Double logarithmic stability in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map Текст научной статьи по специальности «Математика»

MSC 35R30

DOI: 10.14529/ mmpl50305

DOUBLE LOGARITHMIC STABILITY

IN THE IDENTIFICATION OF A SCALAR POTENTIAL

BY A PARTIAL ELLIPTIC DIRICHLET-TO-NEUMANN MAP

M. Cfumlli, University of Lorraine, Metz, France, [email protected],

Y. Kian, Aix-Marseille University, Marseille, France, [email protected], E. Soccorsi, Aix-Marseille University, Marseille, France, [email protected]

We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Sehrodinger equation in a bounded domain, from a partial elliptic Diriehlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H_1-norm) of H for t > 0, and bounded (resp. L2) potentials.

Keywords: inverse problem; stability; Sehrodinger equation.

This work is dedicated to the memory of Alfredo Lorenzi.

1. Introduction

1.1. Settings and Main Result

In the present paper Q is a bounded domain of Rra, n > 3, with C2 boundary T. We denote by v(x) the outward unit normal to r, computed at x E r. For £ E S^1 fixed, we introduce the two following subsets of r

r±(£) = {x E r; • v(x) > 0}, (1)

and denote by F (resp. G) an open neighborhood of r+(£) (resp. r_(£^^n r. In what follows r+(£) (resp. r_(£will sometimes be referred to as the ^-shadowed (resp., illuminated) face of r Next, given q E L^(Q), real-valued, we consider the unbounded self-adjoint operator Aq'm L2(Q), acting on his do main D(Aq) = H^ (Q) H H 2(Q), as

Aq = -A + q.

0 Aq 0

Aq

Q = {q E L^(Q; R); 0 is not an eigenvalue of Aq}.

We establish in Section 2 for any q E Q and g E H_1/2 (r) that the boundary value problem (abbreviated to BVP in the sequel)

( (-A + q)u = 0 in Q, (2)

\ u = g on r,

7g Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 78-94

admits a unique transposition solution u £ HA(Q) = {w £ L2(Q); Aw £ L2(Q)} and that the so-called Dirichlet-to-Neumann (DN in short) map

Лд : g м dv u

(3)

is a bounded operator from H 1/2(r) into H 3/2(r). For qj £ Q, j = 1, 2, we denote by Uj the solution to (2) where qj is substituted for q. Since u = u1 — u2 satisfies

!

(—A + qi)u = (q2 — qi)u2 'т П, u = 0 on Г,

and (q2 — q1)u2 £ L2(Q), it holds true that u £ D(Aq1). Therefore dvu £ H1/2(Г) and

Лд1;д2 = Лд1 — Лд2 £ B(H^(Г), H 1/2 (Г)),

hence the operator

Лд1,д2 : g £ H-1/2 (Г) n E'(F) м Лдъд2 (g)\

о,

(4)

(5)

is bounded from H 1^2(r) fi E'(F), endowed with the norm of H 1/2 (r), mto H1/2 (G). We denote by \\Aquq21| the norm of Ag^ in B(H-1/2(r) f E'(F),H1/2(G)).

In the present paper we examine the stability issue in the inverse problem of determining the potential q £ Q appearing in (2) from the knowledge of Aqo,q, where qo is a priori known suitable potential of Q.

Upon denoting by BX the unit ball of any Banach space X, we may now state the main result of this article as follows.

Theorem 1. For any 5 > 0 an d t > 0 we may find two con stants c> 0 an d c > 0, both of them depending only on 5 and t, such that we have

(

||qi — q2|U2(n) < С !Лд1 ,д2 || + In С In |Лд1 ,д2 ||

1

(6)

for any qi,q2 £ Qn 6BL^(Q) satisfying (q2 — qi)xn £ SBnt(vn), and

(

||qi — q2|H-1(n) < c д1, д2 Н + lln С lln |Лд1, д2!

-i

(7)

for any q1 ,q2 £ Qf 5BL2{n).

Let us now briefly comment on Theorem 1.

qj j = 1 , 2

not hard to see that the statement can be adapted at the expense of greater technical difficulties, to the case of complex-valued potentials. Nevertheless, for the sake of clarity, we shall restrict ourselves to real-valued potentials in the remaining part of this text, (b) For s > n/2 and e £ (0, s — n/2) we recall from the interpolation theorem [1, Theorem 12.4, page 73] that Hn/2+%Q) = [Hs(tt),H-1(n)]0 with 9 = (s — (n/2 + e))/(s + 1). Therefore we have

I\q1 — ta\\L~m < C(s)\\q1 — q2\\H-(n) \\q1 — q2\H-i(n) < C(s)Sl-d\\q1 — qtXH-1(n),

for any q1 ,q2 E Q such that q2 E q1 + 6BHand some constant C(s) > 0, depending only on s. From this and (7) then follows that

Ikl - q2||L~(Q) < c

0

■-91,92 1

II + ln 5 ln ||Aq 1,92 ||

(8)

(c) Fix t E (0, Then, arguing as in the derivation of (8), we find two positive

constants c and c such that the estimate

|qi - q2||L2(Q) < c 11Aqi,92 11 +

o

ln C

ln

91,92

-A t+1

l

holds uniformly in qi,q2 E Q obeying q2 E q1 + 6BHt(n)- However in the particular case where SQ is C[i]+1, we point out that the above estimate is weaker than (6).

