Asymptotic behaviour of solutions of semilinear hyperbolic systems in arbitrary domains

Asymptotic behaviour of solutions of semilinear hyperbolic systems in arbitrary domains

F.Jochmann

Institut fur angewandte Mathematik, Humboldt Universitat Berlin Unter den Linden 6 10099 Berlin Abstract:

In this paper the long time asymptotic behavior of solutions of semilinear symmetric hyperbolic system including Maxwell's equations and the scalar wave-equation in an ar- bitraty domain are investigated. The possibly nonlinear damping term may vanish on a certain subset of the domain. It is shown that the solution decays weakly to zero if and only if the initial-state is orthogonal to all stationary states. In the case that the nonlinear damping is in addition montone, also strong local L^q-convergence is shown.

AMS:

35B40, 35Q60, 35L05, 35L40, 78A35.

1 Introduction

The subject of this paper the long time asymptotic behavior of solutions of semilinear hyperbolic systems of the form

@t

E

=E⁽¹⁾

"

3

X

k⁼¹H_k@k

F

^!;

S

(tx

E

)

#

(1.1)

@t

F

^{=E(2) X3}

k⁼¹Hk@k

E

^(1.2)

with the initial-condition

E

^{(0x) =}

E

^0(x)

F

^{(0x) =}

F

^{0(x): (1.3)}

Here

E

²C(0¹)L²(IR^M)) and

F

²C(0¹)L²(IR^N)) are the unknown functions depending on the timet0 and the space-variablex ². IR³is an arbitrary domain.

Hk ²IR^NM are constant matrices, E^{(1) 2}L¹(IR^MM) and E^{(2) 2}L¹(IR^NN) are 1

positive symmetric variable matrices, which depend on the space-variable x ² and satisfy E⁽¹⁾ = 1 and E⁽²⁾ = 1 on ^{0 def}= ⁿGwith some subset G.

The generally nonlinear function

S

^{: 01)IRM !IRM satises}

S

^(tx

y

) = 0 for allx²⁰ = ⁿG

and

S

(tx0) = 0 for allx²t ²(0¹):

That meams that the damping-term

S

^(tx

E

) is only present on a certain subsetG. The following coerciveness-assumption is imposed.

yS

^(x

y

^{)(x) min fj}

y

^jpj

y

^{jgfor all}

y

^2IRMx2G:

Here p ² 2¹) and ² L¹(G) is a positive function on G, which does not necessarily have a uniform positive lower bound on G.

This means that

S

(tx

y

) is allowed to be bounded as ^jy^{j ! 1} and ^j

S

(tx

y

)^j behaves like ^jy^jp;1 for small ^jy^j. For example

S

(tx

E

)^def= (x)^j

E

^jq(1 +^j

E

^jq)^;1

E

with q ² 0¹) is possible.

A domain D(B) L²(IR^M+N) containing C⁰¹(IR^M+N) is chosen, such that the operartor

B(

E F

)^def=

E⁽¹⁾

3

X

k⁼¹H_k@k

F

^!E⁽²⁾

3

X

k⁼¹Hk@k

E

^!!

is skew-adjoint on D(B), i.e. B = ^;B with respect to a weighted scalar-product. The choice of D(B) involves boundary-conditions on @ supplementing 1.1-1.2.

A physically important example for this system are Maxwell'sequations describing the propagation of the electromagnetic eld

"@t

E

^{= curl}

H

^;

S

^(tx

E

^{) and @}t

H

^=;curl

E

^(1.4)

supplemented by the initial-boundary conditions

~n^{^}

E

^{= 0 on (01});¹~n^{^}

H

^{= 0 on (01});² (1.5)

E

(0x) =

E

⁰(x)

H

(0x) =

H

⁰(x): (1.6)

In 1.5 ;¹ @ and ;^{2 def}= @ⁿ;¹.

E

H

denote the electric and magnetic eld respectively which depend on the time t 0 and the space-variable x ² and

S

(tx

E

) describes a possibly nonlinear resistor. The dielectric and magnetic susceptibilities "²L¹() are assumed to be uniformly positive.

