Rend. Sem. Mat. Univ. Poi. Torino. Fascicolo speciale 1989

P.D.E and Geometry (1988).

N. Garofalo

A NEW RESULT ON THE POMPEIU PROBLEM

§ 1. The problem of reconstructing a function from the knowledge of its integrai averages on a given family of sets belongs to the classical and fascinating subject of inverse problems. The coming of recent technologies, such as for instance computerized tomography, has raised a new wave of interest in the subject.

In this exposition I will discuss an inverse problem that has been proposed by the Rumanian mathematician D. Pompeiu [Pl-2] in 1929.

The Pompeiu problem consists in characterizing those bounded sets DC IR2 for which, if for two unspecifìed continuous functions / and g on IR2 we know

/ f(x)dx = \ g(x)dy for every rigid motion <r of JR? , Ja(D) Jo(D)

then must be / = g. This problem can be rephrased as follows: For which bounded sets DC IR2

is it true that f = 0 is the only continuous function on IR2 such that

(1) / f(x) = 0 for every rigid motion of IR2? . Jo(D)

In spite of its simple formulation the Pompeiu problem has resisted the efforts of many mathematicians and, although several partial results are known, there is not to present date a complete solution.

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In the first part of this exposition I will give a survey of what is already known and discuss the connection of the Pompeiu problem with questions of symmetry in partial differential equations, complex function theory and harmonic analysis. In the second part I will discuss a result on the asymptotics of the Fourier transform of a class of curves, which constitutes recent work in collaboration with F. Segala, see [GS]. This result yelds a contribution to the Pompeiu problem.

It is worth remarking at this point that the Pompeiu problem, like some of the results that I will discuss in the first part of the exposition, admits a multi-dimensional formulation. Here, however, I will only concern myself with the two-dimensional case.

To following result, due to Cramér and Wald [CW], and subsequently rediscovered by Newman [N] and Besicovitch [Be], shows that if in (1) we replace the bounded open set D with a straight line L, then the problem can be given a simple solution.

THEOREM 1. Let f e L*(IR2) and suppose the for a straight line L.

I f(x) di = 0 for every rigid motion <r of IR2. Jo{Lo)

Then, / = 0 a.e.

Proof. Let ( = (&, f2) / 0, and for a = y/(* + $ > 0 set

« = —x -f —y, v = - ^ + ^ , , (x,y)eM2. a a a a

The map (x,y) »-• <f>(x,y) = («,v) has Jacobian equa! to one, so that we have

/ ( 0 = / / e-i{Ux>y))f(x,y)dxdy =

= L€'iau(L n+~l(u>v))dv)du = °> since for every ti 6 IR the inner integrai in the r.h.s of the above equality equals

/ fdl for some a = <ru. Jo(Lo)

By the arbitrariness of f G 1R2\{0}, we conclude / = 0 a.e. Henceforth, I will say that a bounded set ZX7IR2 has the Pompeiu

property if / = 0 is the only continuous function on IR2 for which (1) holds.

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Pompeiu himself did not realize that rotation invariance causes failure of the Pompeiu property. He conjectured, in fact, and even incorrectly proved in [P2], that every disk has the Pompeiu property. He also showed that every square has the property, although he did so under the assumption that / —• 0 at oo. Subsequently, Christov [Ch] removed this assumption and also settled the questiòn for parallelograms.

The following example due to Chakalov [C] brought a deeper insight into the nature of the problem.

Chakalov's example. For x = (xi,a?2) G IR2 let BR = {x|a?J + ar| < R2} and f(x) = sin(axi), where a e IR2 is fixed, and TXo denotes the translation TXo(x) = XQ + x, then a calculation and [GR, formula (10) on p. 401) yield

f 2xR (2) / sin(axi)dx = sìn(a£o)J\(aR) ,

M0(B«) « where J\ is the Bessel function of the first kind and order one. Since BR is rotation invariant, if we choose a € IR such thataft is a zero of J\ we can make the integrai of f(x) — sin(a«i) over TXo(BR) zero.

Therefore, disks do not have the Pompeiu property. One may surmise that what makes the things go wrong for the disk is precisely tha fact that this set is not afTected by rotations. The following example shows, however, that, even imposing that the domain be simply-connected, symmetry must be interpreted modulo sets of measure zero. Removal of the latter does not affect, in fact, the presence or absence of the Pompeiu property.

© Fig. 1

The slit disk in fig. 1 does not have the Pompeiu property. In absence of simple connectedness the situation becomes more intricated.

