Math. Ann. 259, 107-110 (1982) tthmatische Annalen �9 Springer-Verlag 1982

A Note on the Pompeiu Problem for Convex Domains*

Leon Brown I and Jean-Pierre Kahane 2

1 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA 2 Math6matique, Bfttiment 425, Universit6 de Paris-Sud, Centre d'Orsay, F-91405 Orsay C6dex, France

As the Pompeiu problem - named after the late Rumanian mathematician Dimitric Pompeiu - we shall denote the following question. Let S denote the group of all rigid motions of the plane. Given a bounded measurable set D in the xy-plane,

suppose that f is a continuous function satisfying ~ ~ f (x , y) dx dy = O, ~re S. Does ~(O)

this imply f_= 0? If it does, we say that D is a Pompeiu set. The purpose of this note is to give a geometric condition on a set D which implies that it is a Pompeiu set.

Early attempts at the solution settled the question for such regions as discs (No), parallelograms (Yes), and triangles (Yes).

In [2], Brown et al. have obtained, via the theory of distributions and mean periodic functions, a general theorem on spectral synthesis in C(R") which gives a solution to this problem in terms of the Fourier-Laplace transform of the Lebesque area measure on D. Using this result and asymptotic estimates of the growth of certain Fourier-Laplace transforms developed for this purpose, it was then shown, in particular, that any polygonal region, ellipse, or convex set with at least one "corner" is a Pompeiu set, while any rotationally-invariant set is not.

Studying the Pompeiu problem leads to another problem of a similar nature, which we shall refer to as the Morera problem because of its relationship to the classical theorem of Morera. Let F denote a rectifiable closed curve in the plane,

and let f be continuous in the xy-plane. Suppose ~ f(z)dz=O, aeS . Does this ~(r)

imply that f is an entire holomorphic function of z? If this is so, F will be called a Morera set. In [2], it is shown that if F is the boundary of D then the Pompeiu problem for D is equivalent to the Morera problem for F.

In this note we will restrict ourselves to convex regions in R 2. It has been conjectured by many people that if D is a bounded convex region and is not a Pompeiu set then D must be a disc.

The following proposition holds for more general regions. However we state it for the regions we are considering.

* The first author was supported in part by the National Science Foundation

0025-5831/82/0259/0107/$01.00

108 L. B r o w n a n d ].-P. K a h a n e

Lemma 1 (see [2, Theorem 4.1]). Let D be a bounded convex region. For 04:a~C, let m ~ = { z = ( z l , z2)6C z :z12+z~=0~}. I f # is Lebesque area measure on D and Z = [t-1(0) then the jbllowing assertions are equivalent:

(1) D is a Pompeiu set. (2) F is a Morera set. (3) M , df_Z for all c~4:0.

Lemma 1 leads to another interesting formulation of this problem.

Lemma 2 (see [4, Theorems 1 and 2] or [1, pp. 129 and 133]). D is not a Pompeiu set if and only i f there is a nonzero ~ in C and a function uECI(R 2) such that

(*) A u + a u = - 1 in D and u vanishes outside o f D.

This is equivalent to the existence o f a nonzero ~ in C and uE C1(R 2) such that

(**) A u + a u = O in D and u = c = c o n s t , ~nn = 0 on OD,

where n is the exterior normal to c~D.

Note that Berenstein in [1] proves the equivalence of the Pompeiu problem and the eigenvalue problem (**) with the assumption that D has a C 2 +~ boundary. However Williams in [5] proves that if D is not a Pompeiu set and the boundary of D is locally Lipschitz then D has a real analytic boundary. Thus we may assume we have a convex region with a real analytic boundary.

Evidence for the conjecture above comes from a theorem of Berenstein.

Proposition 1 (see [1, Proposition 2]). I f the eigenvalue problem (**) has nonzero solutions for infinitely many choices o f ~, then D is a disc.

We have further evidence for the conjecture in the following result. Let D be a convex set. For 0 < 0 < ~, let w(O) be the distance between the two

support lines for D making an angle 0 with the positive x-axis. Let m(D) = inf w(O) and m ( o ) = sup w(O).