For 1.4, 1.5 the operator B is dened in the space X ^def= L²(CI⁶) by

B(

E

F

^{)def= (";1 curl}

F

^{;;1 curl}

E

^{) for (}

E

F

^{)2D(B)def= WE WH:}

2

Here WH is the closure of C⁰¹(IR³ⁿ;²CI³) inHcurl(), where Hcurl(), is the space of all

E

²L²(CI³) with curl

E

²L²().

WE denotes the set of all

E

²Hcurl(), such that

Z

E

curl

F

^;

F

curl

E

dx= 0 for all

F

²WH

which includes a weak formulation of the boundary-condition~n^{^}

E

^{= 0 on ;1, see 5].}

Another example for 1.1-1.2 is the rst-order system corresponding to the initial- boundary-value-problem of the scalar wave-equation with nonlinear damping, see 3], 4], 7].

@_t²'= div (E^r')^;S(x@t') (1.7)

supplemented by the initial-boundary-onditions

'= 0 on (0¹)@ (1.8)

'(0x) =f⁰(x) and @t'(0x) =f¹(x) (1.9) for initial-data f^{0 2}H⁰¹ () and f^{1 2} L²(). Here E ² L¹(IR³³) is a symmetric matrix-valued function satisfying E = 1 on ⁰ = ⁿG.

Note that

u

^{def= (@}t'E^r')²C(0¹)L²(IR⁴)) solves the system

@t

u

^{= ( div (}

u

^2::

u

^4)Er

u

^1);(S(tx

u

^{1)000) (1.10)}

which is of the form 1.1-1.3.

The set ^N of stationary states for 1.1-1.3 is the set of all

u

^{2D(B) with}

B

u

+F(t

u

) = 0 for all t 0, where the nonlinear operator F : (0¹)X ^! X is dened by

F(t

u

)^def= ^;E⁽¹⁾

S

(t

u

¹( ))0: From the assumptions on

S

it follows

f(

E F

)² ker B :

E

= 0 on G^gN:

Conversely assume

u

= (

E F

) ² D(B) with B

u

+F(t

u

) = 0. Since B is skew-adjoint this yields 0 =<

u

B

u

>X= ^; <

u

F(t

u

) >X= ^R

uS

(tx

E

)dx which implies

E

= 0 onG by the coerciveness assumption. Hence the set of stationary states is given by

N =^f(

E

F

^{)2 ker B :}

E

^{= 0 onGg:}

The aim of this paper is to show

(

E

^(t)

F

^{(t))t!1;! 0 in L2() weakly (1.11)}

3

if and only if the initial-data (

E

⁰

F

^{0)2L2() obey}

Z

E^(1);1

E

⁰

e

^+E(2);1

F

⁰

f

^dx= 0 for all (

e

f

^{)2N: (1.12)}

In the case of Maxwell's equations 1.4-1.6 the condition 1.12 on (

E

⁰

F

^{0) implies}

div ("

E

⁰) = 0 on ⁰ and div (

H

⁰) = 0 on (1.13) since ^N contains all elements of the form (^r'^r) with '²C⁰¹(⁰) and ²C⁰¹().

The proof of 1.11 is based on a suitable modication of the approach in 3], 11] for the case that the operator B does not necessarily have purely discrete spectrum. The basic idea is to show that for each f ² C⁰¹(IR^nf0^g) and

g

²!⁰(

E

⁰

F

⁰) the function f(iB)

g

is real-analytic and vanishes on G, where !⁰(

E

⁰

F

⁰) denotes the !-limit-set with respect to the weak topology of the orbit belonging to the initial-state (

E

⁰

F

⁰). This implies f(iB)

g

= 0 for allf ²C⁰¹(IR^nf0^g) and hence

g

^{2 ker} B. (Here the operator f(iB) can be dened by the spectral-theorem, since iB is self-adjoint in L²(CI^M+N).)