II

1.2. State of the Art and Comments

The celebrated inverse problem of determining q from the knowledge of A9 was first proposed (in a slightly different setting) by Calderón in [2]. The uniqueness issue was treated by Sylvester and Uhlmann in [3] and a log-type stability estimate was derived by Alessandrini in [4]. As shown by Mandache in [5], this log-type estimate is optimal.

All the above mentioned results were obtained with the full data, i.e. when measurements are performed on the whole boundary T. Taking the Neumann data on r_(£), while the Dirichlet data is imposed on the whole boundary T, Bukhgeim and Uhlmann proved in [6] that partial information of A9 still determines uniquely the potential. Their result was improved by Kenig, Sjostrand and Uhlmann in [7] by measuring the Dirichlet data on the shadowed face of r and the Neumann data on the illuminated one. Moreover a reconstruction result was derived by Nachman and Street in [8] from the same data as in [7].

Stability estimates with partial data go back to Heck and Wang's article [9], where the L^ (n)-norm of q is log log stably recovered from A9 with partial Neumann data. The same type of estimate was derived in [10]. Both papers require that the Dirichlet data be known

on the whole boundary. This constraint was weakened by Caro, Dos Santos Ferreira and

q

associated with Dirichlet (resp. Neumann) data measured on a neighborhood of r_(£) (resp. r+(£)) where N in a subset of Sra-1. Their result, which is similar to (6), is established for the Lp-norm, p E (1, of bounded and WA'p-potentials q with A E (0,1/ p]. Therefore (7) is valid for a wider class of allowable potentials than in [11].

The derivation of Theorem 1 relies on complex geometrical optics (CGO in short) solutions to (2) and the Carleman inequality established by Bukhgeim and Uhlmann in [6]. These are the two main ingredients of the analysis carried out in [9]. But in contrast to [9], we use here the above mentioned Carleman estimate to construct CGO solutions vanishing on a definite part of the boundary T.

Notice that usual stability estimates in the inverse problem of determining a potential from the full DN map are of log type, while they are of log log type for partial data. Indeed, it turns out that the low frequencies of the Fourier transform of the potential are bounded uniformly in all directions by the DN map, but that this is no longer the case with the partial data. This technical issue can be remedied by using the analytic properties

of the Fourier transform. The additional log in the stability estimate for the potential may thus be seen as the price to pay for recovering this analytic function by its values in a subdomain, which is an ill-posed problem.

1.3. Outline

The paper is organized as follows. In Sect. 2 we introduce the transposition solution associated with the BVP (2) and rigorously define the various DN maps required by the analysis of the inverse problem. Sect. 3 is devoted to building CGO solutions that vanish on some part of the boundary T. These functions are useful for the proof of Theorem 1, given in Sect. 4.

2. Transposition Solutions

In this section we define the transposition solution to the BVP associated with suitable data (f,g),

f (-A + q)u = f in Q, \ u = g on r,

which play a pivotal role in the analysis of the inverse problem carried out in this paper. To this purpose we start by recalling two useful results for the Hilbert space Ha(Q) = {u E L2(Q); Au E L2(Q)} endowed with its natural norm

( \ 1/2

HuHffA(fi) = (jlullL2(fi) + llAullL2(fi)J •

The first result is the following trace theorem, borrowed from [6, Lemma 1.1]. Lemma 1. For j = 0,1, the trace map

tju = dju\r, u E D(Q),

extends to a continuous operator, still denoted by tj, from HA(Q) into H-j-1/2(r). Namely, there exists Cj > 0, such that the estimate

¥j ullH-j-i/2(r) < cj ||u||Ha(q),

holds for every u E Ha(Q).

Let us denote by (•, •)j+1/2, j = 0,1, the duality pairing between Hj+1/2(r) and H-j-1/2(r), where the second argument is conjugated. Then we have the following generalized Green formula, which can be found in [6, Corollary 1.2].

Lemma 2. Let q be in L^(Q). Then, for any u E Ha(Q) and v E H2(Q), we have

j (A - q)uvdx = j u(A - q)vdx + (ti u, t0v)3/2 - {t0u, tiv)i/2• n n

Let q E Q. By the usual H2-regularity property for elliptic BVPs (see e.g. [1, Theorem

5.4, page 165]), we know that for each f E L2(Q) and g E H3/2(r), there exists a unique

solution Sq(f, g) E H2(Q) to (9). Moreover the linear operator Sq is bounded from L2(Q) x H3/2 (r) into H2(Q), i.e. there exists a constant C > 0 such that we have

\\Sq(f,g)Wn2(h) < c (\\f |L2(Q) + \\g\\H3/2(d) . (10)

For further reference we put Sq,o(f) = Sq (f, 0^d Sq>1(g) = Sq (0, g), so we have Sq (f, g) = Sq,o(f) + Sq>1(g) from the linearity of Sq.

