AI -n IDETIICRTION OF SRRL IMMO NOONEITIES OF EXTREME 1/1I CONDUCTIVITY mY BOUWOR. (U) MINESOTR UNIV MINNERPOLISI INST FOR NATNENR1ICS RNO ITS RPPLI. A FRIEDMAN ET R.

UWCR~F1lD DEC 67 INA-PREPRINT-SER-373 F/S 26/3 MuruuuuruSOMEO"E

44

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: r. FILE G,."',

IDENTIFICATION OF SMALL INHOMOGENEITIES

OF EXTREME CONDUCTIVITY BY BOUNDARYMEASUREMENTS: A CONTINUOUS DEPENDENCE RESULT

By

Avner Friedman

and

Michael Vogelius

IMA Preprint Series # 373

December 1987

INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS

UNIVERSITY OF MINNESOTA

514 Vincent Hall LTII '206 Church Street S.E. D I

Minneapolis, Minnesota 55455 r ."ECT-.1MAY311988

v'A~uu-f1JN ErATEcM 1L

Ap~ae fo pb~ I*e

Db".-1tnjt.m cUqnte

IDENTIFICATION OF SMALL INHOMOGENEITIESOF EXTREME CONDUCTIVITY BY BOUNDARY

MEASUREMENTS: A CONTINUOUS DEPENDENCE RESULT

By

A-, cr Fried iian

and(

Michael Vogelius

IMA Preprint Series # 373

December 1987

DTIC.

AftELECT

MAY 3 1198

rXH-

D111M~ON STAENUM

Avp~aed !r pubic rlo5u

DisDTbtiC Uaint.Np

Lnc I t)

SECURITY CLASSIFICATION OF THIS PAGE (When ridN iant4

REPORT DOCUMENTATION PAGE BEFORE COMPLETs G FORM

REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBERIMA Preprint Series # 3731

4. f[ITLrk (in?.S Subtitle) S. TYPE OF REPORT & PERIOD COVE[RED

Sent fication of Small Inhomogeneities ofExtreme Conductivity by Boundary Measurements: Final life of the contract

A Continuous Dependence Result s. PERFOMING ORO. REPORT NUMBER

7. AUTHOR(e) 1 2. CONTRACT OR GRANT NUMB(e)

Avner Friedmani and Michael Vogelius2

S. PERFORMING ORGANIZATION NAME ANO AODESSnst. for Math. 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUMBERS

and its ApplicationsUniversity of Minnesota

Minneapolis, MN 55455

I1. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Department of the Narvy December 1987

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14. MONITORING AGENCY NAME I AOORESS(It different from Controllng Office) IS. SECURITY CLASS. (e thle report)

ISa. DECLASSIFICATION/OOWNGRADINGSCHEDULE

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Approved for public release: distribution unlimited

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IS. KEY WORDS (Continue on revere aide It neceesary mad Identify by block nuember)

'. L

20. ABSTRACT (Continue on reverse aide i neceeeeay nd Identify by block mayber) We consider an electrostatic

problem for a conductor consisting of finitely many small inhomogeneities of

extreme conductivity, embedded in a spatially varying reference medium.

Firstly we establish an asymptotic formula for the voltage potential in terms

of the reference voltage potential, the location of the inhomogeneities and

their geometry. Secondly we use this repr""'ntatien formula to prove a

Lipschitz c,,t ,,¢us dcpcoidence estimate for the corresponding inverseproblem. This estimate bounds the difference in the location and in the

relative size of two sets of inhomegeneities by the difference in the (over)

DO I jOMAN73 1473 EDITION OF I NOV 65 IS OBSOLETE

S/N 0 102- F- 014- 6601 SECURITY CLASSIFICATION OF THIS PAGE (heon Date Entore)

N~ %-

~I.

-V

20. boundary voltage potentials corresponding to a fixed current distribution..Iv

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IDENTIFICATION OF SMALL INHOMOGENEITIES OF EXTREME

CONDUCTIVITY BY BOUNDARY MEASUREMENTS:A CONTINUOUS DEPENDENCE RESULT

AVNER FRIED MANt AND MICHAEL VOGELIUSt

Abstract. we consider an electrostatic problem for a conductor con.sisting of finitely inany smallilolliogeiieities of extreme condctictvity, embe(lde(d in a spatially varying reference medium. Firstlv we

establish all asymptotic formula for the voltage potential in terms of the reference voltage potential, thelocation of the inlioniogexeities and their geometry . Secondly we use this representation formula to provea Li)sclhitz cont inuous dependence estimate for the corresponding inverse problem. This estimate boundsIe ldill'e.rence in tIe location and in the relative size of two sets of inhiomegeneities by the difference in theboundary voltage l)ote tials corresponding to a fixed current distribution.

!jl Introduction and statement of the main result. The determination of con-ductivity profiles from knowledge of boundary currents and voltages has recently received

;i lot, of attention in the literature. In the biomedical community a common name appears

to have ('ueirge(l for such work: electrical impedance imaging [6]. It is usually assumed

t'hat t he (direct current) voltage potential u satisfies the differential equation

V.(y(x)Vzu)=O in Q1, Q C R",n >2

where (x) is the positive, real valued conductivity, to be determined. The extra informa-

ti, b 1)asel ipoin which it is sought to determine -y(x), consists of knowledge of currentsDii.

1- aid the, corresponding voltage l)otentials u at the bounda'y, DQ.

Let A., denote the linear operator from H 1/ 2 (0Q) into H -/ 2 ()Q) which takes Diriclilet-

to) Ne'unian-(ta:

A()= - with 7-(i'Vu)=0 inV ,- (T\ona),l

(oiiplete knowledge of ,- is known to determine the function I uniquely under quite

g,'Imeril assiiuniptions: Thi.s was verifie(d for analytic and piecewise analytic 3, in [11] and

[12], f, C' -I zind diiiiisioi n > 3 in [18]j and under the assunl)tion that - be sufficiently

close to a, coistwit it was verified for C' "r and dimension = 2 in [17] For a laycrcd-coiidictoi, i.e., - = (xt), two sets of Dirichlet- and Neumann-data suffice to determine

(well, ;I b~oundd,l mcaslrl'he - (cf. [131).

tli,.t itoit,' for Mathemat ics amd it.s Alpplications, University of Minnesota, Minneapolis Minnesota 55-1551 )epar ieit of Mat henati s, University of , arvland, College Park, Maryland 207.12

WI

Because of inevitable error in measurements, it is at least as important to study the

continuity of the depedellnce of -" on A- as it is to verify uniqueness. Alessandrii [2]

has recently examined this (lllestion for n > 3 and shown that the mapping A., -+ Y lias S

a mni IIiiis of c(nt.,Iility of 1ogaritlnhic type, provided -" is a priori known to belong t.o

a 1imlndel set iII some Sol)olev space; his proof exploits the continuity of the mnappings

A-, - y7s I and A-, - 0- / - jj established in [19]. The assertion of Alessandrini is, more

Secifica lly, that if A-t, and A -2 deviate by 6 (in the operator norni on B(H 1/2 H- 1/2))

then -y and -y2 deviate at most by C (log )- (in L'(Q)) for some 0 < a < 1. While such

a reslt is theoretically very interesting, it is at the same time somewhat disappointing.

since it, predicts quite a weak form of continuous dependence. It does not explain the

a1pparclt practical success of various numerical algorithms to recover - from only partial

knowledge of A-, (cf. [4], [14], [20] and [21]). To bridge the gap it seems relevant to analyze

the c,ltillous dependence for interesting classes of conductivities, where the functional

dependence on .r is further restricted; one example of such analysis is found in [5], [9]. It is

of practical importance to seek continuous dependence estimates in terms of only finitely

many sets of Dirichlet- and Neumann- data.

