On stability of a transmission problem

(1)

P[D,g]

ON STABILITY OF A TRANSMISSION PROBLEM

HYEONBAE KANG AND JIN KEUN SEO

ABSTRACT. We investigate the behavior of the gradient of solutions to the refraction equation div((1

+

(k - I)XD)\7U) = 0 under perturbation of domainD. IfU and ^Uh are solutions to the refraction equation corresponding to subdomainsD and Dh of a domain

n

in 2 dimensional plane with the same Neumann data on

an,

^respec-

tively, we prove that 1I\7(u - uh)IIL2(fl) ~ CVdist(D,Dh) where dist(D,D_{h )} is the Hausdorff distance between D and D_{h .} We also show that this is the best possible result.

1.

Introduction and statement of results

Let 0 be a simply connected bounded domain in

jRn

(n 2: 2) with the C

²

smooth boundary and let D be a simply connected C

²

subdomain of 0 with closure in O. Let k i= 1 be a positive number. Consider the following Neumann problem

div((l + (k -

l)XD)\7u)

= ° ⁱⁿ ⁰

ou

⁼

⁹ on 00, r ⁹

⁼

^0, ⁹

^E

^L

²

⁽⁰⁰⁾

ov ian

L ^u=O,

where

v(x)

is the unit normal to the boundary,

~~ = v . \7u,

and

XD

is the characteristic function of D.

In this paper we study stability of the solution to the transmission problem P[D,g] under the perturbation of D. This study is moti- vated in relation to the inverse problem to P[D, g], namely the inverse conductivity problem.

Received May 17, 1997.

1991 Mathematics Subject Classification: 35B35.

Key words and phrases: conductivity problem, stability, layer potential.

The both authors are partially supported by GARC-KOSEF, BSRl'97, and KOSEF 95-0701-01-01-3.

(2)

(1.4)

Let Dh be a bounded simply connected subdomain with C

²

smooth boundary defined by

aDh : f(s) + hwh(S)V(S) (aD: x

=

f(s»

where s is 1 dimensional local parameter,

^Wh

(s) is a Cl function on aD whose Cl norm is bounded uniformly in h, and v(s) is the outward unit normal to aD. Let U and Uh be solutions to P[D,gJ and P[Dh,gJ, respectively. In [2J and [1J, it is proved that

(1.1) lIu - uhIlLP(o)

~

Ch

if p < 4 and aD and aDh are only Cl,l. (This result is for n-dimension (n

=

2,3).) In [7J, (1.1) is proved for

p = 00.

Both results were used strongly in the study of the inverse problem (see [2, 1, 7]).

In this paper we investigate the behaviour of\7u under perturbation of D. We prove that

1I\7{U - uh)IIL2(o) _~ CVh.

We also prove that Vh is the best possible result one can expect. To be precise, we have the following theorem:

THEOREM

1.1. Let n be a simply connected bounded domain in

]R2

and D and Dh ^be as above. Let Db,.Dh ^be the symmetric difference of D and Dh. Then, there exists a constant C such that

(1.2) tim

-hI {

1V'(u - Uh)/2dx

=

0,

h->O lO\Dt::..Dh

(1.3)

limsup ~ ^{ ^{1\7(u -} ^uh)l2dx ~ ^{C {} ^laa:

¹²

^der.

h->O

h

lDt::..Dh lBD

v

Here, der is the line element on aD and

a a: (P) = lim (\7u(P ± tv(P», v(P».

v

^t->O+

Moreover, if

Wh

converges to

W

uniformly as h

---+

0, then

I 1

1 k-1 au

2

lim -h { 1\7(u - uh)1 2 dx

=

- k - { a + Iwlder.

h->O

1

^Dt::..Dh

1

^BD

^v

If the Neumann data 9 is not zero, then cfz,\

^is

not zero and hence

we have

(3)

COROLLARY

1.2. There exists a constant 0 independent of h such that

1I\7(U -

uh)IIL2(n)

:S CVh.