Next, applying Lemma 2, we get for all (f,g) E L2(Q) x H3/2(r^d v E H = H1(Q) n H2(Q) that

— J gdvvda(x) + J fvdx = J u(—A + q)vdx, u = Sq(f,g). (11)

r H H

In view of the left hand side of (11) we introduce the following continuous anti-linear form on H

m = —j gB^daix)^ f~vdx,v EH. (12)

r H

In light of (10), the operator L = 1 o Sq>o is bounded in L2(Q). Further, with reference to (12), we generalize the definition of the anti-linear form I to (f,g) EH* x H _1/2(r), upon setting

l(v) = —{g,dvv)1/2 + {f,v), v EH,

where {•, •) denotes the duality pairing between H and H*, conjugate linear in its second argument. For any h E L2(Q), L(h) = 1 (Sq,0(h)) satisfies

|L(h)| < \\g\H—1/2 (r) \ Sv Sq,o (h) \ \HV2(r) + \\ f \\h \\Sq>o(h)Wн < C (\\g\\H-i/2(r) + \\f \\h ) \h\L2(H),

according to (10). Hence L is a continuous anti-linear form on L2(Q). By Riesz representation theorem, there is a unique vector Stq(f,g) E L2(Q) such that we have

— {g,dvSq,o(h))1/2 + {f, Sq,o(h)) = j S\(f,g)hdx, h E L2(Q).

H

Bearing in mind that Aq is boundedly invertible in L2(Q) (since 0 is in the resolvent set of Aq) and that Sq,o = A-1, we obtain upon taking h = Aqv ^n above identity, where v is H

— {g,dvv) 1/2 + {f,v) = J sq(f,g)(—A + q)vdx, v EH. (14)

H

Moreover (13) entails

\\sq(f,g)\\L2(H) < C (\\g\\H—!/2(P) + \\f \\h). (15)

(f, g) E H* x H-1/2(r), Sq(f,g) will be referred to as the transposition solution to the BVP (9). As a matter of fact we deduce from (14) that

(—A + q)Sq(f,g) = f in the distributional sense in Q. (16)

Let us establish now that the transposition solution Slq(f,g) coincides with the classical H2(n)-solution Sq(f, g) to (9) in the particular case where (f,g) E L2(Q) x H3/2 (r).

Proposition 1. For any (f,g) E L2(Q) x H3/2(r) we have Stq(f,g) = Sq(f,g).

Proof. Put u = Sq(f,g).\Ne have u E Ha(H) directly from (16) hence t0u E H_1/2(r). Further, given p E H1/2 (r), we may find v E H such that t1v = p by the usual extension theorem (see e.g. [1, Theorem 8.3, p. 39]). Applying Lemma 2 for such a test function v, we find that

J fvdx = J (-A + q)uvdx = J u(-A + q)vdx + (t0u,p)1/2•

n n n

From this and (14), it then follows that J fvdx = J fvdx - (g, p)1/2 + (t0u, t1 v)1/2, which

nn

entails

(g - tou, p) 1/2 = 0^

Since the above identity holds for any p E H1/2 (r) we obtain that t0u = g and hence u = g on r. This yields the desired result.

For g E H-1/2(r) we put S* 1(g) = Sq(0,g).We have Sq 1(g) E HA(Q) by (16), with°

||55>1(^}|нд(п) < с (1 + IMlL-(fi)) \\g\\n-1/2(P),

from (15). Hence S¡л E B(H-1/2(Г), HA(Q)) so we get

Лд = ti ◦ Si, E B(H-1/2(Г),Н-3/2(Г)),

with the aid of Lemma 1.

Finally we have Sq(f,g) = S^f, 0) + S\(0, g), by linearity of S^. Put S^f) = S\(f, 0) and Stql(g) = SI(0,g). Since Sqo E B(H*, НА(П)) and S^ E B(H-1/2(Г), HA(Q)), it follows from Lemma 1 that

Лq = ti ◦ Si, E B(H-1/2(r),H-3/2(Г)).

Further, as StqA(g) = SoA(g) + Sq,o(f^ with f = -qS0A(g), we get that

Л1 = Ло + Rq,

where the operatorRq

Rq : g ^ t1Sq,o (^Ag)) E H 1/2(Г), H -1/2(Г) into H1/2 (Г). Remark 2. Let us denote by Eq the (continuous) DN map

Eq : g E H3/2(Г) ^ t1Sq1 (g) E H 1/2(Г).

q

J gEq(h)da(x) = j Eq(g)hda(x), g,h E H3/2(Г), г г

which entails that (Е*)|Я3/2(Г) = Eq. On the other hand, we deduce from Lemma 2 that E* = Л

Eq = Л q'

3. CGO Solutions Vanishing on Some Part of the Boundary

In this section we build CGO solutions to the Laplace equation appearing in (2), that vanish on a prescribed part of the boundary r. This is by means of a suitable Carleman estimate borrowed from [6]. The corresponding result is as follows.

Proposition 2. For 5 > 0 fixed, let q £ 5BL^^ny Let (,n £ Sra_1 satisfy ( • n = 0 and fix t > 0 so small that r_ = r_ (() = {x £ r; ( • v(x) < —t} = 0. Then there exists t0 = To(5) > 0, such that for any t > t0 we may find ^ £ L2(Q) obeying < Ct-1/2

for some constant C > 0 depending only on 5, Q and t, and such that the function u = eT(z+in>x(\ + £ Ha(Q) is solution to the BVP

{

(—A + q)u = 0 in Q, u = 0 on P

Proof. The proof is made of three steps.

