Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 323–341

SOME RESULTS ON THE POMPEIU PROBLEM

Peter Ebenfelt

Royal Institute of Technology, Department of Mathematics S-100 44 Stockholm, Sweden; ebenfelt@math.kth.se

Abstract. A domain Ω ⊂ R^{2} is said to have the Pompeiu property if f ≡ 0 is the only continuous function in R^{2} such that the integral of f over σ(Ω), for every rigid motion σ of R^{2} , vanishes. It has been conjectured that the disc is the only bounded simply connected domain, modulo sets of Lebesgue measure zero, in which the Pompeiu property fails. In this paper we obtain some results which support that conjecture. In the first part of the paper we show that the disc is the only quadrature domain in which the Pompeiu property fails. In the second part of the paper we prove a result claiming nonexistence, under certain conditions, of solutions to a family of overdetermined Cauchy problems. This result is used to obtain the Pompeiu property for a wide variety of domains, including “k th roots” of ellipses and domains which are mapped conformally onto the unit disc by a rational function other than a Möbius transformation.

1. Introduction

In this paper we consider the Pompeiu problem: characterize the bounded domains in R^{2} which have the Pompeiu property (see Definition 1.1 below). It was (almost?) conjectured in [GS1] that the disc is the only bounded simply connected domain, modulo sets of measure zero, without the Pompeiu property. Our main results, Theorem 2.1 (although this is not just a statement about simply connected domains; see the remark at the end of Section 3), Theorems 3.1, 3.2 (or perhaps Corollaries 3.1 and 3.2) and 3.3, support this conjecture. We refer the reader to [B], [BST] and [GS1] for a more extensive introduction to the problem and a discussion of previous results. Also, see [Gu] and [S] for a discussion of quadrature domains.

Definition 1.1. A domain Ω ∈ R^{2} has the Pompeiu property if f ≡ 0 is the only continuous function in R^{2} such that

(1.1)  ∫_{σ(Ω)} f(x, y) dx dy = 0  for every rigid motion σ of R^{2}.

Note that the Pompeiu property is invariant with respect to adding or taking away sets of Lebesgue measure zero.

1991 Mathematics Subject Classification: Primary 30E15; Secondary 35J25, 35N10.

By a theorem of L. Brown, B.M. Schreiber and B.A. Taylor (see [BST]) a bounded domain Ω has the Pompeiu property if and only if μ̂, the Fourier–Laplace transform of the area measure μ of Ω, does not vanish identically on

(1.2)  M_{α} = {(ζ_{1}, ζ_{2}) ∈ C^{2} : ζ_{1}^{2} + ζ_{2}^{2} = α}

for any α in C^{∗} = C \ {0}. Also, S.A. Williams showed (see [Wi1]) that if Ω is a bounded simply connected Lipschitz domain then it has the Pompeiu property if and only if there is no solution to the following overdetermined Cauchy problem

(1.3)  \{ \begin{matrix}Δu + αu = 1  in Ω \\ u(x, y) = |∇u(x, y)| = 0  on ∂Ω, \end{matrix}

where the Laplace operator Δ is (∂^{2}/∂x^{2}) + (∂^{2}/∂y^{2}), for any α ∈ C^{∗} (in fact, it suffices to consider α > 0, cf. [B]). As the referee so kindly remarked, in this formulation the problem essentially goes back to the original book on “The theory of sound” by Lord Rayleigh. It later became known as the Schiffer problem and in this context the conjecture mentioned above is known as Schiffer’s conjecture (see [GS1]). Another result of Williams is that any Lipschitz domain in which the Pompeiu property fails must have a nonsingular analytic boundary (see [Wi2]). Consequently, as regards the Pompeiu problem, assuming that the boundary of a domain is nonsingular and analytic is not excessive.

In Section 2 we use the formulation of Brown, Schreiber and Taylor to show that the disc is the only bounded quadrature domain without the Pompeiu property. In Section 3 we obtain, using an idea due to H.S. Shapiro, a result claiming nonexistence of solutions to a family of overdetermined Cauchy problems, which includes (1.3), under certain conditions (see Proposition 3.1). This idea was used successfully by G. Johnsson in [J] to prove that ellipsoids in R^{n} have the Pompeiu property (Definition 1.1 immediately extends to arbitrary dimensions). We also find that k th roots of ellipses, loosely speaking, and a family of domains, the conformal maps onto the unit disc of which have certain properties (see Theorem 3.2), satisfy these conditions and therefore, by the result of Williams, have the Pompeiu property. We conclude the paper by showing some examples of domains that satisfy the hypotheses of Theorem 3.2.