After Chakalov's example it is naturai to accept that the annulus on the l.h.s. of fig 2 does not have the Pompeiu property. The asymmetric "annulus" on the r.h.s. does not either, however, and this seems a less obvious fact.

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An aspect which is far from being obvious is the intimate link existing between the Pompeiu problem and differential properties of functions. A well known instance of the striking connection between integrai and differential properties is Koebe's theorem. The latter states that is a continuous function ti in Ut2 coincides at every point with its average on ali sufficiently small balls centered at that point, then « is harmonic.

Fig. 2

A lewss known, but beautiful, result of Delsarté [D] states that the same conclusion is true if at every point x € Hn

(3) « ( « ) = / u(y)dy j = l , 2 ,

provided that the quotient — does not belong to the set of the points z e<C\{0} »*2

such that

(4) z ? r ( | + i ) z - 5 j ? ( 2 ) = i .

The notation / in (3) stands for "average", whereas in (4) r and Jn respectively denote the gamma function and the Bessel function of the first kind and order - .

Zalcman [Z] in 1971 first realized that the Pompeiu problem is connected with a problem in complex function theory that generalizes the content of a classical theorem of Morera. A collection of closed, rectifiable curves in C, {r}, is said to have the Morera property if it is true that, given / € C(C), the condition

(5) / /(*) àz = 0 for every rigid motion o of M2 and every T e {T} ,

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implies that / is entire. It can be proved, see [Z] and [BST], that: A set D with rectifìable

boundary dD has the Pompeiu property if {dD} has the Morera property.

Zalcman was the first one to use the Fourier transform and a theorem of L. Schwartz, see [E, p.319] to prove the following sophisticated version of Delsarte's result.

THEOREM 2. (see [Z]). Let f e L}oe(<C) and assume that for a.e. z €<C

(6) / /Wrfui = 0 i = l,2, Jcr.(z)

wliere Cr(z) = {tv € <C I \w - z\ = r}. If — is not a positive zero ofJl} condition Ti

(6) implies that f coincides a.e. with an entire function,

As a consequence of Theorem 2 we see that the family {T} = {Crj(z)\j = 1,2,2 e<C} has the Morera property provided that — avoids the exceptional

r2

set of the quotients of the positive elements of Jx 1(0). In the light of the above

cited equivalence with the Pompeiu property we see that one circle only does not suffice for Theorem 2 to hold. Here is an explicit counterexample.

Zalcman's example: let r > 0 be given and f l x o > 0 such that J\(ar) = 0. Define g(z) = g(x\ + ix2) = e,aa?a. By the Cauchy-Riemann equations g is not analytic at any z €<C. Yet, for any z — x\ + ixi we have as in [Z,p.244]

f g(w) dw = r f g{z + rw) dw = re*'0*' / eiraìmv} dw = ( 7 ) JCr(z) J\w\ = l J\w\=l

= 27rre,,<u?9.7-i(ra) = 0 ,

by the above choice òf a > 0. I remark explicitly that Zalcman's example, can be used to show that the disk does not have the Pompeiu property.

Using ideas closely related to those in [Z] in 1973 Brown, Schreiber and Taylor [BST] proved the following

THEOREM 3. Let DCYR? be a simpìy-connected bounded set. Then D has the Pompeiu property if for no a > 0 the complexified Fourier transform of the characteristic function of D,

(8) X D ( 0 = / e'*«^>dxt C = (Ci,C2)G€2, JD

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vanishes identically on a set ofthe type

(9) Ma = K = (0,C2)€C2|<12 + <1

2 = a } .

The proof of Theorem 3 in [BST] relies crucially on a result of spectral synthesis which the authors establish with the aid of the above cited theorem of Schwartz. Namely, that every rotation and translation invariant closed subspace of C°°(IR2) is generated by the polynomial exponential functions it contains.

Theorem 3 has been an important tool in giving examples of domains that do or do not have the Pompeiu property. For instance, if BR is as in Chakalov's example a computation yields

VSl +S2

from which it is clear, via Theorem 3, why disks do not have the Pompeiu property. On the other hand, let

Eah = {(*!, *2)G1R2 x\ t x\ ^ + ^ < 1 } , a , 6 > 0 .

Then Eab has the Pompeiu property. This was proved in [BST] as a consequence of the formula

(io) fa,»(Ci,6)-=2imè v/ 2 _ A2/2 ' , G.fcec,

from which it is clear that for no a > 0 can X£?.fc = 0 on Mtt, unless a = 6.