Step 1: A Carleman estimate. For notational simplicity we write r± instead of r±(Z), which is defined in (1), and recall from [6] that we may find two constants t0 = t0(5) > 0 and C = C(5) > 0 such the estimate

Ct2 J e~2TX<M2dx + t J \Z • v(x)\e_2TX<\dvv\2da(x) < n r+

< J e~2TX<\(A — q)v\2dx + t j \Z • v(x)\e_2TX<\dvv\2da(x), n r_

holds for all t > T^d v £ ^^ce r±(—() = ), the above inequality may be equivalently rewritten as

Ct2 J e2TX•z\v\2dx + t j \Z • v(x)\e2TX • z\dvv\2da(x) < (17)

n r_

< J e2TX•z\(A — q)v\2dx + t J \( • v(x)\e2TX z\dvv\2da(x), v £H, t > To. n r+

In view of more compact reformulation of (17) we introduce for each real number t the two following scalar products:

(U'v)T = / ^ ^ ^^

n

and

= J j(x)e2TX'zfopda(x), where j(x) = \Z • v(x)\, in L2(r±). r±

We denote by L^(Q) (resp., Li^r^)) the space L2(Q) (resp., L2(r±)) endowed with the norm || • ||T (resp. || • ±) generated by the scalar product (•, ^)T (resp., (•, •)T,^,,±)- With these notations, the estimate (17) simply reads

Ct2||v||2 + tWdAH^_ < ||(A — q)v||2 + tRv^ v £H, t > To. (18)

(y. | Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 78-94

Step 2: Building suitable transposition solutions. Let us identify L0(Q) = L2(Q) (resp., LO,^^) = L2(r±)) with its dual space, so the space dual to L^(Q) (resp., L'2,»(r±)) can be identified with L—T(Q) (resp. L2_T»-1 (r±)). Next we consider the operator

P : v EH^ ((A - q)v,dv v]r+) E LT (Q) x L^(r+), (19)

which is injective by (17). Therefore, for each (f,g) E L—T(Q) x L-T»-1 (r-), the following anti-linear form

e(w1,w2) = (f,v) + (g, dvv)_, (w1 ,W2) = Pv, v E H, (20)

where (•, •) (resp., (•, -)±) denotes the usual scalar product in L2(Q) (resp., L2(r±)), is well defined on Ran(P). Moreover it holds true for every (w1,w2) = Pv, where v is arbitrary in PL, that

\l(w1,w2)\ < llf ||_T Mr + llgll-T,--lldvvlT»,_ (21)

< (t-1||f ll-T + T-1/2llgll-T,»-1,-) (t2|v|2 + tlldvvllt,»,-)1/2 •

Thus, upon equipping the space LT.(Q) x L^»(r+) with the norm \|(w1,w2)|\T =

1 /2

(|w1|2 + tllw^l2»,+) , we derive from (18) and (21) that

\l(w1 ,w2)\ < C (t-1llf ll-T + T-1/2llgll-T»-i _-) {ll(A - q)vl2 + TlldvvlT» +1/2 < C (T-1llf ll-T + T-1/2llgll-T,»-i__) \llPvll\r, for some constant C > 0 depending only on 8. As a consequence we have

\l(w1,w2)\ < C (t-1llf ll-T + T-1/2llgll-T,»-i,-)\ll(w1,w2)ll\T, (w1,w2) E Ran(P) (22)

Let us identify the dual space of LT.(Q) x L^»(r+) with L2-T(Q) x L2-T»-1 (r+) endowed

1 /2

with the norm \|(w1;w2)|\-T = (|w1|-T + t-llw2l2-T»-1,+) . Thus, with reference to (19)-(20), we deduce from (22), upon applying Haim Baiiach extension theorem, that there exists (v1 ,v2) E L2-T(Q) x L2-T »-1 (r+) obeying

\ll(v1 ,v2)ll\-T < C (t-1llf ||-T + T-1/2llgll-T,»-1,-) , (23)

C > 0

(v1, (A - q)v) + (v2,dv v)+ = (f,v) + (g,dv v)-, v EH•

Bearing in mind that f EH* and gxr_ - v2Xr+ E H-1/2 (r), where Xr± is the characteristic function of r± in r, the above identity reads

((-f),v)-(gxr- - v2Xr+ ,dvv)1/2 = J v1(-A + q)vdx, v eH•

n

v1

(-A + q)v1 = -f in Q,

{

Г (24)

v1 = gxr- - v2Xr+ on Г.

Step 3: End of the proof. Set p = t(Z + in)- The last step of the proof involves picking p E D(Rn; [0,1]) such that p(x) = 1 for x E ri and supp(p) H r C r^2, and considering the transposition solution v1 to the BVP (24) associated with f = qep• ^d g = —pep•x. Thus, putting 0 = eip• xv1y we derive from (23) that

lo = \\vi\\-T < C (T^Wqem + T^Wpe^lU^i)

< ct i1/2( iTo 6i1/2 + )

where |ri| denotes the (n — 1)-dimensional Lebesgue-measure of ri. Further, bearing in mind that p• p = 0 it is easy to check that u = epx(1+ 0) E Ha. (Q) satisfies (—A + q)u = 0 in Q. Finally, si nee v1 = g on ri by (24), we obtai n that 0 = —p = —1 and consequently that u = 0 on P.. This proves the desired result.

□

4. Proof of Theorem 1

This section contains the proof of Theorem 1 which consists of a succession of five lemmas.