If

S

is independent of t and monotone with respect to

E

strong L^r-convergence is shown, i.e.

jj

E

(t)^jjL^r(K⁾+^jj

F

(t)^jjL²⁽K^)t!1;! 0 for all 1 r <2 and compact setsK (1.14) if the initial-data (

E

⁰

F

⁰)²L²() obey condition 1.12.

Finally 1.11 is used to prove that the solution the wave-equation 1.7-1.8 in an arbitrary domain IR³ decays with respect to the energy-norm on each bounded subdomain of . The case of a bounded domain has been considered in 3] and 11]. Here the domain is not necessarily bounded. For all R² (0¹), f^{0 2}H⁰¹ ()) and f^{1 2}L²()) it is shown that

jjr'(t)^jjL^{2( \}B^R)+^jj@t'(t)^jjL^{2( \}B^R)

t^!1

;!0:

2 Notation, Assumptions

For an arbitrary open set K IR³ the space of all innitely dierentiable functions with compact support contained inK is denoted by C⁰¹(K).

Let IR³ be a (connected) domain and let ⁰ be an open subset, such that G ^def= ⁿ⁰ has nonempty interior. The variable matrices E^{(1) 2} L¹(IR^(MM)) and E^{(2) 2}L¹(IR^(NN)) assumed to be symmetric and uniformly positive in the sense that y^? E⁽¹⁾(x)yc^0jy^j2 and z^? E⁽²⁾(x)z c^0jz^j2 (2.15) for allx²y²IR^M and z ²IR^N with some c^{0 2}(0¹) independent of xyz.

Next,

E⁽¹⁾(x) = 1 and E⁽²⁾(x) = 1 for allx²⁰ = ⁿG: (2.16) 4

The assumptions on

S

^{: 01)IRM !IRM} are the following.

S

^(tx

y

^{) = 0 ifx20 = nG (2.17)}

S

⁽

y

) measurable for xed

y

²IR^M (2.18)

and Lipschitz-continuous, i.e. there exists L²(0¹), such that

j

S

(tx

y

)^;

S

(tx

y

~)^j L^j

y

^;

y

^~j for all

y y

~ ²IR^M and x²: (2.19)

j

S

^(tx

y

^{)j2 C0 <}

y

S

^(tx

y

^{)> for allt 0x2G}

y

^{2IRM (2.20)}

with some C^{0 2}(0¹). Moreover,

yS

⁽x

y

⁾(x) min ^fj

y

^jpj

y

^{jgfor all}

y

²IR^Mx²G: (2.21) Here ² L¹(G) with > 0 and p ² 2¹). The function does not necessarily have a uniform positive lower bound on G. It follows from the two latter assumptions that

S

(tx

y

) = 0 if and only if

y

= 0 for allx²G.

In the sequel L^q(K) denotes for a measurable subset K G the weighted L^q-space endowed with the norm

jju^jjL^q(K^{) def}= ^Z_K^ju^jqdx^1=q where q²1¹) and as in 2.21.

The matricesHj ²IR^NM obey the following algebraic condition, which is fullled in the examples 1.4-1.6 and 1.7-1.9.

3

X

k⁼¹kHk

!

3

X

k⁼¹kH_k

!

3

X

k⁼¹kHk

!

=^jj2

3

X

k⁼¹kHk

!

for all ²IR³ (2.22) LetW⁰ L²(CI^M) be the space of all

e

^{2L2(CIM) with P3}k⁼¹@k(Hk

e

^{)2L2() in the}

sense of distributions endowed with the norm

jj

e

^jj2W^{0 def}= ^jj

e

^jj2L² +^jjX3

k⁼¹@k(Hk

e

^)jj2L²:

Furthermore, let D(A) with C⁰¹(CI^M) D(A) be closed subspace of W⁰ with respect to the above norm and

A

e

^{def= X3}

k⁼¹@k(Hk

e

^{) for}

e

²D(A): (2.23)

Then the adjoint operatorA obeys C⁰¹(CI^N)D(A ) and A

F

^=;X3

k⁼¹@k(H_k

F

^{) for all}

F

^{2D(A ): (2.24)}

5

For a vector

w

^2CIM+N we denote by

w

¹ the rst M and by

w

² the last N components of

w

.