2. Quadrature domains

Let Ω be a bounded quadrature domain, i.e. Ω is a bounded domain such that there is a distribution ν with finite support in Ω and with the property that μ − ν, where μ is the area measure on Ω, annihilates the space of integrable holomorphic functions in Ω (see e.g. [Gu] or [S]). It is known that Γ, the boundary of Ω, is algebraic. Furthermore, there is a meromorphic function S in Ω, called the Schwarz function of Ω, which extends analytically across every nonsingular point of Γ and which satisfies

(2.1)  S(z) = z̄

on Γ. We assume, as we may since the Pompeiu property is invariant with respect to adding or subtracting sets of Lebesgue measure zero, that Γ has no isolated points. By Stokes’ theorem, we have

(2.2)

μ̂(ζ_{1}, ζ_{2}) = ∫_{Ω} e^{−i(ζ_{1}x+ζ_{2}y)} dμ(x, y)

= ∫_{Ω} exp ( − \frac{i}{2}((ζ_{1} − iζ_{2})z + (ζ_{1} + iζ_{2})z̄)) dμ(z)

= \frac{1}{2i(ζ_{1} + iζ_{2})} ∫_{Γ} exp ( − \frac{i}{2}((ζ_{1} − iζ_{2})z + (ζ_{1} + iζ_{2})z̄)) dz,

where z = x + iy. Pick α in C^{∗}, parameterize M_{α} by

(2.3)  \{ \begin{matrix}ζ_{1} − iζ_{2} = −ζ \\ ζ_{1} + iζ_{2} = −\frac{α}{ζ} \end{matrix}

and let f(ζ; α) be the restriction of 2i(ζ_{1} + iζ_{2})μ̂(ζ_{1}, ζ_{2}) to M_{α}, i.e.

(2.4)  f(ζ; α) = ∫_{Γ} exp (\frac{i}{2}(ζz + \frac{α}{ζ}z̄)) dz.

Clearly, f(·; α) is analytic in C^{∗}.

Thus, by the theorem of Brown, Schreiber and Taylor mentioned above, Ω fails to have the Pompeiu property if and only if there is an α ∈ C^{∗} such that f(·, α) is identically zero. Following [B] and [GS1] we wish to study the asymptotic properties of f(ζ; α) as ζ → ∞ to deduce one of our main results (Theorem 2.1 below).

Proposition 2.1. Suppose that S has a simple pole and nonconstant regular part at a point z_{1} and suppose that z_{1} is an extreme point of Δ, the convex hull of the set of poles of S. If Re(ie^{iθ}z) = c is a supporting line for Δ through z_{1} —we may assume that Re(ie^{iθ}z) < c for every other point of Δ—then, as r → ∞,

(2.5) \begin{matrix}f(re^{iθ}; α) = \frac{2πi\sqrt{a_{−1}α}}{re^{iθ}} exp (\frac{i}{2}(re^{iθ}z_{1} + \frac{a_{0}α}{re^{iθ}} ))· \\ · (J_{1}(\sqrt{a_{−1}α}) + (\frac{i}{re^{iθ}} )^{k+1} \frac{a_{k}α}{2}(\sqrt{a_{−1}α})^{k+1}J_{k+1}(\sqrt{a_{−1}α}) + O(\frac{1}{r^{k+2}} )), \end{matrix}

where the a_{j} ’s appearing are the coefficients in the Laurent series expansion of S at z_{1} , k is the least positive integer such that a_{k} ≠ 0 and J_{l} is the lth order Bessel function of the first kind.

Remark. By a complex number C = |C|e^{i arg C}, arg C ∈ [0, 2π), raised to a noninteger β we mean |C|^{β}e^{iβ arg C}.

The proof of Proposition 2.1 is based on the following

Lemma 2.1. If γ is a closed curve homologous (relative C^{∗}) to a circle around the origin then

(2.6)  \begin{aligned}_{∫γ} & exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−1}}{z} + a_{k}z^{k}))) dz \\ & = ∑_{j=0}^{∞} \frac{2πi^{jk+1}}{j!} (\frac{ia_{k}α}{2})^{j}(\sqrt{a_{−1}α})^{jk+1}J_{jk+1}(\sqrt{a_{−1}α})\frac{1}{ζ^{j(k+1)+1}} . \end{aligned}

Proof. Clearly, we have

(2.7)  \begin{aligned}_{∫γ} & exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−1}}{z} + a_{k}z^{k}))) dz \\ & = ∑_{j=0}^{∞} \frac{1}{j!}(\frac{ia_{k}α}{2ζ})^{j} ∫_{γ} z^{jk} exp (\frac{i}{2}(ζz + \frac{αa_{−1}}{ζz} )) dz. \end{aligned}

We may choose γ to be the circle \{|z| = |\sqrt{a_{−1}α} |/|ζ|\}. If we parameterize γ we obtain

(2.8)

∫γ exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−1}}{z} + a_{k}z^{k}))) dz = ∑_{j=0}^{∞} \frac{i}{j!}(\frac{ia_{k}α}{2ζ})^{j}(\frac{\sqrt{a_{−1}α}}{ζ})^{jk+1}×

× ∫_{−π}^{π} exp (i\sqrt{a_{−1}α} cos θ)( cos(jk + 1)θ + i sin(jk + 1)θ) dθ.

Since sin(jk + 1)θ is odd and cos(jk + 1)θ is even the lemma follows from an integral representation of J_{jk+1} (see [O], p. 360).

This lemma will be useful in the proof of the proposition.

Proof of Proposition 2.1. Using the Schwarz function of Ω we may write

(2.9)  f(ζ; α) = ∫_{Γ} exp (\frac{i}{2}(ζz + \frac{α}{ζ}S(z))) dz.