4.1. A Suitable Set of Fourier Variables

For t> 0 we choose n = n(t) E (0,n/2) so small that

((1 — sin 0)2 + 4 cos2 0)1/2 < t, 0 E (n/2 — n,n/2 + n), (25)

and we define Be as the set of vectors f = (f^,..., ffn) E Rn with the following spherical coordinates

f1 = s cos 01,

fij = s cos 9j sin , j = 2,... ,n — 2

_ —

/3n-i = s sin ( (f[sin 9k) , k fin = s cos ( (nl-2 sin 9k) ,

where s E (0,1) 91 E (n/2 - n,n/2 + r¡), 92(0,n/3), 9j E (0,n) for j = 3,...,n - 2, and ( E (0, 2n). Notice th at Be has positive Lebesgue me asure in Rn given by:

1 n/2+n n/3 n n 2n ni2 v

m = j J J J...J jsni1li[ sink 0ni1iJ dsd01... d0ni2d^> 0. (27)

0 n/2iV 0 0 0 0 \k=1 '

For further reference, we now establish the following result.

Lemma 3. Let T be any orthogonal transformation in Rn that maps £ onto e1 = (1,0,..., 0). Then for all t > 0 mid k E T*Be, there exists Z E Sni1 satisfying k • Z = 0 and IZ — £| < t.

Proof Let f = (f1,...,fn) E Be be given by (25)-(26). Introduce ( = Z ,<2> 0,..., 0), where

. Q cos 01

sin 01 ? cos 02

Cl = (-TTT2, C2 = —

1/2 ' j2 1/2 *

(sm2 Oi + CoSfi) (sm2 Oi + CoSf)

Evidently we have ( E S" ^d в ' Z = 0. Moreover it follows from (25) that

~ (sin&1 1)2 + ^Og2h . -W2 x A 2a ^ 2

\Z - e1\ = -. cos2 2 < (sin 91 - 1) +Acos 01 <e •

Sin2 91 + COS27T

1 cos2 02

Finally, bearing in mind that k = T* ft and e1 = T£, we obtain the desired result upon taking Z = T*Z-

□

4.2. Alessandrini's Identity and Consequence

For e > 0 put F"(£) = r \ n(£) and Ge(£) = T \ Tl(-£), where T-(±£) is the same as in Proposition 2. Since Fe(£) = {x E r; £ • v(x) > -e} (resp., Ge(£) = {x E T; £ • v(x) < e^^, it holds true that n>0Fe(£) = {x E T; £ • v(x) > 0} = r+(£) (resp., ne>0Ge(£) = {x E r; £ • v(x) < 0} = ^^)). Thus, from the very definitions of F and G, we may choose e0 = e0(£, F,G) > 0 so small that

F2e(£) C F and G2e(£) C G, e E (0, eo]- (28)

Having said that we turn now to proving the following statement.

Lemma 4. Let T be the same as in Lemma 3, let t E [t0, +<x>), where t = t0(8) is defined in Proposition 2, and let e E [0,e0). Then, there exists a constant C > 0, depending only Q 8 e

(q2 - qi)e-iKXdx

< c(e2dT||Л,ж1| + r-1/2)

holds uniformly in k E rT*Be and r E (0, 2t). Here ||Aq1,q21| is the B(H-1/2(T) n E'(F),Hl/2(G))-norm of the operator Aq1 ,q2 defined in (3)-(5), and d = d(Q) = maxxen \x\ < ro.

Proof. Fix r E (0, 2t), k E rT*Be, and 1 et Z be given % Lemma 3. Pick 1 E Rra such that I • k = 1 • Z = 0 (which is possible since n > 3) and \k +1\2 = \k\2 + \l\2 = 4t2. Set

j = (-1)'tz - j = 1,2,

and let Uj = ePj'x(l + vpj) G HA(Q) be defined in accordance with Proposition 2, in such a way that we have

(—A + qj)uj = 0 in Uj = 0 onn ((2j —

{

(29)

and

ФПод < CT-1/2. (30)

Put u = Sfqi, 1(t0u2) so we have w = u — u2 = Sqi,0((q1 — q2)u2) E H and

t1W = Aqi (toU2) — Aq2 (toU2). (31)

Upon applying the generalized Green formula of Lemma 2 with u = «1 and v = w, we derive from (29) that

J(92 - qi)ui«2dx = (ioUi,iiw)i/2• (32)

n

Notice from the second line of (29) that the trace t0u2 (resp., t0ui) is supported in Fe(() = r\r-(Z) (resp., Ge(() = r\r-(—()), where r-(±Z) is defined in Proposition 2. Otherwise stated we have Fe(() = {x E r; ( • v(x) > —e} (resp., Ge(() = {x E r; ( • v(x) < e}) and hence Fe(() C F2e(£) (resp., Ge(() C G2e(£)) since \( — < e. From this and (28) then follows that supp(t0u2) C F (resp., supp(t0ui) C G), which together with (31)-(32) yields

(92 — qi)uiu2dx

< ||Aqi ,q2 lllt0ui|H-1/2(r)|t0u2|H-1/2(r)- (33)

Moreover, by (29) - (30) and the very definition of uj, j = 1, 2, we get that

lt0uj||H-i/2(r) < cj (|luj^&(n) + 1 Auj||L2(n^ < Cj (luj||L2(n) + huj¡L2(n)) (34) < CedT (1 + t ~i/2).