Now, the following operators are dened.

LetD(B⁰)^def= D(A)D(A ) and

B⁰

w

^{def= (;A}

w

^2A

w

^{1) for}

w

^{2D(B0) = D(A)D(A ):}

Next, B ^def= EB⁰ with E ^def= diag (E⁽¹⁾E⁽²⁾), i.e. D(B)^def= D(B⁰) and

B

w

^def= EB⁰

w

=^;E⁽¹⁾A

w

²E⁽²⁾A

w

¹ (2.25) for

w

^2D(B). It turns out that B is a densely dened skew self-adjoint operator in the Hilbert-spaceX ^def= L²(CI^M+N) endowed with the scalar-product

<

F G

>X^def= ^Z E^;1

FG

dx

This follows from the closedness ofA, which implies thatA =A=A. (It is advantageous for following considerations to consider a complex space X. But whenever the

S

⁽tx

E

⁾

occurs in an equation, the function

E

is of course assumed to be real-valued.) Now, let^N be the set of all

a

² ker B with

a

¹(x) = 0 for all x²G.

Moreover, letX^{0 def}= ^N? be the space of all

w

^{2X with <}

u

w

^>X= 0 for all

u

^2N.

For

w

²L²(IR^M+N) a function

u

²C(IRX) is called a weak soution to the problem 1.1-1.3, if

dtd^h

u

^(t)

a

ⁱX =^;h

u

^(t)B

a

ⁱX +^hF(t

u

^(t))

a

ⁱX for all

a

^{2D(B) (2.26)}

Here F : (0¹)X ^!X is dened by F(t

u

)^def= ^;E⁽¹⁾

S

(t

u

¹( ))0:

2.26 is equivalent to the variation of constant formula

u

(t) = exp (tB)

w

+^{Z t}

0

exp((t^;s)B)F(s

u

(s))ds (2.27) where (exp (tB))t²IR is the unitary group generated by B. Since F(t ) is assumed to be Lipschitz-continuous in X by assumption 2.19, it follows from a standard result that this integal-equation has a unique solution

u

²C(0¹)X), see 8], ch.7.

2.27 yields the energy-estimate 12 d

dt^jj(t)^jj2_X =^hF(t

u

^(t))

u

^(t)iX =^;Z_G

S

^(tx

u

^(t)1)

u

^{(t)1dx 0: (2.28)}

In the sequel T( )

w

^2C(01)X) denotes the unique solution to 1.1-1.3 in the sense of 2.26.

6

3 Weak convergence for

^{t ! 1}

In the following lemma it is shown in particular that T( )

w

² L¹((0¹)X) , i. e.

jjT(t)

w

^jjX is bounded ast ^!1.

Lemma 1

^Suppose

w

²X and

u

(t)^def= T(t)

w

^{. Then}

Z

1

0

h

u

(t)F(t

u

(t))ⁱdt+^jj

u

(t)^jj2_X ^jj

w

^jj2X (3.29)

Z

1

0

jjF(t

u

⁽t))^jj2_Xdt C^0jj

w

^jj2X

with some C^{0 2}(0¹) independent of

w

. Moreover,

u

^{1 2Lp((01)L1(K})) for all bounded measurable subsets K G: (3.30)

Proof:

^Let

u

^{(t) = (}

E

^(t)

F

^{(t))def= T(t)}

w

. By the assumptions 2.20 on

S

^{one has}

jjF(t

f

^)jj2X C^0hF(t

f

⁾

f

^{i for all}

f

^2X

with some C⁰ >0 independent of

f

. Therefore, the energy-estimate 2.28 yields 12 d

dt^jjT(t)

w

^jj2X ^hF(t

f

)

f

^{i ;}C^0;1jjFT(t)

w

^jj2X: This implies 3.29 by Gronwall's lemma.