III this paper we consider conductivities that correspond to a finite number of small in-

hio(og,','ities, with extreme conductivity, imbeded in an n-dimensional reference medium.

The reference conductivity -y(x) satisfies

(1.1) 0 < cO < (x) <Co < .

We asslmime that each inhomnogeneity has the form zk + EpkB where B is some boundeddo~main III Rq" wvith i )

-1.2) 0 E B and OB of type C 2+ 0 for some 0< < 1 .

The points {Zk}1'I belong to Q and satisfy:

Zk - Zj I >_ do > O , Vj : k , and "•S(list (:k, Q) > do > 0 , 'Vk

'Fh' )laramneter f determines the common length scale of the inhomogeneities, and the Pk•

~do Pk </0 Do;:(1.4) d P ,

deterllen their relative size. We always assume that e is small enough that the sets

''A J- 2( )B are disjoint anm tm their distance to R"\Q is larger than d .

,= U(z:k + pk B)k=l 2 4

9

(lnte the total Collection of inhomogenieities. If they all have infinite cond~uctivity. tlhen

the voltaige potential it =it given the boundary current ',is the solution to

(1.5) 1i ll( VI 12 (1 -J ifL.

11 OSI

If aill the Holiogeiieities have conductivity 0, then the voltage potential it, I uSolves

(1.6) iun I [Vudr - f udj

XNazsslille t ht

(1.7) Q is bounded with Q E

f Jall he 1.1. For prolems (1.5), (1.6) to have a solution it is necessary and sufficient

.

(1.9) = 0 .J7

(1.7) t mad i my be obtained as limits from an electrostatic problem iN- ) vigfinite and niolnzero conductivity: Let y,, +. alo hav , , > 0, and denote by i" the

sott~ o, D -x,( c)wih

f 2" '

in I 71 I x - it =ds. SPCT :

u~El'l Q~) 2-aQ

5,,

Then

(1.) lin u and = i n l I/'

0a1o

tie first liiiit is relaitive to H ( Q), the second relative to H ( Q \L,;). _______

h lil of this pprwe sn" all focl s our a ttenltion Oin tle problem wit l ihologeit i CS

ii finite tnd te of zero conductivity is very s=i"iir and is treat briefly

ill lic filiial sect ion. .~Cods

fAvai11 Ar~/or

3n Dist Spec al3, o

*; a 0j ;

We shaill hencefor'th wr"-ite u, instead of it' . Our main tool of analysis is to (xpi(ss u,

iII tcrms of U, the solilt ion of

li1II 11 2

U." is t1o( )ltg(e, l)ot(',tial (corresponding to the referece imedium alone, and it is normalize(d

hy

(1.13) / ds -0

) \VWe assure(' that

))(1.141) "7 C 2+ (

;a,(d that. th fo)llowing nton-dcqcn( racy condition holds:

(i,) VU(x) $ 0 Vx Q.

III )ractic(e this ineans that, we (-,in choose any -- harmonic function U, with VU( x) 5$ 0 in

S? ;l(1 tl (n apply the )oundary current ', =yOU/Ov on OQ.

Consi("er two arb itrary collections of inhomogencities

h% h"

Ll,= U (Zk + fPk B) and>,,, U(4-+ cp.B)k=I k=l

1both satisfying (1.3), (1.4) an(l denote by it, and u' the correspon(ding voltage potentials

(wit d, fixed 1)omidary ('irrent, I,). Our main result is the following continuous dependence

TiI I. (i1 Em; 1.1. Let F he a given nonemptv open subset of OQ. There exist constants

0() and C and a fhn'tirn 1(c) .lin 1(f) = 0, such that iff < Co and

(i) A, = ', ald, after l)plpO)riate Ceorderlnrg,

(ii) :. - I . + 1k - I'Ik

C- + ,1 1 < A < K)

4

,.;, € .. :.., .:.. , .,, ,:.:v :v~ .:,:..,.:.,. ._ ... _ .. _ .:.:.."- _. ."." ..,,, . ._ _ . _ __,s.,. .-., ':a..'y;. ,,-.,. a-.r--,', ;,;i:

4.

Tho coi.stajIt. fo, bo MRl C adl the function il depend on do, Do F. Q, -y and )' liht ;t,othrwise ililhJelCit of the two sets of inhoiaogeneities.

Tle factor e: " in front of I1u, - ,1L rm is best possible; it follows immediately fromor aalysis (cf. LCmma 3.3) that cven if Izk - Z.! and JPk - P',. is of order 1 then the(liscrel)Iilcy in tile boundary data is of order c". For fixed (and small) c our theorem1 4msically shows that the locations of the inliomogeneities, -4, and their relative sizes, Pk,d'pend Lilpschitz-continuouslv on the rescaled boundary deviation E-" --uL(r).

In the formulation of the theorem we used the L' norm of (u, - u ,) r ; this is note'ssclit i'll, in fact tile analysis in Section 4 shows that other norms can be used, such as tileL 11i11'n.

A brief outline of this paper is as follows. Theorem 1.1 is proved in section 4. The

proof is based on an asymptotic represent.ation formula for u,, as derived in section 3. Toestablish this representation formula we require some energy estimates of U - ?,; these('stmiates are found in section 2.

R E;MAR K 1.2. Recently Friedman [S] has shown that the presence of an inhonogeeity "-Mr ;I C, llect ioll of inhmog)oelities can be detected by boundary measurement of the voltagelmtential (on r) corresponding to a single current distribution. For the case of smallilihoilmogellities of extrene conductivity Theorem 1.1 goes much further. It shows that

the exact locations and the relative sizes are determined, in a continuous fashion, by the11a111eC iliasurenent.

P I-;MItN 1.3. Theorem 1.1 can be extended to the case where each inlhomogeneitv isof tlme form -k + EOkBk and Bk are different, fixed domains with smooth boundaries. %

2 Energy estimates. In this section we estimate the difference between u, and tle14,fem, 4ice potential U. The first result concerns the H1 (Q)-nornm of U - ii,.

Li _\i,\1 *~ 2.1. There exists a CoI]stait C Stuch that

.J(; - i,)12 + I - 11, 1) dx < C " l 1 -1, 1

Pl,,of. SinelC

12 (I.< C -. 2 (. . 2'/ ,,' dx _ C (/ ,,- d.,'+,- ,ds,-')

e" 0 its ficst ro c hi

% . J,, I

'.4

.

(2.1) 1 V(Uj 1,') 12 dx < C £1k It 2 (j4.

SetV, - vcH'(Q) Vv 0 Oill~ jvds O}

FromUX the (lcflIltlollS of if' U we get

(2.2) Vit u dx u, ds 7U Vw dx V11, E V,

S1S

(2. 3) 1 -L 1,o)1 dx iiiii V(U - v) 12 IX.

Sinice y i I uiided from abovec aild away fromn 0, it suffices to show that tlc:-e exists a

V,~ s 511(11:t t

(2.) 1 I V(U -v,) 12 cx C II-1/ 2 (D)) %

SIC*)ISIIifist iccaeof one iomogeneity (,with 0Oand p 1 Let

C( L dx(2c)"I1B I

At (e i 2,( f t I i Po Iiiicarc6 inlequality ( on 2B ) gives that

(.)JIU-C, 12 (1., < C, f2 J 7 1 2 dx.