Upon possibly substituting max(1, t0(^)) for t0(5) (which does obviously not restrict the generality of the above reasoning) we deduce from (33)-(34) that

(92 — qi)uiu2dx

< Ce2dT ||Aqi,q2 ||. (35)

Now the desired result follows readily from (30) and (35) upon taking into account that uiu2 = 6-^(1+ ^i)(1+ 02 )■

□

4.3. Bounding the Fourier Coefficients

We introduce the function q : Rra ^ R by setting q = (q2 — q^Xn where xn denotes the characteristic function of Q in Rn. We aim to upper bound the Fourier transform q of q, on ^te unit ball B of Rra, by means of the following direct generalization of [12, Theorem 4] for complex-valued real-analytic functions.

Theorem 2. Assume that the function F : 2B ^ C is real-analytic and satisfies the condition

M!

\daF(k)| < K^J, K E 2B, a E Nra, 1 wi - p\a\

for some (K,p) E R+ x (0,1]. Then for any measurable set E C B with positive Lebesgue measure, there exist two constants M = M(p, \E\) > 0 md 9 = 9(p, \E\) E (0,1) such that we have

||F||l~cb) < MK1-8 ^^ J\F(K)\d« The result is as follows.

Lemma 5. For all e E (0,e0] there wist two constants C = C(Q,5,e) > 0 mid 9 9(Q,e) E (0,1), such that we have

\q(K)\ < Ce(1-e)r (edT\\Лqi,q2\\ + t-1/2) , к E rB, r E (0, 2t), t E To, +ю).

Proof. Let t E [t0, r E (0, 2t) be fixed. By Lemma 4, we have

\q(rn)\ < C (e2dT \ \ Л qi, q2 \\ + T-1/2) , к E T*Be. (36)

In view of (27) we apply Theorem 2 with E = T*B€ and F(k) = q(rK) (or k E 2B. Indeed, in this particular case it holds true for any a = (a1, • • •, an) E Nn and k E 2B that

daF (k) = f q(x)(-xr)a ( TT xe-irKXdx1 ••• dxw

(й

whence

, , , , / rl«K U|! Ш!

\cTF(к)\ < Мщп)^ < ^\q\Li(n)< Ы^) j-jR, ^

where we recall that d = d(Q) = max^q \x\. Thus, with reference to (36)-(37), we obtain that

\q(rK)\< Ce(1-e)r {edT\\kquq2\\ + t-1/2)' , к E B, (38)

which immediately yields the result.

□

4.4. Stability Inequalities

We turn now to proving the stability inequalities (6)-(7). We start with (6). For t > 0 fixed, we assume that q E Н*(Жп) and put M = \q\nt(R")- By Parseval inequality, it holds r>0

1Ы\Ь(П) = \\qlW) = j \q(n)\2dk + J \q(n)\2dk (39)

|«|<r |re|>r

< J \q(n)\2dk + J \(1 + \к\2УШ\Чк < j \q(n)\2dk + M,

|«|<r |re|>r M<r

since J \(1 + \к\2)1 \с{(к)\2(1к = \Ы\я4(п)- From this and Lemma 5 then follows that

Rn

\\q\L2(q) < Crne2(1-e)r (edT \ \ Л qi, q2 \\ + t-1/2У + M E (00, 2t ),t E [to , +<x>). (40) Let us suppose that

\ \ Л qi, q2\\ E (0,Y0), (41)

where y0 = u(t0) and v(t) = t-1/2e-dT for t E (0, Notice that y0 E (0,1)

since t0 E [1, Moreover, и being a strictly decreasing function on [t0, there

exists a unique t* E (t0, +o) satisfying u(t*) = ||Aqi, q21|. By elemen^ computation

we find that t* = I 2

ln

q1 q2 |

— ln t*) /(2d), which entails t* < ln ||Aqi, q21|

since

(* E (1, +o) and d E [1, +oo). As a consequence the real number t* is greater than

ln ||Aqi,q2 ||

ln

ln ||Aqi, q21| ) /(2d), so we get that

t* >

ln ||Aqi,q2 ||

2d

(42)

upon recalling that lnx < x for all x E (0, +o). Further, taking t = t* in (40) we obtain for each r E (0, 2t*) that ||q||L2(n) < C228rne2rt- + M2/r2t < C'e(n+2)r(2dT*)-8 + M2/r2t, where C' = C238d8. This and (42) entail

~ -8 M2 &2(n) < C 'e(n+2)r ln 11A qi, q2 || + M, r E (0, 2t*).

(43)

The next step of the derivation involves finding r* E (0, 2t*) such that both terms ~ -8

C'e(n+2)r* ln ||Aqi,q21| and M2/r*t appearing in the right hand side of (43) are equal. This can be achieved upon assuming in addition to (41) that ||Aqi, q21| E (0,7i), where

m2 A i+2t-e

Yi = e V ("+2)cv ; jn such a way that we have

(n + 2) ln 11A qi, q2 ||

i+2t-8 M2 > -.