To prove 3.30 let

f

² X and dene

a b

²L²((GIR^M) by

a

(x)^def=

f

¹(x) if ^j

f

¹(x)^j 1 and

a

^{(x) def= 0 if j}

f

^{1(x)j >} 1. Moreover,

b

^{(x) def=}

f

^{1(x) if j}

f

^{1(x)j > 1 and}

b

^{(x) def= 0 if}

j

f

¹(x)^j 1.

Then it follows from assumption 2.21 that

a

⁽x)

S

⁽tx

a

⁽x))(x)^j

a

⁽x)^jp and

b

⁽x)

S

⁽tx

b

⁽x))(x)^j

b

⁽x)^j for allx²G. Holder's inequality yields

jj

f

^1jjL¹⁽K^{) jj}

a

^jjL¹⁽K⁾+^jj

b

^jjL¹⁽K⁾ (3.31) CK^1jj

a

^jjL^p(K⁾+^jj

b

^jjL¹⁽K⁾=CK¹

Z

G^j

a

^(x)jpdx1=p+Z_G^j

b

^(x)jdx

CK¹

Z

G

a

(x)

S

(tx

a

(x))dx^1=p+^Z_G

b

(x)

S

(tx

b

(x))dx CK¹

Z

G

f

^(x)

S

^(tx

f

^(x))dx1=p+Z_G

f

^(x)

S

^(tx

f

^(x))dx

=CK¹(^h

f

F(t

f

^)i)1=p+h

f

F(t

f

⁾ⁱ CK²

1 +^jj

f

^jj2;2X ^=p

(^h

f

F(t

f

^)i)1=p

Finally, the assertion follows from 3.29 and 3.31

2

7

Lemma 2

^X0\D(Bn) is dense in X^0\D(B^m) for all mn²IN with m < n.

Proof:

Let

w

²X^0\D(B^m) and dene

w

^def= ⁿ(^;B)^;n

w

²D(Bⁿ) for > 0. Then

jjB^k(

w

^;

w

^)jjX (3.32)

=^jjB^k

w

^;( ^;B)^;1]ⁿB^k

w

^jjX ^!1

;! 0 for allk ^2f01::m^g: Suppose

a

^2N. Then

<

w

a

^>X=<

w

^{n( +B);n}

a

^>X=<

w

a

^>X= 0: Hence

w

²X⁰. By 3.32 the proof is complete.

2

Lemma 3

ⁱ⁾

w

=B⁰²

w

^on for all

w

²(ran B⁰)^\D(B²⁰), in particular

;

e

^{=A A}

e

^and;

f

^=AA

f

^{on for all}

e

^{2(ranA )\D(A) and}

f

^{2(ranA)\D(A ).}

with A

e

^{2D(A ) and A}

f

^2D(A).

ii)

w

=B²

w

^{on 0} = ⁿG for all

w

²X^0\D(B²).

Proof:

Let

u

² C⁰¹(CI^M+N)D(B⁰ⁿ) for alln ²IN. Then it follows from 2.22 using Fourier- transform that

F(B⁰³

u

)1() =^;i

0

@ 3

X

j⁼¹jH_j

1

A

3

X

k⁼¹kHk

!

3

X

l⁼¹lH_l

!

F(

u

²)()

=^;i^jj2

3

X

l⁼¹lH_l

!

F(

u

²)() Analogously,

F(B⁰³

u

)2() =^;i^jj2

3

X

l⁼¹lHl

!

F(

u

¹)() and hence

B⁰³

u

=B⁰

u

for all

u

² C⁰¹(CI^M+N): (3.33) Now, assume

w

^2(ran B⁰)^\D(B⁰²), i.e.

w

⁼B⁰

v

^{with some}

v

²D(B⁰³). Then

Z (B⁰²

w

⁾

u

^dx=hB03

v

u

ⁱL² =^h

w

^B30

u

ⁱL²

8

=^h

v

^B0

u

ⁱL² =^h

w

u

ⁱL² =^Z

w

u

^dx

for all

u

^2C01(), which means B⁰²

w

⁼

w

in the sense of distributions.