D 'fill

(2.G) ~ ~Iv z(X) @()C, (-c()

1,; C' litoffI fiict ion:

foII

C)() 0 for q R"\2B

%

< 1i21

I .iia

n (2 ) . lir( 2.6 ), 2.7) it flV,,s tl;

,

J2.8) 2 (h. < C 1(U.-w ,12"

ami t., clsinat (2.5w), (2.8) tipev gt2--)

(2.S) a ( c / '-e) ,,12 d.,. <C, (' U 12 dIII Ct

2 2(/) o th

fov elliotic estihitat('s (ot .

I U 2" ,t, <! B[ " 2 l71 l !,.,?), < C'ellll-,/ca)".

ajl ii (')IliIati()l with (2.) this l)ov(s (2.4). :

()far wve hi;ive co(nside'red one: inhioiogeueity. Iii the case of A inhiomogeiieit ics th 1.wi ,si iit. f lh)ws from the' l)revious lproof and a localizat ion argument . E

Lemnmma '2.1 asserts that 1[I, - u 'q (12 O( 0( "/); fr'om the trace theorem it t herefore".

f,,lh ws t lI mt I1l - u, I! 1/2(s = O( "/2 ) • This however is not thlie 1)est possihile ('st Iimiat ('. .

iII f,(t we have:

I.i':MmA 2.2. There '.\ XIss a coistant C such that

- , II--(a 11- " 1 + luil /I2(,i2)} •

1)u, m t L't II' aii (I ', 1) s(luiti( n1s t( the sati(' niliiinlizatiol prol)hems as U a111(1 t,.

just with ,' I,"11('ue( ) y 1.". From the Sclhwar/i inequiality and Lemmia 2.1 we get that

(2.M) V , • V(() -It,) dxd= 0 .

.--

Ililt grathN J'U vv .Iwts vi)'hls

(1,, ?1,) t

S2S

I.V1 . - ' d -(I

s i V ( - VI - )1 0 ill Q and " 'I on 0Q. A colmbina t ion of (2.9), (2.10) and (2.11) S

an l t lois Iv t ki ig t l l(- l iniXil l over 1" silJ(t to IIll- /2(a) - flind f In( 0. we

get tihe ;ut'srtion of the liliia. L

v I.;M \I{ I 2.1. The estiiiate in Lenma 2.2 follows directly from the representation

forinilda for u, which we shall derive in the next section. Indeed, it follows that U - u, is

of order (", near oQ. in the (+;-'- norm, 0 < 0' < 0; however the present proof is simpler

a id we( hiyi, ilch(led it for comleteness.

!*3 A re)resentation formula. We now proceed to find an asymptotic expression for

This ,xr(ssion will involve the functions 4j (jn 1 n) which solve the exterior

A 0 in R"\,.

(Di 'on OB()I is uniformly bounded in R"\B and

if,, > 3: Oj(y) -- 0 as y -, oo

51 j'] 'I' exist a11(1 ar liluique, [13]. For n > 3 we shall also need the function <, whichsat i sti,.s

A6 =0 in R"\B, S

(3.2) = on OB"

6(y) --0 a, --* .

The fullowiig l('l u is well known, ;(l is only statedl here for the convenience of the

relad r.

1.1 l \ , 3. 1. Let ,1,(!/) be a mifornilv hounded harmonic function in R"\B. If , > 3.

;,"11111c that (l'(!/) -- 0 a., ! -- C. Then thtre exists soe constant c such that

I 1 2 -n' +(( / -iij,I(,) C jv y " 0(1 :/'") .d

VI,(,y) :-V(e I , I1 i ") + 0(1 ,7!'') aS I-- CM. ;

5

+. ;is-

/, .. ; . ..- 'L -,.- K -K - * = S - - - - -" w ,- . , +,,

Briefly the heiiiina follows by considering the Kelvin transform

(1 0 el(I!I'1

;t11l noting that the siniglarity of (I)' at y' 0 is reinovablc siince

{,(l) 0 (log if

o(1y1n) if n > 3.

From Lcinia 3.1 it follows that

(C, O - 4j --_) ds -0 as R-+

M 1id (;r,.'ii's formula therefore yields

f &f o+(3.3) I , ds=- if--- ds.

I 1 vOJ v

(3.4)1 y , d(s = ]4- ds -ds, n > 3 .

a ,l) 013 013

N, ic, b v t he maxinimi principle, that 0 < 4 < 1 in R"\B and "

(3.5,) >0 on OB

(v is t Hie tward l1,rnlal relative to B).

\\c how prove two crucial estimates, needed for establishing the representation formula.

I;N. MM, 3.2. For an i:, E H- 1/2 (DQ) there exists a constant C such that

(3.6) / I ds <_ C< ' - and

-f P-

S(:.7) . ~ -?, -- ,r - .) <1 = (Ep )"?I~(z)AVU(sk) ± o(")

Zj( : -t i' a, 1

5) H

00b OD i ds 04jJDF 0, Dd1

w!,ie,' is tli outiward normal rlative to B and 6, = 0 if n = 2, b, 1 if n > 3.

lroof The f;t't that A is svinimetric follows from (3.3). To verify that A is positive1.fiiiite we compute

I 1 2 1 Dsa B

,. (I"), (- , ds 2

Vat, ai) B1 a) 1

Xith I = - > ,'), and note' that

J --a I'. J V 12 dy >0 ( 2 2), and

di R2

\I/ \a\+(/ y(1(,-J2 d) -

/ ' 7 (Iy)2( J VO 12 (l)-' > 0 ( > 3).

*R', \I W"\11 R- \l34 this )iiLlhi i . aration by lpart's anii(d usc of Le n ma 3.1.

T,) , i' (3.G) and (3.7) (')nsi(ler first the case in which there is only one inhomogeneitv.with - () anI d p 1 B. Let d be a fixed positive number (chosen sufficientlyuiall Without loss of geinevr;ity we may assume that

(cH C)) I) C • < (I I C {. < 2 1} c .

l.t \ 1 I)(. tlic s,,ll tio)n to) f

. (.S ,) :0 ill {I.,1 < ,t}\I ,

I, oi- D(f1B) a md \, -0 onl {II d ,}.

1(0

Se~t.C, 11(I1-VB (a Constant)

ad ll t I()lic( the1( finit iOns

(3.10) V.x) u(x)-x)(c, t U(0)), x.i JIY <( (l}\f

I"E)- UOFY) (ffj<dI \B

011 OB:

~7)-U(O) - U(cy) -- UO

(3.11)

-- VU(O) -y as e--*O0

muiformily ini (1+O', for axy0< 0' < 0. On jyI=d/E:

= lEY) - U(Ey) ?_ n(X) - UWr

With xi cy, IX-1 d. By Leniiiia, 2.1

J (IV 1_-U~2 + 1~ 1 2 O(I <It<2d)

VV (uf =)0 , in £\{xld/2),

NVe Can a1)1)I ellip)tic estimates to deduce that

u((x) - U(.r) -(3.12) 'flaX <Cc 2%

(3.13) 1 t" (Y) I< CCT 2 Oi -

;11(l ;it the( saluic tiliiic 11" solves,

dI -(3,.1)V()1~y) 0 in f{I y I< -}\B.

z1 I I1

PiT' .(.suit in this heinina diff'rs slightly depending on whet her n= 2 or it > 3. We

thu,, fir-t to t1he cas it = 2. From (3.11), (3.13) and (3.14) we obtaiin, using the miaximumii .

plincil , that 1,(y) is 1oinl(fl in {1 y j< R}\B uniformly in R and e, provided R < (/(.