- C'

(44)

Indeed, we see from (42) that the function id : r M- (dr)2te(n+2)dr satisfies

id (2t*) > ln 11A qi, q21|

2t

>(n+2) \ln ,i

>92 11 \

>

2t

ln 11A qi, q2 || + (n + 2) ln 11A qi, q2 ||

i+2t

according to (44). Thus there exists r* E (0, 2t*) 8

and hence id(2T*) > (M2/C') ln |Aqi,q2 such that we have id(r*) = (M2/C') ln ||Aqi, q21| . This entails that (2t + n + 2)dr* —

2t ln(dr*) + (n + 2)dr* is greater that l^ (M2/C') ln ||Aqi, q21| the estimate 11q |L2(n) < 2M2/r*t, yields

>

which combined with

||9lU,<n> < a+ft2)'Mr)

2 \ i/8

ln ||Aqi ,q2 ||

)

t

(45)

Summing up, we obtain (6) with c = 2M((2t + n + 2)/9)t and c = (M2/C')i/8 provided l|Aqi, q21| is smaller that 7 = minj=0 ;i Yj-

On the other hand, in the particular case where ||Aqi, q21| — y,^ have || q | L2<n) < (M/ Y)|Aqi, q21 Putting this together with (45) we end up getting (6) upon possibly enlarging c.

2

8

Finally, we obtain (7) by arguing as in the derivation of (6) upon preliminarily substituting the estimate

IMlH-Hn) = J 1(1 + \K\2)-1\q(K)\2dK + J |(1 + \K\2)-1\q(K)\2dK <

|K|<r |re|>71

< J \q(K)\2dK + 1 J \q(K)\2dK < j \q(n)\2dK + M

|«|<r |^|>r |^|<r

for (39). This completes the proof of Theorem 2. 5. Application to Conductivity Problem

In this section we examine the stability issue in the inverse problem of determining the conductivity coefficient a appearing in the system

i -div(aVu) = f mQ, [ u = g on r,

from the partial DN map. The strategy is to link this inverse problem to the one studied in the first four sections of this paper and then apply Theorem 1 in order to derive a suitable

a

Assume that a e W+ ™(Q) = {c e W1 '^(Q; R); c(x) > c0 for some c0 > 0}. Then for any (f,g) e L2(Q) x H3/2(r), we know from the standard elliptic theory that (46) admits a unique solution Sa(f,g) e H2(Q), and that the linear operator

S& : (f,g) e L2(Q) x H3/2(r) ^ S&(f,g) e H2(Q)

is bounded. In the more general case where (f,g) eH* x H-1/2(r), we obtain by arguing in the exact same way as in Section 2 that there exists a unique u e L2(Q) obeying

- J udiv(aVv)dx = (f,v) - (g,adv v)i/2, v e H. (47)

n

u

denoted by Sta (f,g)-

Let us introduce the Hilbert space Hdiv(aV)(Q) = {u e L2(Q), div(aVu) e L2(Q)},

( \ 1/2

endowed with the norm ||u||ffdiv(ffV)(n) = (JM^n) + ||div(aVu) ^(n) J • % a sliSht modification of the proof of Lemma 1 (e.g. [6]), the trace map

ju = ajdju|r, u e D(Q), j = 0,1,

is extended to a linear continuous operator, still denoted by j from Hdiv(aV) (Q) into H-j-1/2(T). Thus, bearing in mind that St(0,g) e Ha(Q) for g e H-1/2(r), we see that the DN map

Aa : g e H-1/2(r) ^ tlSt(0,g) e H-3/2(r),

is a bounded operator.

Assume that a E W+ ~(Q) = W2 '~(Q) n W+ ~(Q). Taking into account that —div (aV(a-i/2v)) = ai/2 (—Av + a-i/2(Aai/2)v) , v EH, we get upon substituting (0,a-i/2v) for (f, v) in (47), that

/^FSTT9"^ = ^^v),v E H,

n

where qa = a-i/2Aai/2. As a consequence we have ai/2Sfa(0,g) = Siqa (0,ai/2g), and hence

al/2Sta(0, a-l/2g) = S^(0,g).

From this and the identity ti(ai/2w) = a~ ~i/2 tl w +1 a- ~i/2(dva)t0w, which is valid for every w E H2(Q), and generalizes to w E Hdiv<aV)(Q) by duality, we get that

AqCT = 2 a-i(dv a)I + a-i/2Aa a-i/2. (48)

Now pick ai,a2 E W^(Q) such that ai = a2 on r and dvai = dva2 on F. Thus, putting qj = qaj for j = 1, 2, we deduce from (48) that

(Aqi — Aq2)(g) = a-i/2(Aai — Aa2)(a-i/2g), g E H-i/2(r) n E'(F). (49)

Let us next introduce

A^ : g e H-i/2(r) n E'(F) m (ACTi — ACT2)(g)|c e H1/2(G). We notice from (49) that

Aqi,q2g = a-i/2Aai,V2 (a-1/2g),

gE H-i/2(r) n E'(F), provided ai = a2 on ^d dvai = dva2 on F n G. In this C>0

11A qi ,q2 ll < C ||A ^ ll, (50)

where || • || still denotes the norm of B(H-i/2(r) n E'(F), Hi/2(G)). Here we used the fact that the multiplier by ai-1/2 is an isomorphism of H±i/2(r).