To prove ii) let

w

² X^{0 \}D(B²). Suppose

u

² C⁰¹(⁰CI^M+N). and dene ~

u

^def= (B^02;)

u

^{2C01(0CIM+N) D(Bn0}). Then 3.33 yieldsB⁰

u

~ = 0 and hence ~

u

^{2N. In}

particular 0 =^h

w u

~ⁱ, because

w

²X⁰. Since E = 1 on ⁰, it followsB

u

~ =B⁰

u

~²D(B) and ~

u

= (B^2;)

u

. With

w

²X⁰ and ~

u

^2N one obtains

0 =^h

w

u

^{~iX =h}

w

^B2

u

^{iX ;h}

w

u

^{iX =hB2}

w

u

^{iX ;h}

w

u

^iX

=^Z B²

w

]

u

^;

w u

dx

Since for all

u

²C⁰¹(⁰CI^M+N) is arbitrary, the assertion follows.

2

Remark 1

Due to the facts that generally E^{(j) 6}= 1 and

a

^{1 = 0 on G for all}

a

^{2 N we}

have

w

^{1 6}= (B²

w

)1 on G for all

w

²X^0\D(B²) in general.

For example is the case of Maxwell's equations 1.4-1.6 all

w

²X^0\D(B²) obey (B²

w

^{)1 =;}"^;1 curl (^;1 curl

w

¹). The condition

w

²X⁰ implies

div ("

w

^{1) = 0 on 0 and div (}

w

²) = 0 on , as mentioned in the introduction, but it does not provide any information on the divergence of

w

^{1 on the set} G, since

a

¹ = 0 on G for all

a

^2N.

In the next lemma it is shown that X⁰ is an invariant space ofT(t).

Lemma 4

^{i) < T(t)}

w

a

^>X=<

w

a

^{>X for all}

w

^2X,

a

^{2N and t0.}

ii) T(t)

w

²X⁰ for all

w

²X⁰ and t 0.

Proof:

Suppose

w

² X⁰ and

a

^{2 N}, that means

a

² ker B and

a

¹ = 0 on G. Then 3.51 and 2.27 yield

< T(t)

w

a

^>X=<exp(tB)

w

^{;Z t}

0

exp ((t^;s)B)F(sT(s)

w

^)ds

a

^>X

=<

w

exp(^;tB)

a

>X ^;

Z t

0

< F(sT(s)

w

)exp((s^;t)B)

a

>X ds

=<

w

a

^>X ^;

Z t

0

< F(sT(s)

w

⁾

a

^>X ds=<

w

a

^>X : Hence, i) is proved. ii) follows from i) and the denition X^{0 def}= ^N?.

2

9

In the sequel let !⁰(

w

) denote the !-limit-set of the solution T( )

w

with respect to the weak topology of X, i.e. the set of all

g

²X, such that there exists a sequence tn ^{n!1;! 1}

with T(tn)

w

^n!1;!

g

ⁱⁿ X weakly, that means with < T(tn)

w f

>X^n!1;!<

g f

>X for all

f

^2X.

Since theT( )

w

²L¹((0¹)X) by lemma 1 the weak!-limit-set!⁰(

w

) in nonempty for all

w

²X.

Theorem 1

^{i) Let}

w

^{2X . Then !0(}

w

^)N.