Bv ehliltic estiniates and a coiipactness argument we deduce that 11V IV as e -- 0,

xvllre IlVO(y!) is the solution of

AIlVO =0 in R 2 \BlIT-, = ,>ILTOO), y.on.o

TV 0 is bounded in R 2\B

the ,,,nvrgcc('c is in C +1 4 ( {(y< - R}\B) for any 0 < 0' < 3. In particular

(3.15) Ol -4 0 m uniformly on 0B.

Recalling the definition of IV, we get

(3.16) al, (.) -O°o() 0 on a(EB)al" 0 v F

here I has 1)een llsed simultaneously to denote a smooth unit normal field v(x) on D( _B)as well as its c,,lnterlart, v(ey), on OB. From (3.16) we immediately get, upon inserting

(3.1o),

al7,u, x(3.17)() - ,- )(c" - U(0)) - VU(0) . ,,(x) - 7n0-) 00v

nuifornuyi oi 0( B) as t --* 0. It is easy to see that

allJ --9 5 ds = O,

0((B)

atrl (3.17) thlerefore yie(lds

(3.18) 2 ds + -0(0) ,(y) ds 0

a((B) aB

as + -- . IBut. iiio(y)ds= lir - ds=0

fn {lyl=R}

;I" ; 011sttiiie',l O f the sec((nd identity in Lemma 3.1. so (3.18) gives

(3.1c) U() O\ ds 0 as f-.0.

0((Ii)

12 II

The maxiinum principle shows that OV,/&U has constant sign on O(eB), i.e., (3.19) may

be written

(3.20) - U(O) ds - 0 as c 0 .

/)(, I) p.

In ('onjunction with (3.17) this gives

o9B) B)

which is (,xactly the assertion (3.6).

In terms of the solutions 4 )j of (3.1),

I= EUZI (0) IP

and thus

(3.21) &0 (O) a -I)j on aB.

A c()miiation of (3.17), (3.20) and (3.21) yields

(3.22) I 0 (Is x- (0)Ux(0) (viy + "j y) dsu y ,(0)AVU(0)

O((B) OB

where A (aij) is the matrix with

aij = ( ±jyi + ! Yi) dsy

aB

B I b, - J -g%'&8B

and (3.7), (3.8) follows.

now turn to the case n > 3. The function

2-n

,,,(,) = ii'i(,, - , .- -, ,. = y aI.

sat, isfi's

V. (-( i,)V,,(y )) < -y(y,)(ih" + ni<' - (II V-lIKL')d -

r 7

d<0 if lyl<-,

13

a.~- *~W 1 ~ ~ ~ ~ a' a'/ .. ~ a"%

lrovided d is sufficiently small. Using (3.11), (3.12) and (3.14), we may compare 11" with,'

'm(!) fo)r some positive constant C (independent of e) and conclude that

- ~ d(3.23) IW (y) )I< C I1 I' for y C {I y }\B.

'lcr(ef(re 11, c(nve rges unifoimly ol compact sets towards W0O, the uniqlue' solution of

A17 = 0 in R"\B,(3.24) 1170 =-VU(0) .y on B,

11 0(y) --+ 0 as y 1---* oo.

The stat(eneits (3.15), (3.16) and (3.17) remain valid and, as before,

rf- U(0) xf,,-i s-O- d, -- t(O) fM0(Ma d.s.

OvB(3.25) h .

- ) 1: O)Z (x) ds-;

(011y mix, lO right-Liand side is in general j 0. (If we ce I Y 12-n + Q (1 ( 1--) 'IS

r l - ;,ji> 3, then f (D4)Do {= 0 if and only if c 0). The function wihv ias

a fixed Sign oin (1B), , 1 lid by combining (3.25) wfith (3.17) we deduce that the assertion(3.6) holds.

Fnio (3.17) we obtain a relation similar to (3.22) with the additional term ,

_ _, [TO aF< Ij Iiri C(,) xdlf- 09V0(f B)

appearig 0. oni t Ile right-hland si(e. Using the barrier Ci(y) we see that the functions ec,( cy)

;1,41 all II wjori.el by Cm(y) onl {1 y I }<d\B. Thus Vjey) ' 0(1y) uniformly with its

first derivatives iii (I y 1: R}\B, for any R > 0. Combining this fact with (3.25), we get

'Z ~ d' ''J v J v -9, v

'r'ic foi n li (3.8) follows by use of (3.4) in the above expression for .

We Have llbs compl~eted thle proof of Lemmna 3.2 in case of one ixihoinogeneity, wit 11

-0. p -1: a Siimplie resealiiig and a translation gives the result for thle g-eneral case of oile

14

%I

inhoiiogneity. If there are more than ene inhomogeneity, then we can apply the previous

argiiiiient to the inhiomogneities one at a time, provided we verify that

(3.2G) J t -( ds = 0

Zk+fpk B

for c';l'l 1 < k < K. To prove (3.26) we return to the Ui' defined in the introduction; its ehar that (3.26) holds for all it, and since u, = lim u' (in H'( Q)) the same identity

a-00hol(ls for i'. [

Let N(., y) denote the Neumann function in Q? corresponding to -f, i.e., the solution of

-V .(VN)=6b in Q,ON 1- T-- - on O,

with

Nds =0;

I/ denotes the outward normal to OQ. If y E Q\w then

It,(y) u, V] - ('yV N) dx = aN d

(3.27) Q al

-yVx- N ds + 7 -N ds. %:O f f Ou

an

Since -,(OX/Ov) is constant on 0f) and f u, ds = 0, it follows that f uE7(ON/Ou) ds = 0.aQ aQ

Combining this and the identity -y(Oui/Ov) = ' on OfQ with (3.27), we get

(3.28) u() N- - N ds + fNds

aw, a

Note that v denotes the outward normal relative to w , The first term on the right-hand

si(e of (3.28) may be written as

(3.29) - - ' N s

Sk=d( Zk +(pk /)

- - 5N(zk !) JV- d - VN(Zk, Y) J7 (X - k) d(1

-) k=l

i(Zk +(pk 1) ( Zk +fp. B)+ 0(2 f i,

1iz+~' 1151 1~ (Is).

15

k' ' -' ' '¢ 4 -, ' ,'A., ,,.d, ,P , -,. .

'7"""€d',,. £ ,. , . . -", * . * . .. . ," "" " - -*"

\V alre.dy know that1)i l .,; 01.

! 1sig Lemmaii~ :1.2 andl (3.28), (3.29) we therefore g t:

I:Nm,M :3.3. TJIr( holds:

11'Y) -f"' 57PI(Zk)VzN(Zk, Y) AVU(Zk)

(3.30) ep

+ IC(Yxy) (18. + E"1l(f, Y, {Pkl. f .* SI

where th matrix A (,,j) is given by (3.8) and il(, y, {pk}, {zk} ) as well as Vyv(e, U, {pk },{z.})",',,,,,,.,ig,..S to zero, as f alpl)roaches zero, uniformly with respect to y Q , {Pk} and {Zk}.,

prV(id,', 4,li.t (!/, {z k}) > ( > O

Note that./,(x)N(.r,y) (Is, = U(y)

It general the matrix A cannot be computed explicitly. One exception is the case whenB is the unit ball. In thIt case ipj = -yj/ I y I and (for n > 3) 0 =1 y 12-,. A simple

Colutation gives A = L,I, where I is the identity matrix and w, = meas { y 1}.