Since the integrand now is holomorphic in Ω\{z_{1}, …, z_{n}}, where {z_{1}, …, z_{n}} are the poles of S, we can write

(2.10)  f(ζ; α) = ∑_{m=1}^{n} f_{m}(ζ; α),

where

(2.11)  f_{m}(ζ; α) = ∫_{Γ_{m}} exp (\frac{i}{2}(ζz + \frac{α}{ζ}S(z))) dz

and Γ_{m} is some sufficiently small circle around z_{m}.

We first deal with f_{1}(ζ; α). Write the Schwarz function as

(2.12)  S(z) = \frac{a_{−1}}{z − z_{1}} + a_{0} + a_{k}(z − z_{1})^{k} + h(z),

where h is a holomorphic function near z_{1} which is O((z − z_{1})^{k+1}). We split (2.11) into two integrals and make the change of variables z′ = z − z_{1}

(2.13)

f_{1}(ζ; α) = ∫_{Γ_{1}} exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−1}}{z − z_{1}} + a_{0} + a_{k}(z − z_{1})^{k}))) dz

+ ∫_{Γ_{1}} exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−1}}{z − z_{1}} + a_{0} + a_{k}(z − z_{1})^{k})))(e^{\frac{iα}{2ζ} h(z)} − 1) dz

= exp (\frac{i}{2}(ζz_{1} + \frac{a_{0}α}{ζ}))( ∫Γ̃_{1} exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−1}}{z} + a_{k}z^{k})) dz

+ ∫Γ̃_{1} exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−1}}{z} + a_{k}z^{k})))(e^{\frac{iα}{2ζ} h(z+z_{1})} − 1) dz),

where Γ̃_{1} is some circle around the origin. As ζ tends to infinity we let the radius of the circle Γ̃_{1} tend to zero as |ζ|^{−1} . The imaginary part of ζz +α(a_{−1}/z +a_{k}z^{k})/ζ then remains uniformly bounded and, hence, the last integral in (2.13) will be O(1/ζ^{k+3}). It follows from Lemma 2.1 that, as ζ tends to infinity,

(2.14)  \begin{aligned}f_{1}(ζ; α) & = \frac{2πi\sqrt{a_{−1}α}}{ζ} exp (\frac{i}{2}(ζz_{1} + \frac{a_{0}α}{ζ}))(J_{1}(\sqrt{a_{−1}α}) \\ & + (\frac{i}{ζ} )^{k+1} \frac{a_{k}α}{2}(\sqrt{a_{−1}α})^{k+1}J_{k+1}(\sqrt{a_{−1}α}) + O(\frac{1}{ζ^{k+2}} )). \end{aligned}

Now, we look at f_{m}(ζ; α) for m > 1. Write the Schwarz function as

(2.15)  S(z) = ∑_{j=1}^{p} \frac{c_{−j}}{(z − z_{m})^{j}} + h(z),

where h is holomorphic near z_{m}. The exponent of the integrand in (2.11) can be written as

(2.16)  \frac{i}{2}ζ^{β}(ζ^{1−β}z + \frac{αc_{−p}}{(ζ^{1−β}(z − z_{m}))^{p}}) + \frac{iα}{2ζ} ( ∑_{j=1}^{p−1} \frac{c_{−j}}{(z − z_{m})^{j}} + h(z)),

if we let β = (p − 1)/(p + 1). If we, as above, make the change of variables z′ = z − z_{m} and integrate over a circle Γ̃_{m}, the radius of which tends to zero as |ζ|^{−(1−β)} when ζ tends to infinity, then we get

(2.17)  f_{m}(ζ; α) = e^{\frac{i}{2} ζz_{m}} ∫Γ̃_{m} exp (\frac{i}{2}ζ^{β}φ(z, ζ) + ψ(z, ζ)) dz,

where |φ(z, ζ)| and the real part of ψ(z, ζ) remain uniformly bounded on Γ̃_{m} (in fact, |ψ(z, ζ)| tends to zero) as ζ tends to infinity. We deduce that there is a constant b_{m} such that f_{m}(re^{iθ}; α), as r tends to infinity, is O(r^{−1+β} exp(\frac{1}{2} Re(ire^{iθ}z_{m}) +b_{m}r^{β})). Since β < 1 the proposition follows from (2.10), (2.14) and the fact that Re(ie^{iθ}z_{m}) < Re(ie^{iθ}z_{1}).

We also need an asymptotic expansion of f(ζ; α) in case there are no points z_{1} that satisfy the hypothesis of Proposition 2.1. To this end we use a result due to N. Garofalo and F. Segala. In proving their Theorem 3.1 of [GS1] the authors prove (the argument in Section 3 of [GS1] modulo some unfortunate misprints) that the saddle point method (see [De]) is stable under small perturbations in the following sense:

Lemma 2.2. Suppose that φ(z) is analytic in a punctured disc D_{R}^{∗} = {0 < |z| < R} and that ψ(z; σ), for (z; σ) in D_{R}^{∗} × (−δ, δ), is analytic in z and realanalytic in σ. If there is a curve γ in D_{R}^{∗}, which is homologous to a circle around the origin in D_{R}^{∗}, through a point z_{0} such that z_{0} is a nondegenerate critical point (Morse point) of φ, i.e. φ′(z_{0}) = 0 and φ′′(z_{0}) ≠ 0, and such that