Using Theorem 3 Brown, Schreiber and Taylor also proved that every polygonal region, or, more in general, any convex set with a true corner, have the Pompeiu property. The case of domains with a smooth boundary was, however, left open.

Inspired by the ideas in [BST], Williams [Wil] in 1976 proved a remarkable connection betweeh the Pompeiu problem and a symmetry problem in pde's known an Schifler's conjecture, see [Y, Problem 80 on p. 688]: Let D c E " he a bounded, connected open set with C2 boundary OD. Does the existence of a nontrivial solution of

rA = -A in D, A>0 ,

(H) du u\ = const , —

du OD

= 0 , BD

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imply that D is a bali? Using Rellich identity it is not diflìcult to show that if a = ni , then

for a nontrivial solution of (11) must be

(12) / D " 2 r f * = ! T -

(12) implies, in particular, that a ^ 0. It is then easy to check that, via the transformation

ȓe-')-(11) is equivalent to

(13)

In [Wiì] it was proved that a domain D e IR2 fails to have the Pompeiu property iff there exists a nontrivial solution of (13) and, therefore, one of ( i i ) .

When D is a ball BR there exist infinitely many solutions of (11) of the type

«AiM = Cj^-^Jn^z (N/ÀJ|*|) , i € N , n-2

~2

where Cj are numbers suitably chosen in order to satisfy (12), and \j > 0 is chosen so that y/\jR is a zero of Jn.

In 1980 C. Berenstein [B] proved a two-dimensional converse to this result, thus characterizing the disk as the only domain for which there exist infinitely many nontrivial solutions of (11).

In [Wi2], following the approach of Caffarelli [Ca], Williams proved a free boundary result for (11) which, in the two-dimensional case, has the following implication.

THEOREM 4. (see [Wi2]). Let D e IR2 be a bóìinded, simpìy-connected domain with Lipschitz boundary OD. If OD is not reaì analytic, D has the Pompeiu property.

In the same paper it was shown that any convex domain D e IR2 whose boundary is not real analytic has the Pompeiu property.

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Concerning convex domains with a real analytic boundary the only existing result is due to Brown and Kahane [BK].

THEOREM 5. Let D cM2 be a bounded, convex open set whose boundary is real analytic. If

(14) min diam(D) < - max diam (D) ,

thèn D has the Pompeiu property.

In (14) min diam (D) and max diam (D) respectively denote the minimum and maximum diameter of D.

I dose the first part of this exposition remarking that the results in [BST] and [Wi2] leave open, as far as the Pompeiu problem is concerned, the case of domains with bad boundaries (less than Lipschitz) and that of domains with real analytic boundaries. For the latter the only existing partial result is Brown and Kahane's Theorem 5.

§ 2. In what follows I intend to discuss the above mentioned result in [GS]. I will start with a motivating example.

The conchoid of the circle is the curve whose equation in polar coordinates is given by

(15) p(s) = acoss-f 6, b > a > 0, « € [ - A , T T ] .

When a = b the conchoid has a cusp at the origin of the type x\ — -|z2|2^3, whereas for a < b the curve bonds region D C HI2. It is an easy exercise to show that

(16) min diam (D) > - max diam (D) ,

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see (14). The picture below represents the possible geometrie configurations

a = b a<b<2a 2a<b

b+a b+a

Fig. 3

Whether a domain bounded by a conchoid has the Pompeiu property or not cannot be decided by the existing results discussed above.

Theorem 4 does not apply, in fact, to the case a = 6, while (16) prevents to use Theorem 5 for the case 2a < 6. The intermediate case a < b < 2a remains uncovered as well.

e'* + c~" Using the formula cos = —-r I can rewrite (15) as follows

(17) p(s)e" = V " + be'' + § = p(e"), s e [-*, *} ,

where p(w) = -w2 + bw + £, w €C. In [GS] we prove the following result

THEOREM 6. Let DCJ&2 be a bounded, simply-connected domain whose boundary dD is a cìosed, simple curve parametrized by xi(s) = (a?,(s),a:2(s), 8 q [_*, * ] . Suppose that there exists a polynomial with complex coefficients

N

(18)

such that

(19)

p(w) = ^ajku;* , w GC, TV € IN Jfc=0

xi(s) + ix2(s) = p(ei$) , s e [-*, TT] .

Then D has the Pompeiu property, unless p(w) = aw + b, for some a€<C\{0}, ÒG<C.