§4 Continuous dependence. The Neumann function, introduced earlier is symiuet-

tic in its arguinents in 'a

(SI x Q) \ diag (Q x Q). It furthermore has the form

1 lo y z+,yz fn 9':..

N(z,y) =N(!,z) - Iy- z ±?(yZ) if t 2.(. 1) 2 ?(z)

-Y - 2- +R(yz) if n >3

yev 1i(q!, 7) solves

&Y 1IOQI ~ + In-I ~ yfuL

I V(y). (Y )- • l ('(jVR01..) 0 (X2.{_ + - z I" Y E l

? is ii t yv'I1l fltric il Its arg tients. Since the function

Y vl(/). ( 0 -Z)-4 - z

1,0

.- w J?'V U

4

is in LI(Q) for any 1 < p < ,, it follows from (4.2) and (illi)ti(" estimat(,s that R(.. zisin IV2 'P(f) for any z E Q, 1 < p < n-n. From (4.1) we thus get

N(,) C I'V1 P(Q) for 1 < p < z E Q( 4 3 ) - n - 1 ''(.,.) t ,V2 '1(Q\{z}) for e .

It is ,asy to verify that the function R(y, z) is differentiable with respect to z E Q. Infact, for any fixed vector a 5 0, the function R,(., z) = a. VzR(., z) is the solution to(4.4o) - VE(-}(y)VR <(y,z)) 1 -Q V-().(y-z)

o 1

(4.41)) -y)-R (y,z)-aVZ ,YY( y C) DQV

R0(1 ,z wi -YZ I -,

with the normalization ¢

(4-5)

- - z 2-n ds if, i> 3.

The right hland side of (4.4a)

,-

- -- < .0, (nt 2),( ) 2 y - z 1

VZ -YY)( Z) V-,(Y), Vy a i,_ (n > 3),

is Iin ( | lqf ) it < q , since both"

V g (n 2) and V. (n 3),(- ) - (z) Ily -Z In- 2 n > )'

;irv in LP(fQ), 1 < p < - and are continuous on &Q (z C Q). The right hand side71

(of (4.41)) is continuous on OQ (z C 9). By elliptic regularity and duality it now follows

that

(4.6) R,(., z) C IV ,(fQ) for 1 < p < z C Q

Tlie ftiictiois

n V:(log I y- z ) = (z - y) (n=2)

' 111d

17

*-'- it ",I u" .,r 4 -, -,-, , -.- , - '.-

.-...-.--- -"- ,i v, .-~ ,, .- *I ,,'- .- .- -.- , - .5 . ,( S" ,,s "i" P," ' "! l """ ,d ""

\7 ( - y(7 > 3)zI Y - z

arC III L"(Q) but not i In '(\z Consequently, it follows front (4.1) and (4.6) tha~t

oI V..Y(_- U.~L(Q) for I 1) < E Q ,and1.7) i- 1

(l V.N(z, ) E W'(Q }for Z c Q

for- a11\, v(tor / 0. 1

Pr ( .',ling once more 1y the method used to prove (4.6) one can show that

(-t.S) DzR(.,) L"(Q) for 1 <p < C Q.

Diffi',-vlliati)n1 of log I Y - gives

a .3 2((z - y). a)((z - y)-%(D' log y n-- - 3 =)' 2- zi _ y

I y - Z 121zI Y I

fi Vm whi,'h w c(ichie that, in the case ti = 2,

(D'2 h, z Ia) '3 is not in L (Q\f{z})

f,,r any vctors 0v )4 0,/3 )4 0 . The same statement holds in dimension n > 3 for the

fil ictiP ) -- z 2. A combination of this with (4.1), (4.8) yields

(4t.9) (D.(z, .)o) 3 is not in L'(Q\f{z})

for ay vectors (k $ 0, 3 ) .0

hitm liice the finction F:

(4.1.0) F( {Pk} k {-}")(Y) = 'p -(Zk)V V\ (Zk, Y) AVU(k)k=1

T] le FrclIet d hrivat ive of F wit Ih respect to {pk} , {k Z k is the linear expressioli

DF( {Apk'. I{A k)(Y)

I.'.

= M''.:k)'D : ,,)2-1VU(k)). '-4(. 11) A=I

III4 ,(Zk)V.V(Z..I). (A4D.'(Zk)( A)

,, VL(:k)] ,l\ , \ k

is

N

I

LrMMA 4.1. If

L 1 kI A~Zk 0) , Ilk > 0 al]I'\7V(ZA) /U

tij.l the fi ,'ti m DF( {Ap.k} '. {Aa.} ")(.) is 1nt idC'ticall, z,'', iII \{z.j, Q\14

Proof If at least one Azk is not zero, then it follows from (4.9), (4.11) and (4.7) (with1) 1) that DF is not in L'( Q\{zk} I' ) and thus not identically zero; here we used the factthat A is positive definite and VU(zk) 5 0. If all the A Zk equal zero then at least one !kPk

is not, equal to zero, and DF is not in TV'l(Q\{Zk}j1j), by (4.7); thus again DF : 0. 0

For the following lemma we need all the assumptions made earlier in the introduction,

in marticular (1.3),(1.4) and (1.15). Let •

H-({p(}", {:k}) ; , ({p'' •}, {Z".}I

a S=1i({- [f))}, { -}) -F( {p }, {k)11 L(r)0,+ 11N [F({m. 1, 1 -k }) -r({pk, I { .]11L- (F)

where F is a fixed nonempty, open subset of OQ.

LvMMA% 4.2. There exists a positive constant 6 such that if

%H ~ ~ j '' f ~ '' f k1' {Zk-l )

< 5' .

V..,

then

(i) [ = K' and, after appropriate reordering,

(ii) Ik--kI + Pk - P' CH({Pk}', {z } ",{p'Pk"{Z'}(" ),1 <1 k < K,,

,stants 6 and C depend on the same parameters as 60. (: anid C in Theorem 1.1. .

Prloof Supl)ose the assertion K K' is not true. Then there exist sequences

(~n { )} , ,( ) =1 { } , ,"

fn '~ m I_ {. ) }A(?f)r{"f),}~

with Kt K /t.' such that the corresponding H converges to 0 as in -- oo (since K. K' <

IO =1 Q / I [ I < (/u/2} , we can choose K and A' to be independent of m). Passinght) a si i1)i"c(lieice w(' get p

I - , - p _ -4p an p -- p.

19 I.'-* ~ UJ!BY A~.?..i

rum rx I

Sinice II11 poele eo

I'm~-) Fq) m= liiii .F)pt"', y C(

;11( snllIryfor +') F Ie

-F(p.:(y - (p'. '(y y G F

Tl-I Jtsh le fmiictioii G( !J) E Np. z)( y) - .F(p', Z' )(!'/) is a ,,(liltioin of

V.((Ij)\G) =(0 in Q\( {Zk I I U I{Zk)

whIichI v.IiiiisI es t ( )et.Il(l Nvi t i it', first derivatives on F. B unqueniess o)f the solutmlol to

the ( ch ieliv ii Q en(:3], [7]1) it f diows t hat G Is ident ically zero,ie.

(12) F(p. z F( p', V' in Q (Zkj U f )-k II

The c millilc114uts o)f f) a id ,,/ are all Iositive, so F( p, z) has a sinigularity at precisely' each

()f t he p iilt s zj andI F( K ' )ha a siiigiilariitN at precisely each o~f the p)oints zj(by (4.7)).

'F Icicfrc (-112) is, a c( lt raicimilll and~ Nve conclude that K =A'. The same argumnt

sla vs t hat , ;Iftcr rcoI.lllel10.