(2.18)  Re (φ(z)) < Re (φ(z_{0}))

for all z ∈ γ \ {z_{0}} then, as σ → 0,

(2.19)  \begin{aligned}_{∫γ} & exp (σ^{−ϱ}(φ(z) + σψ(z; σ))) dz \\ & + σ^{ϱ/2} exp (σ^{−ϱ}(φ(z_{0}) + ν(σ))) (\frac{\sqrt{2π}}{i\sqrt{φ′′(z_{0})}} + O(σ^{τ})), \end{aligned}

where ν(σ) → 0 as σ → 0 and τ = min(\frac{1}{2}ϱ, 1), for all ϱ > 0.

From now on we will use the notation

(2.20)  φ(z; θ) = e^{iθ}(z + \frac{1}{z^{p}} ),

(2.21)  β = \frac{p − 1}{p + 1}

and

(2.22)  ε = \frac{1}{p + 1}.

To keep the notation reasonably simple we do not indicate the dependence on the integer p. This should cause no confusion. Using Lemma 2.2 we can prove

Proposition 2.2. Suppose that S has a pole of order p > 1 at a point z_{1} and suppose that z_{1} is an extreme point of Δ, the convex hull of the set of poles of S. If Re(ie^{iθ}z) = c is a supporting line for Δ through z_{1} —as in Proposition 2.1 we assume that Re(ie^{iθ}z) < c for every other point of Δ—and if θ′, defined by

(2.23)  θ′ = βθ + \frac{π}{2} + ε arg(αa_{−p}),

where a_{j} is the j th coefficient in the Laurent series expansion of S at z_{1} , is such that

(2.24)  θ′ ≠ \frac{(2k + 1)π}{p + 1}

for all k ∈ Z—such θ can always be found if z_{1} is an extreme point of Δ—then, as r → ∞,

(2.25) \begin{aligned}f(re^{iθ}; α) & = \frac{|αa_{−p}|^{ε/2}e^{iω}}{r^{1−β/2}} exp (\frac{i}{2}re^{iθ}z_{1} + r^{β} \frac{|αa_{−p}|^{ε}}{2} (φ(z_{0}, θ′) + ν(r)))× \\ & × (\frac{\sqrt{2π}}{i\sqrt{φ′′(z_{0}, θ′)}} + O(\frac{1}{r^{τ}} )), \end{aligned}

where ω = θ′ − θ − \frac{1}{2}π, ν(r) → 0 as r → ∞, τ = min(1 − β, \frac{1}{2}β) and z_{0} = p^{1/(p+1)}e^{−2πik/(p+1)} if

(2.26)  \frac{(2k − 1)π}{p + 1} < θ′ < \frac{(2k + 1)π}{p + 1} .

Proof. We proceed as in the proof of Proposition 2.1 and write f(ζ; α) as (2.10). First, we look at f_{1}(ζ; α). We write

(2.27)  S(z) = \frac{a_{−p}}{(z − z_{1})^{p}} + ∑_{j=−p+1}^{∞} a_{j}(z − z_{1})^{j}.

We make the change of variables z′ = z − z_{1} in (2.11) and obtain

(2.28)  f_{1}(ζ; α) = e^{iζz_{1}/2} ∫Γ̃_{1} exp (\frac{i}{2}(ζz + \frac{α}{ζ}(\frac{a_{−p}}{z^{p}} + ∑_{j=−p+1}^{∞} a_{j}z^{j}))) dz,

where Γ̃_{1} is some circle around the origin. Now, make the change of variables

(2.29)  z′ = \frac{ζ^{1−β}z}{(αa_{−p})^{ε}} .

We get

(2.30)  \begin{aligned}f_{1}(ζ; α) & = \frac{(αa_{−p})^{ε}}{ζ^{1−β}} e^{iζz_{1}/2} ∫_{γ} exp (\frac{i}{2}ζ^{β}(αa_{−p})^{ε}× \\ & × (φ(z; 0) + \frac{α^{1−ε}}{(a_{−p})^{ε}ζ^{1−β}} ψ̃(z, ζ^{1−β}))) dz, \end{aligned}

where γ is some circle around the origin and

(2.31)  ψ̃(z, ζ) = \frac{1}{ζ^{p−1}} ∑_{j=−p+1}^{∞} a_{j}(\frac{(αa_{−p})^{ε}z}{ζ} )^{j}.

Denote the integral in (2.30) by I. Setting ζ = re^{iθ} and

(2.32)  σ = \frac{2^{(1−β)/β}}{|αa_{−p}|^{ε(1−β)/β}r^{1−β}}

we get

(2.33)  I = ∫_{γ} exp(σ^{−β/(1−β)}(φ(z; θ′) + σψ(z, σ)) dz,

where θ′ is given by (2.23) and

(2.34)  ψ(z; σ) = Cψ̃(z, \frac{|αa_{−p}|^{ε(1−β)/β}e^{−i(1−β)θ}}{2^{(1−β)/β}} σ).