If

Wh

converges to

W

uniformly as h

---+

0, then for small enough h

Corollary 1.2 says that ..,ffi is the best possible.

The proof of Theorem 1.1 is based on our earlier result on the rep- resentation of the solution to P[D,g] ([9]). So, we first review the representation formula in Section 2 and then prove Theorem 1.1 in Section 3.

One comment on a notation: the constants 0 in estimates may differ from one step to another. However, those constants do not depend on the quatities to be estimated.

2. Representation of solutions

For this work, we assume that 0 is a simply connected bounded 0

²

domain in R

²

and D be a simply connected subdomain with C

²

boundary which is compactly contained in O. The single and double layer potentials on D is defined by

SDf(X) = 2 1

{ log IX - Qlf(Q)dl7Q, X

E

R

²,

1r laD

1 { (vQ,X - Q)

DDf(X) = 21r laD IX _ QI

2

f(Q)ooQ, X

E

R

².

The following trace formula is well known (see [F] or [8]):

(2.1) where

o~± ^{SDf(P) =} (±~I ⁺ ^KiJ)f(P) ^(P

^E

^aD)

K f(P) =* ~

^{

^(V?, P - Q) f(Q)dl7 .

D

21r laD IP - Qln Q

We denote by KD the dual of KiJ.

(4)

Let £5(on) = {f

E

£2(on) : fan fdu = D}. Then the representation formula for the solution to the problem P[D, g] is as follows.

Representation Formula [9].

If

^U

is the weak solution to the Neumann problem P[D,g], then there are unique harmonic function H

E

W

^l

,2(n) and 'PD

E

£5(aD) so that u can be expressed as

(2.2) u(x) = H(x) + SD'PD(X) for x

E

n.

Moreover, if f = ulan, (2.3)

and

H(x)

=

-Sng(x) + 'Dnf(x)

k+ ¹ * oH

(2.4) (2(k _1)1 - KD)'PD = av ^laD on oD.

See [9] for proof. We remark that the representation formula holds for Lipschitz domains in

]Rn, n _~

2.

LEMMA

2.1 [9].

If

u is the weak solution to the Neumann problem P[D,g], then

(2.6) ou k-10u

'PD

=

(k - 1 ) -

= - - - - .

ov- ^k av+

3. Proofs

Let D and Dh be as in Section 1. Write oDh ^as aDh : (+ hWh(()V(() , (

E

oD

if slight abuse of notations is allowed, where Wh (() is a Cl function on oD whose Cl norm is uniformly bounded and v(() is the outward unit normal to oD at (. Let u and

^Uh

be the weak solutions of P[D, g]

and P[Dh' g], respectively. By the representation formula (2.2), the solutions u and

^Uh

can be expressed uniquely as :

(3.1) u=H+SD<PD and uh=Hh+SDh'PDh inn

where H, <PD, H

h ,

and 'PDh satisfy the relations (2.3) and (2.4). To make the notations short, we put

<P=<PD, S=SD, K=Kv, <Ph = <PDh, Sh=SDh'

K~=K*vh·

(5)

LEMMA

3.1. There is a positive constant C such that (3.2)

Lemma 3.1 is proved in [7].

By identifying the real 2-D vector

(VI,

V2) with the complex number

VI

+ iV2, we can see that

(3.3) \7Scp(z)

=

-1 r ~((~dl(1

27r l;:w ^{z - (}

and

(3.4) JCcp(z)

=

~~ r ^{CP(() d(.}

27r laD z - (

Here

~

is the imaginary part. Let cPh be the diffeomorphism from oD onto oD _h defined by cPh(()

=

(+ hwh(()V((),

LEMMA

3.2. There is a positive constant C such that Ilcph

0

cPh - cpIlL2(aD) ::; Ch

if h is small enough.

Proof. Let A

= 2tk~\).