The pi ( (11) pill )((1 al(ig thle saica lilnes. Assulil t hat thec assert ion is not t rile.

T11411 t i Xist sc((lelies p . an 111P( ... (each eeent, a A -Nvector.) suchl

44) (11). )('71) n

1.4 04 z k -, k P

,ilMcc lie delioilliiiator Ill (4.141) is bounded, it follows that H -*0. and thus by (4.13)

KI(m)) 00 1_ +(1)11) ))k ~ Z k ). -

11 ) 0

a ft 4 m 140 g Palssi ll toI a subse5elile we Iliay' assumlie that .

hill Z 01 hll ~ 44

(011) (444) /11

hilli k -4V..

20(~

L'-rd-'A

lot c t hatA

~3(I AZk + IAPJk ) 1.

Vrmi (.1-1) aml the (lefinition of fl it follows that

+ Dr({Ap.} -, { }, )lL I') ---- 01 ',

+ IJD(fA k L('

whr Dt'" (,n)tes the Fre'liet derivative of F at (p. z). However, DF({Apk}. {f -Zk} )(y)is c1c htiv a s()lutloll of tle Iolli(ogeneous equation

V'.(-.(jlV (DF)(yj))=0 in Q\{zk}"

(se (4. 11 )). ;d lby ullitquil'lcs of the sohuitiol to the Cauchy )robleim

DF( jAIIk { -\-Zk} I)(y) 0 in Q\{zk} 1

1,i-, i;1 c(ItI-ra lict ill to Le mma -. 1.• .5,

Mond (Wt Thewm 1. 1. If < 6-( e" then the representatiou formula in

II'( { /'k I k I'" F F '( I" z' I )lli. u - _ II' -- i llz>u -<,(c)

< 60 + J(f)

aild sinCe )- on F it also follows that "09,

K5~ I(wj.z 1,') -, (r) p

lie ,', ) lenotes the iliaxiliiniii of 2(1 11(f :, {p .}, {z .}) I + I V ,,( . y. {p! }. {k }) i) (floiui-

Lci. ia, :3.3),over !(Fe. p} < [d. D0 ] and {zk} C Q, subject to (1.3). Il terl-is of .

II{p~. Zk 1 {Ik1' "{4}fll) <o (' i 0() '< (

IMJ~k I"- < lo"+

, if" wx, cak,, ,s = t/2 aid e fe so that 2)(() < 6/2. Applying Le'mma 4.2 ve conclude that

A A" 1 a1("

I zk Zki+ I P -&' I< CH -"ll. -CIhi. + i,( )

whiert' 11(e) -0 as f .

21

SF S .P*~ ~ ~ ~ .5 ~ %%,~9%%%

1 ri Alt 1'* -4.1. If it 2 itntd If U Ia)Q Ia only 01 finlitelN manly rela t ivxe eXt 1exIt t l(In

tIL~~~lI'~ ilI t11(s anly poinits In Q heeVL canl vanlish (cef. [1) ). If t lie, pits

..... :, aIrv restriced( to AIV criica 1mi's-' l-I l

C(4J (IIt io)n (1. 15) nay 1e 44'( 4141((.

ismi\ -1.2. li Ht pcacaeoa msatrfenecndclockwis - integral 2

tieIcilltIII(' the fixet iIhav l i nilrt tote es tsa ut he oc inof140

vr4 ;I ist14 llil .'I f Q tI(I till 12'lyi w udx Iiea111 stm t o h

1 Lipci t csi t -) Le m 4(1o.2 -eq al

-im.3. One pioiiem raTseot r qetion 1.1 "btesto posble"i (choic to F r ,,O

%l;tt('II ile 4-V pe B4 that asI'0 . the st rmst fct o fcnd eaills 1nlyi woexuelld~ )

be ill thre air~(~eit at [10] altoughi foresults weul ditart with hdfern

( I)vesI 1.± .(L

?a 5 ,au% (-;1" w -ci it'. I.. re e by 11) Fo rt =. 2' tills folw fro the fac tha problems

(,f. (3.27)) where v is the outward normal relative to , Suppose there is only ole-

iilioniogcieitv, with z = 0. p = 1. We assume that B is star-shaped with respect to the I

(5.2) ,- 0 along OB (r =1 I).

(1 1 dL,'t q" !y) and d'(!) l'otc' the uillique soliutions to I

0in {I y I< -)\B.E%

(5.3) 50 , - 1/ i l on 1 [.

T, (Y) =0 oil, yI, = )

alld

Akpi =0 inl R"\ B .

DPv

(5.4) 0 - viOi a

01(1

I

k,! -+ O as I Y 1- 0

w 1icy I, is tlie outward normal relative to B. Notice that the function r(y) i i(r) =

r , r1 i . satisfies

< 0 on OB (by (5.2)),

;md t herefore Cm,(y) can be used as a barrier for the T[. We conclude that T' -' I in .

C, ill aiv~\ Set f I yi j< R})\1 as f--+ 0.

C( -,lsid 'r the flllct iollI

W ,(y) _ - - . (o)'I'( ) . .haU -

'1l lien rgy estilliate's it follows, as ill section 3, that

I '(!) I < C 2 I

=C--+ ,l- <o Y 2- on lY= -

%.'

r,,vhd f. d6/1C, 0 < ( < 1/2. At the same timelI

0 umifornflh on oB, as c 0O

923

-d!

I¢. . . . . . ., - - • ",, "* , " "" , " " - "* ' , ,'- "" " ," " '

T

' M

' " ,, '"

" ' " ''*, .T

"'

U

flieiefui I I~ Soi~ai~i

(1 J1

UI() L(x) + tT, E 1(OP,( + cul(c. r) xe9(cB),

wiej ~( * )-~0 its -*0. 1iiiifor-iilfl £ E ((C ) .Inlsertilig thIis into (5.1) weC get

ii,(,1 (q) vsx)~ x J) I. ) (r) !

B.ll

Thc frst Htcgra I?)ithrih-adsdcnbewitl s

(3. 2v(;) )d

[I (0 ,ezi( ____ z )d~ Oe

I~ OI))

wijete 1,, 1. th) - h 0 iiiifoilen ofY ~yI>c ii the outward normal reaivt

11c ie I ,v It egral on te right -hand side oft (5e6 writ ten asiItla

a +1'I +Of

24a

%"%'

III slinilnary

c~y0)VN(0y)AVU(O)an a5

+ j '(.x)-,(', y) d.sx + e nq(e, g

wvithi ..1 (uj) giveli by

57) =Ii J i'(Zj + 'I'j(z))ds~z

i, is th1w ollt"ard normal relative to B, and i(e, y) -- 0 as 6 - 0, uniformly with respect

,I i I ,> 0.Simiilrly, for the case of K rescaled inhomogei:eities, we get the representation formula

t(Y) Pk' (Zk)VN (Zk, Y) -AVU(Zk)

(5.8) k=l

S+ Y"(, ,,{pkj}{Z})

the only differelnce between this and (3.30) is the sign in front of the first term and the

Inatrix (ai i), which here is given by (5.7). 9.

FromI the defilit ion of 'I'., (5.4), and integration by parts (using Lemma 3.1) it follows

a lI I + B +V'I' 2 dz

wvithi TI I conseyetyA snnercpstvwthI ns1ntly A is symmetric positive definite. In general A cannot be('"niltitcd explici, 1 , cr in case B is the unit ball

.9

1 zi\ I'(z) - ,.. I

au 1 wve coniLlite that-wi

(= 1) wy, = mneas I = 1•

(Mr lliiit result, Theoreiml 1.1., now imnmediately carries over to the case of v?.-

IA( K NO\VIL.I)C 1KM .;NT. The first author is partially supported by NSF grant DNIS-

S61288(; the second author is partially supported by ONR contract N00014-85-K-01609,

NSF grant DMS-8601490 and the Sloan Foundation. Work on this project was begun dur-ilu a smecster which the second author spent in the Mat hematics Department at StanfordI. iiiversity.