C is some suitable constant which is of no interest to us. The important thing is that ψ(z; σ) is analytic in z and real-analytic in σ in D_{R}^{∗} ×(−δ, δ) for any R > 0, provided δ is small enough. Suppose that we could find a curve γ, homologous to a circle around the origin relative C^{∗}, which passes through a nondegenerate critical point z_{0} of φ and which is such that

(2.35)  Re (φ(z)) < Re (φ(z_{0}))

for all z ∈ γ \ {z_{0}}. Lemma 2.2 would then give the asymptotic behavior of I as σ tends to zero. Using (2.30) and (2.32) we would obtain

(2.36) \begin{matrix}f_{1}(re^{iθ}; α) = \frac{|αa_{−p}|^{ε/2}e^{iω}}{r^{1−β/2}} exp (\frac{i}{2}re^{iθ}z_{1} + r^{β} \frac{|αa_{−p}|^{ε}}{2} (φ(z_{0}, θ′) + ν(r))) \\ × (\frac{\sqrt{2π}}{i\sqrt{φ′′(z_{0}, θ′)}} + O(\frac{1}{r^{τ}} )), \end{matrix}

where ω, ν and τ are as stated in the proposition above. Thus, we need

Lemma 2.3. The points

(2.37)  z′_{j} = p^{1/(p+1)}e^{2πij/(p+1)}

are nondegenerate critical points of the function φ(·; θ). Let γ be the circle {|z| = p^{1/(p+1)}} and suppose that

(2.38)  \frac{(2k − 1)π}{p + 1} < θ′ < \frac{(2k + 1)π}{p + 1} .

for some integer k. Then

(2.39)  Re (φ(z)) < Re (φ(z′_{−k}))

for all z ∈ γ \ {z′_{−k}}.

Proof. The statement about the points z′_{j} being nondegenerate critical points is easy to verify. To prove the second part of the lemma we set

(2.40)  g(t) = Re (φ(p^{1/(p+1)}e^{it}; θ))

and t_{0} = −2πk/(p + 1). The proof is finished once we show that g, restricted to any interval of length 2π that contains t_{0} as an interior point, attains its maximum at t_{0} and nowhere else. Note that

(2.41)  g(t) = p^{1/(p+1)}( cos(t + θ) + \frac{1}{p} cos(pt − θ)).

We find that there are two sets of critical points of g

(2.42)  \{ \begin{matrix}t_{m}^{1} = \frac{2mπ}{p + 1} \\ t_{m}^{2} = \frac{2θ + (2m + 1)π}{p − 1} , \end{matrix}

where m ∈ Z. The important thing to notice is that cos(t_{m}^{1} + θ) = cos(pt_{m}^{1} − θ) and cos(t_{m}^{2} + θ) = − cos(pt_{m}^{2} − θ). Hence, at these points we have

(2.43)  \{ \begin{matrix}g(t_{m}^{1}) = p^{1/(p+1)} cos(t_{m}^{1} + θ)(1 + \frac{1}{p}) \\ g(t_{m}^{2}) = p^{1/(p+1)} cos(t_{m}^{2} + θ)(1 − \frac{1}{p}). \end{matrix}

Now we see that g attains a local maximum at t_{0} and that the only points, inside any interval of length 2π containing t_{0} as an interior point, at which the value of g can be larger than or equal to g(t_{0}) are the points t_{m}^{2} . However, it is easy to check that if g(t_{m}^{2}) > 0 then g′′(t_{m}^{2}) > 0 so g must have a local minimum at such a point. Consequently, since g(t_{0}) obviously is larger than zero, if g would assume its maximum at a point t_{m}^{2} then this point would be a local minimum. This is clearly a contradiction and, hence, the proof is finished.

To conclude the proof of the proposition we note that Re(ie^{iθ}z) = c, by hypothesis, is a supporting line for Δ which passes through z_{1} , so the exact same argument that concluded the proof of Proposition 2.1 is applicable here.

Now, we are ready to prove one of the main results of this paper. If Δ, the convex hull of the set of poles of S, has an extreme point z_{1} which is a pole of order p > 1 then it follows from Proposition 2.2 that f(ζ; α) does not vanish identically for any α ∈ C^{∗}. This means, by the theorem of Brown, Schreiber and Taylor mentioned above, that Ω has the Pompeiu property. Suppose, on the other hand, that all extreme points of Δ are simple poles and let z_{1} be such a point. Unless z_{1} is the only point of Δ the regular part of S at z_{1} is not constant and, hence, we may apply Proposition 2.1. Since J_{1} and J_{k+1} , for a positive integer k, have no common zeros (see [Wa], p. 484) it is clear that f(ζ; α) does not vanish identically for any α ∈ C^{∗} and, consequently, Ω has the Pompeiu property also in this case. A well known fact of quadrature domains is that if the Schwarz function of Ω has a simple pole at some point z_{1} ∈ Ω and no other poles in Ω then Ω is a disc centered at z_{1} (see e.g. [S], p. 51). Also, since the Schwarz function of a disc with radius R centered at z_{1} is S(z) = R/(z − z_{1}) it follows immediately from Lemma 2.1 that a disc does not have the Pompeiu property (this is a well known result, see e.g. [BST]). Hence, we have the following

Theorem 2.1. The only bounded quadrature domains without the Pompeiu property are discs.