Since (AI - JC)* is invertible on L 2 _(oD) _[5],

we have from (2.4) that Ilcph

0

cPh - cpIlL2(aD)

::; CII(AI - JC)(CPh*

⁰

cPh - cp)IIL2(8D)

::; CII(AI - JC'hcph)

0

cPh - (AI - JC*)cpIlL2(8D)

+ CII(JC'hcph)

0

cPh - JC(CPh*

0

cPh)IIL2(8D)

oHh oH

::; CII oV

0

cPh - oV IIL2(8D)

+ CII(JC'hcph)

0

cPh - JC(CPh*

0

cPh)IIL2(8D)'

Since the first term in the most right hand side of the above in- equalities is O(h) by Lemma 3.1, Lemma 3.2 follows from the following

lemma. 0

(6)

SUBLEMMA. For

any

function 9 E

L2(8Dh)

11

(Kj;g)

0

<Ph - K(g*

0

<ph)IIL2(aD) :S Chllgll£2(aDh) if h is small enough.

Proof. By duality and boundedness of

<P~,

it suffices to show that II(Khg)

⁰

<Ph - K(g

⁰

<Ph) 11£2 (aD) :S Chllgll£2(aDh)

for any 9

E

L2(8Dh). By (3.4), (Khg)

⁰

<Ph(Z) - K(g

⁰

<Ph)(Z)

= ~SS ^[ r ^g(() ^{d( _} r ^(g

⁰

<Ph)(() d(]

27r laDh <Ph(Z) - ( laD Z - (

=

:7r SS lD [<Ph(Z~~(~h(() ^{- Z} ~ ^{(] (g}

⁰

<Ph) (()d(

= 2~ ^SS lD [<Ph(Z) ~ <Ph(() - Z ~ ^{(] (g}

⁰

^<Ph) ^(()d(

1 (): r ^h(WhV)'(()

+ 27r ~ laD <Ph(Z) _ <Ph(() (g

⁰

<Ph) (()d(

:=

I(z) + II(z).

Since the Cauchy transform on C

²

-curves (in fact, on Lipschitz curves) is bounded on L

²,

we have

r ^III(()1 ² dl(l:s Ch21Iglli2(aDh).

laD .

Note that

I(z) = ~SS r

^_1_

_~ hj ((WhV)(Z) - (WhV)(())j (g

⁰

<Ph)(()d(.

27r 1

^flD_v ^{Z -} ⁽

~

_J=1 ^{Z -} ⁽ It

is proven in [3] that

i II(()!2dl(l:s f(hCII(whv)'IILoo(aD))2jllglli2(aDh)

aD

^j=1

:S C'h2 I1glli2(aDh)

if h is small enough. This completes the proof. 0

Finally the following lemma leads us to Theorem

1.1.

(7)

LEMMA

3.3. Tbere exists a constant C sucb tbat (3.5)

lim

-hI

f 1\7(Sep - Sheph) (z) !2 dV (z) =

0

h-+O In\DADh

(3.6)

limsup

-hI

f 1\7(Sep - Sheph) (z)1

²

dV(z) :s; C f lep(()1

²

dO"(().

h-+O JDADh JaD

Moreover,

ifWh -+W

uniformly as h

-+

0, tben (3.7)

lim

-hI

f [\7(Sep - Sheph) (z)!2 dV(z)

=

f lep((Wlw(()ldO"(().

h-+O JDADh JoD

Proof.