25

WI

%9'IL M&~ I&

t I"FFIt F"N(EFS '"

I] (. AI,I:SSAN|)ItINI, An ienttitication problem for an elliptic equation in IlwE) variables, 1nlniv. of FI1h-;-.w.c , l'chilial Rept'[ r 198G(.

121 (;. AL,I"SSANI)RINI, Stalh' dlerijoation of Conductivity bI boundary iea.surenients, I.MA Tel.IHeI, rl, I 987.

[3] N. A ItoNSZAJN, A unique continuation theoren for solutions of elliptic partial differential equations

and mequalities of second order, J. Math. Pures Appl., 36 (1957), 235-249.

[I] ).C. BARIBIER AND B.11. BioWN, Recent developments in Applied Potential Tomography - APT. InI IforiEalloEl IhProccssing in Medical Imaging, ed. S.L. Bacharach, 106-121. Nijhoff 1986.

H5 II. If.:.[,LVIr AND A. F-I)MAN, Identification problem in potential theory, Archive Rat. Mech.A ,'\ lo ., apE pear.

[If] I 'lEE 'E'(Eugs Ef othe 1"IX vork.liop on electric,.! iapledance imaging, Sheffield, England, 1986. B.11. Brown

17] I1.0. (C'o)ir;s, lUber ,ite Bes intlheit der L6sungen elliptischer Differential gleichungen durch An-

fangs vorgaben, Nachl. Akad. Wiss. Goettingen Math.-Phys. MI. Ila (1956) , 239-258.

[S] A. I"EII)MAN, 1)iection of mines by electric measuremcnts, SIAM J. Appl. Math, 47 (1987), 201-212.

[] A. I"ItIIhMAN AND B. (I'STAFSSON, Identification of the conductivity coefficient in an elliptic equa-tonE, SIAM .1. Math. Anal., IS (1987), 777-787.

10] ).(. (;hSSlR, 1). ISA,ACSON AND J.C. NEWELL, Electric current computed tomography and Eigenval-

,is 1, I're'l)rit, 1987.

[1I] It. KOIIN AND NI. VOGELIUS, Determining conductivity by boundary measurements, Comm. PureAppl. Math, :17 (198.1), 289 298.

[121 I. K)IiN AND M. Vo;IUIS, l)eternining conductivity by boundary measurements II. Interior Re-

,ilt., ( 'E Itl. lure A p))I. Math., 38 (1985), 643-667.

[13] I. KInN ANI) M. V((;EIIrS, in preparation.

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I1 N .( M. I EVES AND .1. SI.IRIN, The exterior Dirichlet problei for second order elliptic )artial difTr-iti ,il equal ions, J. Math. Mech., 9 (1960), 513--538.

[I ;] I I MILI. it, Stalbilized Ilmierical analytic prolongation with poles, SIAM J. Appl. Math., 18 (1970).31H;:36:1.

117] .1. SYIVI.S]rEit A N) C. IT IfLM ANN, A uniqueness theorem for an inverse boundary value problem inve'trical lro.sectiol, (onim. Pure App. Math., 39 (1986), 91-112.

[Is] .. SYIVITIIIt AND G. [TIIIIANN, A global uniqueness theorem for an inverse boundary value problem,A lsl 4) Math., 12.5 ( 1987), 153- 169. .,

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[21] T..I. YOEIhI I,;Y, 31.(,. \VIISTER AN) W.J. TOMPiKINs, Comparing reconstruction algorithms for Elec-trical imlpedance tonographY., Preprint 11CRL-96627, Lawrence Livermore National Laboratory,

1987. li

26

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280 It. Duran, On the Approximation of Miscible Displacement in Porous Media by a Method of CharacteristicsCombined with a Mixed Method

281 H. Aixiang, The Convergence for Nodal Expansion Method Functional Integrals282 V. Twersky, Dispersive Bulk Parameters for Coherent Propagation in Correlated Random Distributions283 F. den Hollander, Mixing Properties for Random Walk in Random Scenery284 II. It. Jousulin, Nondifferentialile I'utentials for Ntni:quililrimji Steady States285 K. Meyer, G. It. Sell, Ilonioclinic Orbits and Bernoulli Bundles in Almost Periodic Systems286 J. Douglas, Jr., Y. Yuan, Finite P;fference Methods for the Transient Behavior of a Semiconductor Device287 Li Kaitai, Yan Ningning, The Extrapolation for Boundary Finite Elements288 R. Durrett, R. Schoniann, Stochastic Growth Models289 D. Kinderlehrer, Remarks about Equilibrium Configurations of Crystals290 D. G. Aronson, J. L. Vazquez, Eventual C°-Regularity and Concavity for Flows in One-Dimensional

Porous Media291 L. R. Scott, J. M. Boyle, B. Bagheri, Distributed Data Structures for Scientific Computation292 J. Douglas, Jr., P. J. Paes Lerne, T. Arbogast, T. Schmitt, Simulation of Flow in Naturally Fractured

Petroleum Reservoirs293 D. G. Aronson, L. A. Caferelli, Optimal Regularity for One-Dimensional Porous Medium Flow29-1 Haini Brezis, Liquid Crystals and Energy Estimates for S 2 -Valued Maps295 T. Arbogast, Analysis of the Simulation of Single Phase Flow through a Naturally Fractured Reservoir296 H. Yinnian, L. Kaitai, The Coupling Method of Finite Elements and Boundary Elements for Radiation Problems297 T. Cazenave, A. Haraux, L. Vazquez, F. B. Weissler, Nonlinear Effects in Wave Equation with a Cubic

Restoring Force298 M. Chipot, F. B. Weissler, Some Blow-Up Results for a Nonlinear Parabolic Equation with a Gradient Term299 L. Kaitai, Perturbation Solutions of Simple and Double Bifurcation Problems for Navier-Stokes Equations300 C. Zhangxin, L. Kaitai, The Convergence on the Multigrid Algorithm for Navier-Stokes Equations301 A. Gerardi, G. Nappo, Martingale Approach for Modeling DNA Synthesis302 D. N. Arnold, L. Ridgway, M. Vogelius, Regular Inversion of the Divergence Operator with Dirichlet Boundary

Conditions on a Polygon Divergence Operator with Dirichlet Boundary Conditions on a Polygon303 Rt. G. Duran, Error Analysis in LP, 1 < p < oc ,for Mixed Definite Element Methods for Linear and

Quasi-Linear Elliptic Problems304 R.. Nochetto, C. Verdi, An Efficient Linear Scheme to Approximate Parabolic Free Free Boundary Problems:

Error Estimates and Implementation305 K. A. Pericak-Spector, S. J. Spector, Nonuniqueness for a Hyperbolic System: Cavitation in Nonlinear

Elastodynamics306 E. G. Kalnins, W. Miller, Jr., q-Series and Orthogonal Polynomials Associate with Barnes' First Lemma307 D. N. Arnold, R. S. Falk, A Uniformly Accurate Finite Element Method for Mindlin-Reissner Plate308 Chi-Wang Shu, TVD Properties of a Class of Modified Eno Schemes for Scalar Conservation Laws309 E. Dikow, U. Hornung, A Random Boundary Value Problem Modeling Spatial Variability in Porous Media Flow310 J. K. Hale, Compact Attractors and Singular Perturbations311 A. Bourgeat, B. Cockburn, The TVD-Projection Method for Solving Implicit Numeric Schemes for Scalar