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An immediate corollary of Theorem 6 and of (17) is that any domain bounded by a conchoid of the circle has the Pompeiu property.

The proof of Theorem 6 is based on asymptotic estimates of the Fourier transform of the characteristic function of D, when dD is given by (19), and ultimately relies on Brown, Schreiber and Taylor's Theorem 3 above. To give some flavor of what is involved I need to introduce some notation.

For a > 0 I recali that Ma is the algebiaic variety in C2 defined by (9). The points of Ma can be represented as follows

(20) C = r{ + itrj r > 0, t e IR ,

where £ = (cos#,sin#), r)=z (-sin#,costf) , #€[0 ,2TT] .

Setting ¢1 = rcostf — t<sint9f ¢2 = rsìntf + itcoaip ,

we see that C = (Ci, ¢2) € Afa iff

(21) C? + C} = r>-«» = a .

(21) implies that if ( as in (20) belongs to MQ, then

(22) , . . ^ / = ^ ( 1 + 0 ( ^ 5 ) ) •» l » - o o .

To obtain an asymptotic expansion of XD, see (8), under the condition (22) is, in general, a rather difficult problem as one has to deal with a Fourier intnegral with a complex phase, Let me observe that by the divergence theorem we have for C = (Ci,¢2) GC2

<w>(0 = Ki + «a)XD(0-

I exploit this observation and concentrate, from now on, my attention on XBD .

The approach in [GS] is based on Riemann's method of the steepest descent, [R]. Roughly speaking, two types of difficulty may arise: a) There are no criticai points of the phase of the Fourier integrai xaD, or, even if there are some, it is hard to locate them; b) In presence of criticai points, it is difficult to catch good paths of (steepest) descent into which deform the originai contour of integration.

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Denoting by Àf_a the set defined analogously to Ma in (9), and noting that C e Af_0 ifF t'C € Af0, we have

(23) ÌBD(K) = f exp(< *(*),< >)(x[(s) + ix'2(s)) ds ,

where x(s) = (xi(s)fx2(s) is a parametrization of dD.

It is quite obvious that in the case in which x = (a?i,aj2) is defined by (18),(19) the general difficulties a) and b) mentioned above, do not seem to become any easier.

The key idea in [GS] is to exploit the analyticity and periodicity of the integrands in (23) to suitably move the integration to a place where the sun wins over the darkness.

The efFect produced is illustrated by the following identity which we obtain after some reduction

(24) X9D(K) = i > - f exp[r-'+1(v>(*) + !—*(•; r)M«; r) ds ,

where / have set e =

AT+ 1 '

(25) ?(*) = e'< - 2SfiLe-w. ,

yj>(s\ r) is a polynomial in r~e whose coefficients are trigonometrie polynomials in s, and g(s;r) is, modulo small perturbations, of the same type of ^>(s;r). In writing (25) / am implicitly assuming that in(18) I have: N > 2, a0 = 0, ai = 1, apf G IR and as < 0. The general case can be brought within this framework by suitable considerations.

The good feature of (24) is that the phase of the Fourier integrai now appears as

y>+ (perturbation analytic in r~e),

where the term within parenthesis is uniformly small in s e [-ir, ir] as r -• +oo. The gain is that for <p as in (25) we are able to apply Riemann's method of the steepest descent.

The idea, then, is to take advantage of the fact that the perturbation moves the "good" criticai point of y?,

z0 = T In t

f|a//|7VT\ AT+1 /oMAT\

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by little, as r -+• +00 .' Since <p"{zo) £ 0, Dini's implicit function theorem allows to write

<p(z) - ip(zo) + r~<il>(z; r) = /(r"') + (z - z(r~<)) E(z; r"<) ,

where f(<r),z(<T) are analytic functions of a in a neighborhood of <r = 0, and E(z;<r) is an analytic function of (z;<r) in a neighborhood of (z,<r) = (z0>0) satisfying E(z0 ; 0) ^ 0

The analysis from this point on becomes quite technical. Here is the final product

THEOREM 7. (see [GS]). Let 1 < M < N be such that CIM is the first non zero coeffìcient in (18). Let

arg an + 7T

A __ ( cosa; sinuA w ~" \ —sino/ coswy '

*o = T In ( 4M )

Then, if D is as in Theorem 6, C = (r,ì<) € M_a, a > 0,ttr > 0, we have as r —• + 0 0

atw N-M NM

XdD(}\aM\KQ = j^T^rJr *<"+"> exp IrN+M fyfa) + ^(r)

where G and /1 are suitable functions such that

y/2MeiMz° G M ^ — 7 = = = - , /<W-> 0 asr-4+00. \A>"(*o)

Using Theorem 7 we immediately obtain Theorem 6 via Theorem 3. It seems naturai, at this point, to formulate the following

CONJECTURE 1. Let D e IR2 as in the first part of Theorem 6 and suppose that there exists an analytic function f :<C -*<C such that

(26) *i(s) + ica(«) = /(e'') , « € [-ir, ir] .