Remark. If R is a rational function, univalent in the unit disc D and without poles in \overline{D}, then R(D) is a bounded quadrature domain. Consequently, Theorem 2.1 contains the results in [GS1]. Also, there are quadrature domains of any conformal type (see [Gu])—in fact, any domain bounded by a finite number of analytic curves can be approximated, in some sense, by quadrature domains of the same conformal type—so Theorem 2.1 has something to say about multiply connected domains as well.

3. Overdetermined Cauchy problems

We now turn to the study of some overdetermined Cauchy problems. We identify R^{2} with the real plane in C^{2} , i.e. R^{2} = {(X, Y) ∈ C^{2} : Im(X) = Im(Y) = 0}. Following [KS] we make the following

Definition 3.1. If Ω is a domain in R^{2} the Vekua hull Ω̂ of Ω is defined by

(3.1)  Ω̂ = {(X, Y) ∈ C^{2} : X + iY, X̄ + iȲ ∈ Ω}.

Furthermore, the Vekua class V (Ω) of Ω is the class of real-analytic functions in Ω which extend holomorphically to Ω̂.

We let ℒ be a linear partial differential operator of the type

(3.2)  ℒ = Δ + a(x, y)\frac{∂}{∂x} + b(x, y)\frac{∂}{∂y} + c(x, y),

where Δ is the Laplace operator as in (1.3) and a, b and c are real-analytic functions in some domain ω. At this point it is convenient to make the linear change of coordinates

(3.3)  \{ \begin{matrix}z = X + iY \\ w = X − iY. \end{matrix}

In these coordinates a point (z_{0}, w_{0}) is in the Vekua hull of a domain Ω′ if and only if z_{0} and w̄_{0} are in Ω′. Also, R^{2} ≅ C can be identified with the (anti)- complex line {w = z̄}. Note that ℒ is well defined in a neighborhood of ω in C^{2} (acting on holomorphic functions). In terms of the coordinates (3.3) we have

(3.4)  ℒ = 4(\frac{∂^{2}}{∂z∂w} + d(z, w)\frac{∂}{∂z} + e(z, w)\frac{∂}{∂w} + g(z, w)),

where d, e and g are holomorphic in the corresponding neighborhood. Recall from the theory of holomorphic partial differential equations that a point (z_{0}, w_{0}) on a complex analytic curve {φ(z, w) = 0} is called characteristic with respect to ℒ if

(3.5)  \frac{∂φ}{∂z}(z_{0}, w_{0})\frac{∂φ}{∂w} (z_{0}, w_{0}) = 0.

Suppose that Ω is a bounded simply connected domain and suppose that there is a point z_{0} on the boundary of Ω and a neighborhood U_{0} of z_{0} such that Γ = ∂Ω ∩ U_{0} is a nonsingular analytic curve, i.e.

(3.6)  Γ = {z ∈ C : φ(z, z̄) = 0},

where φ is some real-analytic function in U_{0} . It is well known that Γ has a Schwarz function s (see e.g. [S]), i.e. a function, holomorphic in some neighborhood of Γ, that satisfies

(3.7)  s(z) = z̄

on Γ. We have

Proposition 3.1. Suppose that there is a point z_{1} ∈ C \ \overline{Ω} and a simple curve γ: [0, 1] ↦ (C \ \overline{Ω}) ∪ \{z_{0}\} connecting z_{0} to z_{1} such that s extends as a holomorphic function to a neighborhood of γ —by a slight abuse of notation we identify γ with its image—in which the following holds:

(1) s′(z_{1}) = 0 and s′ does not vanish at any other point on γ;

(2) s̄ maps γ \ {z_{0}} into Ω. If a, b, c in (3.2) and f are in the Vekua class of some neighborhood of Ω ∪ γ and f(z_{1}, s(z_{1})) ≠ 0 then the solution to the Cauchy problem

(3.8)  \{ \begin{matrix}ℒ u = f \\ u(x, y) = |∇u(x, y)| = 0 & on Γ \end{matrix}

does not extend real-analytically to Ω.

Remark. Proposition 3.1 cannot be used on quadrature domains, because the derivative of the Schwarz function of a quadrature domain does not vanish outside the domain. This can be seen from results in e.g. [E] or [Gu]. However, we know from the results in this paper that the overdetermined Cauchy problems of Williams fail to have solutions in every quadrature domain which is not a disc. It would be interesting to know if any of the overdetermined Cauchy problems considered in this section has a solution in a quadrature domain other than a disc. In order for this to be nontrivial we should ask that a, b, c extend as entire functions of two complex variables and that f ≡ 1. Because if E is any entire function of two complex variables such that Γ = {E(z, z̄) = 0} bounds a simply connected domain Ω and h is an entire function of one complex variable then u(z) = h(z)E(z, z̄)^{2} extends as an entire function of two complex variables into C^{2} with vanishing Cauchy data on the complexified variety Γ̂ = {E(z, w) = 0}. Consequently, the solution of (3.8) in this case, with f = Δu, always extends into Ω. It is easy to check, however, that this choice of f always vanishes at the characteristic points of Γ̂.

