+^k1k2_H+^k0k2_V)+

C

¹

T

sup

0T^](^k(

t

)^k2_V+^k_(

t

)^k2_H)

C

(

T

+ 1)(^k1k2_H+^k0k2_V) in view of (2.6).

Remark 2.3. If

m

⁶=

n

is follows from (2.18) and (1.3) that

@'

²i

@

²

L

²()

i

= 1

::: m:

(2.19) If

m

=

n

, then (2.18) and (1.4) imply that

@'

²i

@

^{j +}

@'

²j

@

^{i 2}

L

²()

ij

= 1

::: n

so, in particular,

i

@'

²i

=@

²

L

²() for each

i

= 1

::: n

. Since

n

X

ij⁼¹

@'

²i

@

^j+

@'

²j

@

ⁱ

2 = 2^Xn

i⁼¹

@'

²i

@

2+ 2 ^Xn

ij⁼¹

i

j

@'

²i

@ @'

²j

@

and the second sum is in

L

¹(), it follows that (2.19) holds true also in this case.

Theorem 2.4. Assume that ;¹ is star-shaped with respect to some point

x

⁰²¹ and suppose that the coecients

a

^ijk` satisfy (1.14). Let (⁰

¹)²

V

H

. Then the solution of (2.2) - (2.5) satises

(

T

^;

T

(

x

⁰))(^k1k2_H+^k0k2_V)

Z

(x⁰⁾(

h

)

a

^ijk`2

@'

²k

@

@'

²i

@

j

`

d

(2.20) where

T

(

x

⁰) is given by (1.15) and

h

(

x

) =

x

^;

x

⁰.

Proof. Because of (2.18) it su!ces to prove (2.20) for (⁰

¹)²

D

(

A

)

V

. We apply (2.7) with

h

(

x

) =

x

^;

x

⁰. Then

h

0 on ;¹. On the other hand, assumption (1.14) and the standing assumption (1.1) (or (1.2)) imply that

(

a

^{ijk`1 ;}

a

^ijk`2 )

'

¹ij

'

¹k`+

a

^ijk`2

@'

¹i

@

^;

@'

²i

@

@'

¹k

@

^;

@'

²k

@

j

` 0

:

Therefore

2

X

⁼¹

Z

'

_i(

h

^r

'

i)

dx

T

0

+

n

2

Z

Q

'

_i

'

_i

dxdt

+ (1^;

n

2)

Z

Q

a

^ijk`

'

k`

'

j

dxdt

+

Z

Q

a

^ijk`

'

k`

'

ip

h

pj

dxdt

12

Z

(x⁰⁾(

h

)

a

^ijk`2

@'

²k

@

@'

²i

@

j

`

d :

(2.21)

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