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Then D has the Pompeiu property, unless f(z) = az -f- ò, for some aG<D\{0},ÒG<C.

Here is an interesting example. Let

(27) *i(s) = eco8' cos(sin s) ,x2(s) = eco" sin(sin s) .

Then it can be shown that the curve parametrized by (27) bounds a convex region in the piane.

CONJECTURE 2. Saune conclusion as in conjecture 1 if (26) is replaced by

xi(s) + ix2(s) = f(e") + g(e~is) , * G [-^, TT] ,

wiÉh /,</ :(C -+C anaiytic.

This problem is much more difficult than conjecture 1, already at the polynomial level, i.e., when / and g are polynomials.

I remark that for an ellipse with semi-axis a and 6 we have

*i(«) + »*2W = P(e")+ «(«"") .

with p(z) =s z, q(z) = z.z GC,

Work in progress shows evidence, at this point, that a very delicate refìnement of the ideas in [GS] might lead to the solution of conjecture 1.

BIBLIOGRAPHY

[B] C.A. Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, J. Anal. Math. 37 (1980), 128-144.

[Be] A.S. Besicovitch, A uniqueness theorem and a problem on integration, J. London Math. Soc. 33, (2), (1958), 82-84.

[BK] L. Brown and J.P. Kahane, A note on the Pompeiu problem for convex domains, Math. Ann. 259 (1982), 107-110.

[BST] L. Brown, B.M. Schreiber and A.B. Taylor, Spectral synthesis and the Pompeiu problem, Ann. Int. Fourier 23 (1973), 125-154.

[Ca] L.A. CaflFarelli, The regularity offreee boundaries in higher dimensions, Acta Math. 139 (1977), 155-184.

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[CJ L. Chakalov, Sur un problème de D. Pompeiu, Annaire Univ. Sofia Fac. Phys. Math., Livre 1, 40 (1944), 1-44.

[Ch] C. Christov, Sur un problème de M.Pompeiu, Mathematica (Timisoara) 23 (1947-1948), 103-107.

[CW] H. Cramér and H. Wald, Some theorems on distribution functions, J. London Math. Soc. 11 (2), (1936), 290-294.

[D] J. Delsarte, Note sur une proprietà nouvelle des fonctions harmoniques. C.R. Acad. Paris 246 (1958), 1358-1360.

[E] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley -Interscience, N.Y., 1970.

[GS] N. Garofalo and F. Segala, A new result on the Pompeiu problem, Trans. Amer. Nath. Soc. to appear.

[GR] I.S. Gradshteyn and I.M. Ryzhyk, Tables of Integrals, Series and Products, Academic Press, N.Y., 1980.

[N] D.J. Newman, A function summableover the entirepiane, Prob. 4721. Amer. Math. Monthly 64 (1957), 750-751.

[PI] D. Pompeiu, Sur certains systèmes d'équations linéaires et sur une proprietà intégrale des fonctions de plusieurs variables, C.R. Acad. Sci. Paris 188 (1929), 1138-1139.

[P2] — , Sur une proprietà intégrale de fonctions de deux variables réelles, Bull. Sci. Acad. Royale Belgique 15, (5), (1929), 265-269.

[R] B. Riemann, Sullo svolgimento del quoziente di due serie ipergeometriche in frazione continua infinita, Complete works, Dover, N.Y., 1953.

[Wil] S.A. Williams, A partial solution ofthe Pompeiu problem, Math. Ann. 223 (1976), 183-190.

[Wi2] — , Analyticity ofthe boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J. 30 (1981), 357-369.

[Y] S.T. Yau, Seminar on Differential Geometry, Annals of Math. Studies, Princeton Univ. Press 102, 1982.

[Z] L. Zalcman, Analyticity and the Pompeiu problem, Arch. Rat. Mech. Anal. 47 (1972), 237-254.

N GAROFALO

Dipartimento di Matematica, Università di Bologna

Piazza di Porta San Donato, 5

40127 Bologna (Italy)