Proof:

Let

u

^{(t) def= T(t)}

w

^{for t 2 IR. Suppose}

g

^{2 X and t}n ^{n!1;! 1} with T(tn)

w

^n!1;!

g

^{in X0}

weakly. Since

u

(tn+t) = exp (tB)

u

(tn) +^Z_t^tn+t

n

exp((tn+t^;)B)F

u

()d by 2.27, it follows from lemma 1, 3.29 that

jj

u

⁽tn+t)^;exp (tB)

u

⁽tn)^jjX

Z tⁿ⁺t

t^{n jj}F

u

⁽)^jjXd t^1=2Z_t^tn+t

n

jjF

u

()^jj2_Xd^{1=2 n!1;!} 0 for allt² IRand hence

u

^{(tn+t)n!1;! exp(tB)}

g

^{in X} weakly for all t²IR: (3.34) Suppose ² C⁰¹(IR) and dene

f

^{def= R}IR(t)exp(tB)

g

^{dt and}

f

^{(n) def= R}IR(t)

u

^(tn+t)dt Then 3.34 yields by the dominated convergence-theorem

h

f

⁽ⁿ⁾

h

^{iX =Z}_IR^(t)h

u

^(tn+t)

h

^iXdt

n!1

;! Z

IR(t)^hexp(tB)

g h

ⁱXdt =^h

f h

ⁱX

for all

h

²X , i.e.

f

^(n)n!1;!

f

weakly. In particular

f

^{(n)1 n!1;!}

f

^{1 in}L²(G)L¹(K) weakly for all bounded K G: (3.35) On the other hand it follows from lemma 1 iii) that

jj

f

^(n)1jjL¹⁽K^{) jjjj}_L^{p (}_IR⁾

Z b⁺tⁿ

a⁺t^{n jj}

u

^1(t)jjpL¹⁽K⁾dt

!

1=p

n!1

;! 0 (3.36)

10

for allt² IR. Here ab ²IRwith supp (ab). 3.35 and 3.36 yield

f

¹ = 0 onK for all bounded K Gand all ²C⁰¹(IR) where

f

^def= ^R_IR(t)exp(tB)

g

dt. This implies

(exp (tB)

g

^{)1 = 0 on Gfor all t2IR: (3.37)}

SinceiB is self-adjoint inX,f(iB) =^R_IRf()dE can be dened by the spectral-theorem for a Borel-measurable functionf :IR^!CI. Here (E)²IR denotes the family of spectral- projectors of iB. If f ²C⁰¹(IR), then bounded operator f(iB) has the representation

f(iB)

u

^=Z_IR^{f^(t)exp (;tB)}

u

^{dt for all}

u

^{2X: (3.38)}

Here ^f denotes the Fourier-transform of f. To see this let

u v

²X. Then

hf(iB)

u v

^i=Z_IRf()d^hE

u v

ⁱ

= (2)^;1=2Z_IR^Z_IRf^(t)exp(it)dtd^hE

u

v

ⁱ

= (2)^;1=2Z_IRf^(t)^Z_IRexp(it)d^hE

u

v

^idt

= (2)^;1=2Z_IRf^(t)^hexp (^;tB)

u v

ⁱdt

= (2)^;1=2hZ_IRf^(t)exp (^;tB)

u

dt

v

ⁱ

Suppose f ²C⁰¹(IR^nf0^g). Then 3.37 and 3.38 yield

(f(iB)

g

^{)1 = 0 on G: (3.39)}

Moreover,

f~(iB)

g

^=Bf(iB)

g

^{=;E(1)A (f(iB)}

g

^{)2E(2)A(f(iB)}

g

^{)1 on (3.40)}

where ~f() = f(). In particular 3.39 and 3.40 yield in the case that f is replaced by g()^def= ^;1f()²C⁰¹(IR^nf0^g) that

(f(iB)

g

^{)2 =E(2)A(g(iB)}

g

^{)1 = 0 on G}

and hence by 3.39

f(iB)

g

^{= 0 onG (3.41)}

11

Since E(x) = 1 on ⁿG, 3.39 - 3.41 yield

B⁰f(iB)

g

=B(f(iB)

g

) = ~f(iB)

g

for allf ²C⁰¹(IR^nf0^g) (3.42) with ~f() =f().