Proof. Let Ω_{1} be a simply connected neighborhood of γ \ {z_{1}} in which s is holomorphic and s′ does not vanish. Also, choose it so small that Ω′ = Ω ∪ Ω_{1} is simply connected, ∂Ω ∩ Ω_{1} ⊆ Γ and a, b, c and f are in the Vekua class of some neighborhood of Ω′ ∪ {z_{1}}. We make the change of coordinates (3.3) in C^{2} and think of C as the complex line {w = z̄}. Let 𝒞 and 𝒟 be the complex analytic curves defined by \{(z, s(z)) : z ∈ Ω_{1}\} and \{( \overline{s(z̄)}, z) : z̄ ∈ Ω_{1}\}. By (3.7) these fit together to form a complex analytic curve Γ̂, the intersection of which with C contains Γ. In view of hypothesis (2) Γ̂ is contained in Ω̂′. Clearly, Γ̂ is everywhere regular and nowhere characteristic w.r.t. ℒ . By the Cauchy– Kowalevskaya theorem there is a unique holomorphic solution u to the Cauchy problem

(3.9)  \{ \begin{matrix}ℒ u = f \\ u = |∇u| = 0 & on Γ̂ \end{matrix}

in some neighborhood of Γ̂. We claim that the restriction of u to C extends real-analytically to Ω_{1} . To see this we let R(z, w; z′, w′) be the Riemann function for the operator ℒ (see [Ga], [Ha] or [He]). In view of our hypotheses on the coefficients of ℒ the Riemann function R is holomorphic in Ω̂′ × Ω̂′ (see [He], p. 179). We introduce the function

(3.10)  h(z′, w′) = \frac{1}{4} ∫_{z^{∗}}^{z′} ∫_{w^{∗}}^{w′} f(z, w)R(z, w; z′, w′) dz dw,

where z^{∗} and w̄^{∗} are arbitrary points in Ω′. It is clear that h is holomorphic in Ω̂′ and that v = u − h solves the Cauchy problem (see [Ga], p. 141)

(3.11)  \{ \begin{matrix}ℒ v = 0 \\ v + h = |∇(v + h)| = 0 & on Γ̂ \end{matrix} .

To establish our claim it suffices to show that v extends real-analytically to Ω_{1} . Let (z′, w′) be a point close to Γ̂ and let p and q be the points of intersection between Γ̂ and the complex lines {z = z′} and {w = w′}, respectively. By Riemann’s formula (see [Ha], [He] or [K]) the solution v of (3.11) can be represented as follows

(3.12)  \begin{aligned}v(z′, w′) & = \frac{1}{2}(v(p)R(p; z′, w′) + v(q)R(q; z′, w′)) \\ & + ∫_{p}^{q} (P (z, w; z′, w′) dw − Q(z, w; z′, w′) dz), \end{aligned}

where (recall the functions d and e from (3.4))

(3.13) \{ \begin{matrix}P (z, w; z′, w′) & = \frac{1}{2}(R(z, w; z′, w′)\frac{∂v}{∂w} (z, w) − v(z, w)\frac{∂R}{∂w}(z, w; z′, w′) \\ & + d(z, w)v(z, w)R(z, w; z′, w′)) \\ Q(z, w; z′, w′) & = \frac{1}{2}(R(z, w; z′, w′)\frac{∂v}{∂z} (z, w) − v(z, w)\frac{∂R}{∂z}(z, w; z′, w′) \\ & − e(z, w)v(z, w)R(z, w; z′, w′)), \end{matrix}

and the integration in (3.12) is carried out along a curve from p to q on Γ̂ close to the point (z′, w′). Note that the points p and q are well defined and equal to (z′, s(z′)) and (\overline{s(w̄′)}, w′) whenever z′ and w̄′ are in Ω_{1} , i.e. in a neighborhood of Ω_{1} in C^{2} . Also note that the first homology group H_{1}(Γ̂, Z) of Γ̂ is trivial—this follows from the fact that Γ̂ = 𝒞 ∪ 𝒟, where 𝒞 and 𝒟 are homeomorphic to Ω_{1} and 𝒞 ∩ 𝒟 is connected, and the Mayer–Vietoris sequence (see [M], p. 186])—and hence, by Stokes’ theorem, integration of closed 1-forms (in particular holomorphic differentials) from p to q on Γ̂ makes sense, i.e. is independent of the curve connecting p and q. The facts that Γ̂ ⊂ Ω̂′ and v + h = |∇(v + h)| = 0 on Γ̂ in combination with the representation (3.12) now imply that v extends holomorphically to some neighborhood of Ω_{1} in C^{2} and, consequently, the claim above is established. This argument is also sketched in [K]. If we assume that s is univalent in Ω_{1} the claim follows from Theorem 4.1 of [KS].

If we assume, in order to get a contradiction, that u extends real-analytically into Ω then, by Corollary 3.5 of [KS], u extends holomorphically to Ω̂′. Now, since s is holomorphic in a neighborhood Ω_{2} of z_{1} , we may enlarge Γ̂ slightly by adding the complex analytic curve segment (z, s(z)) for z ∈ Ω_{2} . We denote this enlarged complex analytic curve by Γ̂′. Define the curve

(3.14)  γ̂(t) = (γ(t), s(γ(t))),  t ∈ [0, 1].