It

is easy to see that for each 8 > 0

lim

-hI

~ist(Z,aD»oIV'(Sep-Sheph)(z)12dV(z) ^=0.

h-+O Jd

zEn

So, we assume, from the beginning, that n = {z =

(+

tv(() : It I <

8, ( E oD} for some 8. (8 is chosen so that the normal projection from n onto oD is well-defined.) Let

E

> 0 be a fixed number to be determined later and let U be the tubular neighborhood of oD defined by U = {z

=

(+tv(() : ItI <

E,(

E oD}. If zEn \ U, then by (3.3)

V'(Sh'Ph - S'P)(z)

=

~ [r ~(C)_dICI

^- ^{

~h(C2

d1C1]

211" laDz-C laDh z-C

= ~ [r

^'P(C)

^-:!h

⁰ h(C) dlCI

+

h { (Wh

V

_)«() _'Ph₀ h(C)dICI 211" laD z -

C

laD (z - ()(z - h(C))

+ (

'Ph0 h(C)[Ih(C)I-lldlCI]

laD

z -

h(C)

:=h(z)

+

h(z)

+

Is(z).

Suppose z = e ⁺ ^tv(e) ^{E n \} ^D, e ^E ^oD. ^{If N} is the smallest integer such that 2

^Nt

> max{lz - (I: (E oD}, then N:S; Clog t ^and

~ f ^{lep(() -} ^eph

⁰

~h(()1

Ih(z)1 :s; L.J Pj-1t5c1t;-(1<2jt Iz _ (I dl(1

j=l

(EaD

:s; CllogtlM(ep - eph

⁰ ~h)(e)

(8)

where M is the Hardy-Littlewood maximal operator on aD. Since M is bounded on L

²

(aD) ([10]), it follows from Lemma 3.2 that

1 ^\II(Z)12dV ⁼ ^J ^f ^Ih(~ ⁺ tll(~))12dl~ldt

n\u

e<ltl<6

laD

~CJ ^Ilogtl ² ^dt f IM(<p-<PhO<Ph)(~)12dl~1

e<ltl<6

laD

(3.8) ~ CII<p - <Ph

⁰

<Phlli2(aD) _~ Ch 2 . For 1

₂

(z), we have

II2(z)1 ~ ^Ch _{laD z - (} f I 1 12dl(1 ~ ^Chi-I,

and hence

(3.9) 1 _n\u ^II2(z)1 ² ^dV ^~ ^Ch ² ^J

e<ltl<6

^r ² ^dt ^~ ^Ch

²^E-^I^.

Since l<ph «()1-1 = O(h), we have

(3.10) II

3

(z)1 ~ ^Ch f I 1 (I dl(1 ~ ^Ch ^log !.

laD z - t

Thus, we have

(3.11) 1 _nw ^II

³

^(z)1

²

^dV ^~ ^Ch

²^•

Combining (3.8), (3.9), and (3.11), we have

(3.12) ~ f 1\7(S<p - Sh<Ph) (z)!2 dV ~ ^ChE-I.

low

Now suppose that z =

~ +tll(~) E

U. Put

~h

=

~

+

hWh(~)lI(~).

Put se = {(

^E

aD: I( -

~I

< E} and

S~

= {( + hw«()lI«() :

(E

se}. Then,

'l(Sh'Ph -S'P)(z) =

~ [r ~(()-dl(l-

^{

~h((2dl(l]

211"

1aD\s' z - ( 1aD

h\S~

z - (

+ ~ [r

'P(() - 'P(e)dl(1 _ ( 'Ph(() - 'Ph(f,h)

dl(l]

~ 1~ z-( _1~ z-(

+

'Ph(f,h)

[r ~dl(l-

^{ ^_¹

^{_dl(l] .}

211"

1

^S' z - (

1

S~ z - (

+

['P(e) - 'Ph(eh)]

r _

¹ ^-dJ(1

211"

1s. z - (

:=IIr(z) +II2(Z) +II3(Z) +II4(Z).

(9)

In the same way to derive (3.8), one can see that (3.13)

Since 'P is CO: for every a < 1 (see [4]), IIh(z)1 ::; CEO: independently of h. Hence, we have

By Lemma 3.2 and the estimate used in (3.10), we have (3.15)

~ lI114(z)12dV = ~ lE ( _II4(~ ⁺ tv(~))12dl~ldt

U

-EloD

::; _~ LEE _{~D I'P(~)} ^- 'Ph(~hWllogtI2dl~ldt

::; Ch.