0 < 0 < ~ 0__<0<r~

Theorem. Let D be a convex region. I f re(D) < �89 M(D), then D is a Pompeiu set.

Proof. As we stated before we may assume that the boundary of D is analytic. We need to show that for 0 4= a in C the Fourier transform of Lebesque area measure on D, 1i, does not vanish on M~. Using Green's theorem and standard arguments one can prove that �9 > 0 for any ~ for which a nontrivial solution of the eigenvalue problem (**) exists. For a proof of this see Theorem 1 in [5]. This is also pointed out by Berenstein (see [1, p. 129]).

Let rZ= ~ and if k is a positive integer we assume that m < k 2n _-< M, where r

m = re(D) and M = M(D). Since the width w(O) is a continuous function of the angle, after a suitable translation and rotation of D we have

k 2 n r

Ref~(r, O)= Re ~ S eir~ dx dy= ~ f ( x ) c o s r x dx , D 0

P o m p e i u P r o b l e m for C o n v e x D o m a i n s 109

where f(x) is the length of the section of D over x. In tegra t ing by parts we have

k2-

Re/i(r, 0 ) = - S f'(x) sinrx dx<O, 0 r

because the convexi ty of D implies f'(x) is decreasing.

We now have tha t for k = 1, 2 , . . . , if 2~k _> r_> 2rck > 2~k m - - 2m M

r > --, /~ is not identically zero on M~ where ~ = r 2. m

If 0 < r < --, for a suitable t rans la t ion and ro ta t ion of D we have m

or more succintly

m

Im[t(r, O)= ~" sinrxf(x) dx > 0 0

since s inrx >0 . This completes the proof. No te that the function f depends on the app rop r i a t e t rans la t ion and ro ta t ion of D.

Remark. The p roo f of the T h e o r e m can be used to show tha t if D does not have a cons tan t width then ( ,) or (**) can have nontr ivial solut ions for only a finite n u m b e r of eigenvalues. This is a special case of Berenstein 's result (Proposi t ion 1). We wish to thank C. Berenstein for point ing out to us that there are m a n y regions of cons tan t width with analyt ic boundar ies .

Proposi t ion 2 (Hurwi tz [-3]). Let 0 be any positive smooth periodic function such that

O(u)+o(u+rt)=C, 0 < u < 2 r t , (1)

and 2~z 2~z

O(u)sinudu= ~ O(u)cosudu=O. (2) 0 0

Then if R = (x(u),y(u)) is a solut ion of the differential equa t ion

dR - ~o(u) ll(u), where rl(u ) = (cosu , sinu)

du

and R(0)=(0,0) , then R has cons tan t width, is convex, and Q(u) is the radius of curva ture at R(u). The condi t ions on p are equivalent to

Q(u) = a o + ~ as . + 1 cos(2n + 1)u + b2. + 1 sin(2n + 1)u, n = l

where a o is so large so that Q > 0. This result is essentially conta ined in a pape r of Hurwi tz [3-1. See fo rmula (20) on p. 385 as well as p. 386 and fo rmula (23) on p. 386.

110 L. Brown and J.-P. Kahaue

References

1. Berenstein, C.A. : An inverse spectral theorem and its relation to the Pompeiu problem. J. D'analyse Math. 37, 128-144 (1980)

2. Brown, L., Schreiber, B., Taylor, B.A. : Spectral synthesis and the Pompeiu problem. Ann. Inst. Fourier, Grenoble 23, 125-154 (1973)

3. Hurwitz, M.A. : Quelques applications g6om6triques des s6ries de Fourier. Ann. Scuola Norm. Sup. Pisa 19, 357~408 (1902)

4. Williams, S.A.: A partial solution of the Pompeiu problem. Math. Ann. 223, 183 190 (1976) 5. Williams, S.A. : Analyticity of the boundary for Lipschitz domains without the Pompeiu property.

Indiana Univ. Math. J. 30, 357-369 (1981)

Received August 18, 1981