Conservation Laws: A Numerical Study of a Simple Case312 B. Muller, A. Rizzi, Navier-Stokes Computation of Transonic Vortices over a Round Leading Edge Delta Wing313 J. Thomas Beale, On the Accuracy of Vortex Methods at Large Times314 P. Le Talle, A. Lotfi, Decomposition Methods for Adherence Problems in Finite Elasticity315 J. Douglas, Jr., J. E. Santos, Approximation of Waves in Composite Media316 T. Arbogast, The Double Porosity Model for Single Phase Flow in Naturally Fractured Reservoirs317 T. Arbogast, J. Douglas, Jr., J. E. Santos, Two-Phase Immiscible Flow in Naturally Fractured Reservoirs318 J. Douglas, Jr., Y. Yirang, Numerical Simulation of Immiscible Flow in Porous Media Based on Combining the

Method of Characteristics with Finite Element Procedures319 R. Duran, R. H. Nochetto, J. Wang, Sharp Maximum Norm Error Estimates for Finite Element Approximationsof the Stokes Problem in 2-D320 A. Greven, A Phase Transition for a System of Branching Random Walks in a Random Environment321 J. M. Harrison, R. J. Williams, Brownian Models of Open Queueing Networks with Homogeneous Customer

Populations322 Aia Bela Cruzeiro. Solutions ET mesures invariantes pour des equations d'evolution Stochastiques du type

Navier-Stokes323 Salali-Eldin A. Mohammied, The Lyapunov Spectrum and Stable Manifolds for Stochastic Linear Delay Equations324 Bao Gia Nguyen, Typical Cluster Size for 2-DIM Percolation Processes (Revised)325 It. Hardt, D. Kinderlehrer, F.-l. Lin, Stable Defects of Minimizers of Constrained Variational Principles326 M. Chipot, D. Kiniderlehrer, Equilibrium Configurations of Crystals

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Recent IMA Preprints (Continued) 0# Author/s Title

:127 Kiyosi It6, Malliavin's C' functionals of a centered Gaussian system ,.328 T. Funaki, Derivation of the hydrodynamical equation for one-dimensional Ginzburg-Landau model329 Y. Masaya, Schauder Expansion by some Quadratic Base Function330 F. Brezzi, J. Douglas, Jr., Stabilized Mixed Methods for the Stokes Problem331 1. Mallet-Paret, G. R. Sell, Inertial Manifolds for Reaction Diffusion Equations in Higher Space Dimensions332 San-Yih Lin, M. Luskin, Relaxation Methods for Liquid Crystal Problems333 H. F. Weinberger, Some Remarks on Invariant Sets for Systems334 E. Mierseinann, H. D. Mittehnanu, On the Continuation for Variational Inequalities Depending on an

Eigenvalhe Parameter335 J. Hulshof, N. Wolanski, Monotone Flows in N-Dimensional Partially Saturated Porous Media: Lipschitz

Continuity of the Interface336 B. J. Lucier, Regularity Through Approximation for Scalar Conservation Laws337 B. Sturinfels, Totally Positive Matrices and Cyclic Polytopes338 R. G. Duran, R. H. Nochetto, Pointwise Accuracy of a Stable Petrov-Galerkin Approximation to Stokes Problem339 L. Gastaldi, Sharp Maximum Norm Error Estimates for General Mixed Finite Element Approximations to

to Second Order Elliptic Equations340 L. Hfurwicz, H. F. Weinberger, A Necessary Condition for Decentralizability and an Application to

Itemporal Allocation S341 G. Chavent, B. Cockburn, The Local Projection P°P'-Discontinuous-Galerkin-Firite

Element Method for Scalar Conservation Laws A342 I. Capuzzo-Dolcetta, P.-L. Lions, llarnilton-Jacobi Equations and State-Constraints Problems343 B. Sturmnfels, N. White, Gr6bner Bases and Invariant Theory344 J. L. Vazquez, Cm-Regularity of Solutions and Interfaces of the Porous Medium Equation345 C. Beattie, W. M. Greenlee, Inproved Convergence Rates for Intermediate Problems346 H. D. Mittelmann, Continuation Methods for Parameter-Dcpendent Boundary Value Problems 7-347 M. Chipot, G. Michaille, Uniqueness Results and Monotonicity Properties for Strongly Nonlinear -

Elliptic Variational Inequalities348 Aver Friedman, Bei Hu The Stefan Problem for a Ilyperbolic Beat Equation349 Michel Chipot, Mitchell Luskin Existence of Solutions to the Elastohydrodynarnical Equations for

Magnetic Recording Systems350 ft.H. Nochetto, C. Verdi, T'e Combined Use of a Nonlinear Chernoff Formula with a Regularization S

Procedure for Two-Phase Stefan Problems -.

:151 Gonzalo R. Mendieta Two lyperfinite Constructions of the Brownian Bridge352 Victor Klee, Peter Kleinschnmidt Geometry of the Gass-Saaty Parametric Cost LP Alkorithm353 Joseph O'Rourke Finding A Shortest Ladder Path: A Special Case354 J. Gretenkort, P. Kleinschmidt, Bernd Sturinfels, On the Existence of Certain Smooth Toric

Varieties355 You-lan Zhu On Stability & Convergence of Difference Schemes for Quasilinear Hyperbolic Initial-

Boundary-Value Problems:156 Harod Bellout, Avner Friedman Blow-Up Estimates for Nonlinear Hyperbolic Heat Equation357 P. Gritzmnan, M. Lassak Belly-Test for the Minimal Width of Convex Bodies 'A-358 K.R. Meyer, G.ft Sell Melnikov Transforms, Bernoulli Bundles, and Almost Periodic Perturbations359 J.-P. Puel, A. Raoult Buckling for an Elastoplastic Plate with An Increment Constitutive Relation360 F.G. Garvan A Beta Integral Associated with the Root System G2:161 L. Chihara, D. Stanton Zeros of Generalized Krawtchouk Polynomials362 Hisashi Okanmoto O(2)-Equivariant Bifurcation Equations with Two Modes Interactioni:163 Joseph O'Rourke, Catherine Schevoii On the Development of Closed Convex Curves on 3-Polytopes364 Weinan E Analysis of Spectral Methods for Burgers' Equation365 Weinan E Analysis of Fourier Methods for Navier-Stokes Equation a-.

366 Paul LAtenke A Counterexample to a Conjecture of Abbott 031:7 Peter Gritzmann A Characterization of all Loglinear Inequalities for Three Quermassintegrals of

Convex Bodies368 David Kinderlehrer Phase transitions in crystals: towards the analysis of microstructure369 David Kraines, Vivian Kraines Pavlov and the Prisoner's Dilemma:170 F.G. Garvan A Proof of the MacDonald-Morris Root System Conjecture for F4

371 Neil L. White Cayley Factorization372 Bernd Sturanfels Applications of Final Polynonials and Final Syzygies37:1 AvIner Friedmnan, Michael Vogelius ldentification of Small Inhomogeneities of Extreme Conductivity

by Boundary Measurements: A Continuous Dependence Result374 Jan Kratochvil, Mirko KfivAnek On the Computational Complexity of Codes in Graphs375 Thomas I. Seidmnan The Transient Semiconductor Problem with Generation Ternms, II

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