We now deal with Ih(z). Put

For each z

E

U, we may assume that

SE

is a graph by taking

^E

small enough if necessary, namely,

SE : ( =

x + ig(x),

-E

< x <

^E,

g(O) = g'(O) = 0, and z = it(ltl < h). Put J() = 11 + ig'(x)1 and J

h () =

1

1 + ig'(x) + h(WhV)'(x)l.

1 + ig'(x) 1 + ig'(x) + h(WhV)'(X)

Let rh

(r~,

resp.) be the straight line connecting the left (right, resp.)

endpoints of

SE

and S1;. and let Ch be the positively oriented closed

(10)

curve composed of se, ^Sh' rh, and

r~.

Then

By the Cauchy integral formula,

if z

E

DflDh if zEn \ DflDh.

It is easy to see that

Note that

IJ(() - 11:::; Clg'(x) I :::; clxl :::; CI(/' ^(E oD

IJh(() - 11:::; Clg'(x)1 +

Chlw~(x)1

:::; C(I(I + ^h), ^(E ^ODh.

It then follows that for z =

~

+

tv(~),

So far, we proved that

(11)

if z

= _~

+

tv(~)

tt ^Dl1Dh, ^and

if z

= _~

+

SWh(~)V(~) E

Dl1Dh.

It

then follows from (3.16) that

(3.18)

On the other hand, from (3.17), we have (3.19)

I ^-hI _lD~Dh r ^III3(z)1

²

^{dV -} laD r 1<p(~)12Iwh(~)ldl~11

::; * ^lD ^l ^h ^III3(~ ⁺ sWh(~)v(~))12 _1<p(~)121 dslwh(~)ldl~1

::; C laD r l<Ph(~h) ^- <p(~)12dl~1

+ ~ _laDlo ^r ^rh IIWh(~ ⁺ SWh(~)v(~))12 ^- Ildsl<p(~)12Iwh(~)ldl~1

2 1

r ¹

::; C(h +

^E

+ hE- + laD Ihwh(~)llog Ihwh(~)1 dl~l)·

Take

E

= h

²^/³^•

Then (3.12)-(3.15), (3.18), and (3.19) prove Lemma

3.3. 0

References

[1] H. Bellout, A. Friedman, and V. Isakov, Inverse Problem in potential theory, Trans. A. M. S. 332 (1992), 271-296.

(12)

[2] H. Bellout and A. Friedman, Identification Problems in potential theory, Arch.

Rat. Mech. Anal. 101 (1988), 143-160.

[3] R. R. Coifman, A. McIntosh, Y. Meyer, L'integrale de Cauchy definit un operateur bournee sup L² pour courbes lipschitziennes, Ann. of Math. 116 (1982), 361-387.

[4] E. DiBenedetto, C. M. Elliot, and A. Friedman, The free boundary of a flow in a porous body heated from its boundary, Nonlinear Anal. 10 (1986).

[5] L. Escauriaza, E. B. Fabes, and G. Verchota, On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proceedings of AMS (1992).

[6] G. B. Folland, Introduction to partial differential equations, Princeton Univer- sity Press, Princeton, New Jersey, 1976.

[7] E. Fabes, H. Kang, andJ. K. Sea, Inverse conductivity problem with one mea- surement: global stability and approximate identification for perturbed disks, in preparation.

[8] E. B. Fabes, M. Jodeit, and N. M. Riviere, Potential techniques for boundary value problems onCl domains, Acta Math. 141 (1978), _16~186.

[9] H. Kang and J.K. Seo, Layer Potential technique for the Inverse Conductivity Problem, Inverse Problems 12 (1996), 267-278.

[10] E. M. Stein, Singular integrals and differentiability properties of junctions, Princeton University Press, Princeton, New Jersey, 1970.

[11] G. C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, J. of Functional Analysis 59 (1984), 572-611.

On stability of a transmission problem

P[D,g]