In particular it follows by induction

f(iB)

g

²( ran B⁰)^\D(B⁰ⁿ) with B⁰ⁿf(iB)

g

=Bⁿ(f(iB)

g

) (3.43) for allf ²C⁰¹(IR^nf0^g) and n²IN.

The aim of the following considerations is to show that f(iB)

g

is real analytic on . This will be achieved by means of a local integral representation.

Let f ²C⁰¹(IR^nf0^g) and choose ²C⁰¹(IR^nf0^g) with () = 1 on supp f. Dene

F

^{(t) def= exp(;tB)(iB)}

g

^{= (2);1=2Z}_IR^{^()exp((;t;)B)}

g

^d:

Then 3.43 and lemma 3 i) yield

@_t²

F

(t) =B²

F

(t) =B⁰²

F

(t) =

F

(3.44) in particular

@_nt

F

=Bⁿ

F

( )²L¹(IRL²()) and

ⁿ

F

=B⁰²ⁿ

F

( ) =B²ⁿ

F

( )²L¹(IRL²()) for all n²IN which implies

F

^{2C1(IR) and}

@jt@

F

²L¹(IR^K) for all compact ^Kj ²IN⁰ and ²IN⁰³: (3.45) Suppose x^{0 2} and choose R >0 with B²R. Let

K(x)^def= (4^jx^j)^;1f^(^;jx^j) for ²IR and x²IR³ Then 3.45 yields for allx ²BR=²(x⁰)

limr^!0

Z

IR

Z

@B^r(x⁾~n(y)K(x^;y)^ry

F

j(y) (3.46)

;

F

j(y)^ryK(x^;y)]dS(y)d

= (4)^;1_rlim

!0

r^;3Z_IRf^{^}(^;r)^Z_@B

r

(x⁾~n(y)(x^;y)]

F

j(y)dS(y)d

!

=^Z_IRf^()

F

j(x)d=^Z_IRf^()(exp(^;B)(iB)

g

⁾j(x)d 12

= (2)¹⁼²(f(iB)(iB)

g

⁾j(x) = (2)¹⁼²(f(iB)

g

⁾j(x): For all x²BR=²(x⁰) and ally²B²R(x⁰) with y⁶=x one has by 3.44

divyK(x^;y)^ry

F

j(y)^;

F

j(y)^ryK(x^;y)]

=K(x^;y)y

F

j(y)^;

F

j(y)yK(x^;y)

=K(x^;y)@²

F

^j(y);

F

^j(y)@2K(x^;y)

=@K(x^;y)@

F

^j(y);

F

^j(y)@K(x;y)]

and hence

Z

IR

Z

@B^R(x⁰⁾~n(y)K(x^;y)^ry

F

j(y) (3.47)

;

F

^{j(y)ryK(x;y)]dS(y)d}

; Z

IR

Z

@B^r(x⁾~n(y)K(x^;y)^ry

F

^j(y);

F

^{j(y)ryK(x;y)]dS(y)d}

=^Z_IR^Z_B

R

(x⁰⁾ⁿB^r(x⁾ div yK(x^;y)^ry

F

j(y)

;

F

j(y)^ryK(x^;y)]dyd

=^Z_B

R

(x⁰⁾ⁿB^r(x⁾

Z

IR@K(x^;y)@

F

j(y)

;

F

j(y)@K(x^;y)]ddy= 0

since K(x^;y)^j;j!1! 0 and@K(x^;y)^j;j!1! 0, whereas

F

^{and @}

F

remains bounded as ^jj!1 by 3.45 for xedy ⁶=x.

Now, 3.46 and 3.47 yield for all x²BR=²(x⁰) (2)¹⁼²(f(iB)

g

)j(x) =^Z_IR^Z_@B

R

(x⁰⁾~n(y)K(x^;y)^ry

F

j(y) (3.48)

;

F

j(y)^ryK(x^;y)]dS(y)d

Since f ²C⁰¹(IR), there exists a constant C^{1 2}(0¹) with (1 +²)^jf^^(k)()^j C^1k for all ²IRand k ²IN:

13