Finally, taking into account that 0 = a^2 — a2/2 is solution to the system

{

(—A + qi)0 = a^/2(q2 — qi) in Q 0 = 0 on r,

i/2

we get ||0||L2(n) < C||q2 — qilln-i<n) upon taking f = a2 (q2 — qi) and g = 0 in (15). Thus, applying Theorem 1 and recalling (50), we obtain the:

Corollary 1. Let F and G be the same as in section 1, and let 5 > 0 md a0 > 0. Then for any aj E 5BW2,^<n), j = 1, 2, obeying aj — a0 and the condition

ai = a2 on r and dvai = dva2 on F n G, (51)

we may find a constant C > 0 independent of o1 and o2, such that we have:

(

Iki - Под < C ||Ла1>II + ln (7 ln ||ЛСТЬII

1

Remark 3. It is not clear how to weaken assumption (51) in Corollary 1. Indeed, to our knowledge, the best available result in the mathematical literature (this is a byproduct of [13, Theorem 2.2, p. 922 and Theorem 2.4, p. 923]) on the recovery of the conductivity at the boundary, claims for any non empty open subset r0 of r, and for all am, m = 1, 2, taken as in Corollary 1 and satisfying the condition supp (a1 — a2)|r C r0 instead of (51), that

ki - djО2 II(Г) < CI№lt,2 II

|1/(1+j)

j

0,1.

Here Л°1 ^

denotes the operator g e H-1/2(r) n E'(r0) ^ (Aai — Aa2)(g)|r0 e H 1/2(r0), ||A°1;1| is the norm of A°1;CT2 in B(H-1/2(r) n ET),H 1/2(r)) aid C> 0 is a constant

a1 a2

References

1. Lions J.-L., Magenes E. Non-Homogenuous Boundary Value Problems and Applications. I. Springer, Berlin, 1972.

2. Calderón A. On an Inverse Boundary Problem. Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Rio de Janeiro, 1980, pp. 65-73.

3. Sylvester J., Uhlmann G. A Global Uniqueness Theorem for an Inverse Boundary Value Problem. Ann. of Math., 1987, vol. 125, pp. 153-169. DOI: 10.2307/1971291

4. Alessandrini G. Stable Determination of Conductivity by Boundary Measurements. Appl. Anal, 1988, vol. 27, pp. 153-172. DOI: 10.1080/00036818808839730

5. Mandache N. Exponential Instability in an Inverse Problem for the Schrodinger Equation. Inverse Problems, 2001, vol. 17, pp. 1435-1444. DOI: 10.1088/0266-5611/17/5/313

6. Bukhgeim A.L., Uhlmann G. Recovering a Potential from Partial Cauchy Data. Commun. Part. Diff. Equat., 2002, vol. 27, pp. 653-658. DOI: 10.1081/PDE-120002868

7. Kenig C.E., Sjöstrand J., Uhlmann G. The Calderón Problem with Partial Data. Ann. of Math., 2007, vol. 165, pp. 567-791. DOI: 10.4007/annals.2007.165.567

8. Nachman A., Street B. Reconstruction in the Calderón Problem with Partial Data. Commun. Part. Diff. Equat., 2010, vol. 35, pp. 375-390. DOI: 10.1080/03605300903296322

9. Heck H., Wang J.-N. Stability Estimate for the Inverse Boundary Value Problem by Partial Cauchy Data. Inv. Probl, 2006, vol. 22, pp. 1787-1797. DOI: 10.1088/0266-5611/22/5/015

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13. Alessandrini G., Gaburro R. The Local Calderón Problem and the Determination at the Boundary of the Conductivity. Commun. Partial Differ. Equat., 2009, vol. 34, no. 8, pp. 918-936. DOI: 10.1080/03605300903017397

14. Dautray R., Lions J.-L. Mathematical Analysis and Numerical Methods for Science and Technology. II. Berlin, Springer-Verlag, 1988. DOI: 10.1007/978-3-642-61566-5

Received December 17, 2014

УДК 517.9 Б01: 10.14529/ттр150305

ДВОЙНАЯ ЛОГАРИФМИЧЕСКАЯ УСТОЙЧИВОСТЬ В ИДЕНТИФИКАЦИИ СКАЛЯРНОГО ПОТЕНЦИАЛА ПО ЧАСТИЧНОЙ ЭЛЛИПТИЧЕСКОЙ КАРТЕ ДИРИХЛЕ - НЕЙМАНА

М. Чулли, Я. Киан, Э. Соккорси

Исследуется вопрос устойчивости решения обратной задачи определения скалярного потенциала, возникающего в стационарном уравнении Шредингера в ограниченной области по частичной эллиптической карте Дирихле - Неймана. А именно, условия Дирихле ставятся на затененной части границы области и условия Неймана - на ее освещенной части. Установлена оценка устойчивости типа 1од1од для Ь2-нормы (соотв. И-1 нормы) для И*, при Ь > 0 и ограничены^ (соотв. Ь2) потенциалов.

Ключевые слова: обратная задача; устойчивость; уравнение Шредингера.

Мурад Чулли, профессор, Университет Лотарингии (г. Мец, Франция), [email protected].

Явар Киан, профессор, Университет Экс-Марсель, (г. Марсель, Франция), [email protected].

Эрик Соккорси, профессор, Университет Экс-Марсель, (г. Марсель, Франция), [email protected].

Поступила в редакцию 11 декабря 2014 г.