Clearly, γ̂ is a curve on Γ̂′ from (z_{0}, z̄_{0}) to (z_{1}, s(z_{1})). By hypothesis, both γ(t) and \overline{s(γ(t))} belong to Ω′ for t ∈ [0, 1). This means that γ̂(t) belongs to Ω̂′ for t ∈ [0, 1). Consequently, γ̂(1) = (z_{1}, s(z_{1})) is on the boundary of Ω̂′. Since s′(z_{1}) = 0 the point (z_{1}, s(z_{1})) ∈ Γ̂′ is characteristic w.r.t. ℒ . The bicharacteristic of ℒ tangent to Γ̂′ at (z_{1}, s(z_{1})), denoted β_{1} , is the complex line {w = s(z_{1})}. Unless (z_{1}, s(z_{1})) is an exceptional point (in the sense of Leray, see [L], [GKL] or [J]) Leray’s local theory regarding existence of solutions to characteristic Cauchy problems (again see [L], [GKL], or Theorem 2.3 of [J]) states that there is a unique function v, holomorphic in U \ β_{1} for some neighborhood U of the point, that satisfies ℒ v = f with vanishing Cauchy data on Γ̂′ \ {(z_{1}, s(z_{1}))}. Since f does not vanish at (z_{1}, s(z_{1})) it follows from a theorem of Shapiro (see [S], Section 9.2) that v must develop a singularity there. As a consequence of Hartogs’ theorem (Corollary 2.8 of [J]) the whole complex line β_{1} ∩ U must be singular—in two dimensions, as in this paper, this can also be seen using elementary methods (cf. [S], Section 9.3). Since γ̂ \ {(z_{1}, s(z_{1}))} is contained in Ω̂′ it follows that u and v must agree on Ω̂′ ∩ (U \ β_{1}). Also, Ω̂′ ∩ β_{1} ∩ U is not empty—the complex line β_{1} ∩ U is parameterized by (z, s(z_{1})) for z near z_{1} and since z_{1} is a boundary point of Ω′ it follows that there are points on β_{1} , arbitrarily close to (z_{1}, s(z_{1})), that are contained in Ω̂′. We have reached a contradiction to the fact that u is holomorphic in Ω̂′. Consequently, we just need to establish that (z_{1}, s(z_{1})) is not an exceptional point. By Lemma 2.2 of [J] it suffices to show that β_{1} is not contained in a characteristic subvariety of Γ̂′ near (z_{1}, s(z_{1})). But this is clear since any characteristic subvariety of Γ̂′ consists of isolated points—the characteristic points w.r.t. ℒ on Γ̂′ correspond to zeros of s′.

Remark. The author acknowledges helpful correspondence from D. Khavinson in connection with this proof.

Example 3.1. We introduce a family of algebraic curves on which we can use Proposition 3.1. Let P_{k}(z, w) be defined by

(3.15)  P_{k}(z, w) = Az^{2k} + 2Bz^{k}w^{k} + Aw^{2k} − 1,

where k is a positive integer and

(3.16)  \{ \begin{matrix}A = \frac{1}{4}(\frac{1}{a^{2}} − \frac{1}{b^{2}} ) \\ B = \frac{1}{4}(\frac{1}{a^{2}} + \frac{1}{b^{2}} ), \end{matrix}

for some positive numbers a and b (we may assume that a > b). Let Γ_{k} be the intersection between the zero locus of P_{k} and C.

The curve Γ_{1} is an ellipse with halfaxes a and b. For k > 1 the curve Γ_{k} is the inverse image under z ↦ z^{k} of the above mentioned ellipse Γ_{1} . The mapping properties of the algebraic function S_{k}(z), one branch of which is the Schwarz function s_{k} of Γ_{k}, defined by P_{k}(z, w) = 0 were investigated in [E]. If k = 1 the Schwarz function s_{1} is holomorphic outside a line segment connecting the foci F_{−} and F_{+} of Γ_{1} and s′_{1} vanishes at two points z_{−} and z_{+} located on the real axis, symmetrically about the imaginary axis, outside Ω_{1} = {the domain inside Γ_{1}}. Also, s̄_{1} maps the segment of the real axis connecting e.g. z_{−} and Γ_{1} to the segment of the real axis connecting F_{−} and Γ_{1} . Consequently, we may choose z_{1} to be z_{−}, γ to be the segment of the real axis connecting z_{1} to the point z_{0} on Γ_{1} and apply Proposition 3.1 to the ellipse.

For k > 1 the Schwarz function s_{k} of Γ_{k} satisfies

(3.17)  s_{k}(z) = (s_{1}(z^{k}))^{1/k}

for a suitable choice of branch of the k th root. Moreover, the points z_{−} and z_{+} mentioned above are located on a confocal ellipse Γ_{1}^{T} (see [E] for an exact location of Γ_{1}^{T} ) outside Γ_{1} and the only zeros of s_{1} are the points of intersection between this ellipse and the imaginary axis. We deduce from (3.17) that any choice of points and connecting curve among the inverse images of z_{1} , z_{0} and γ under z ↦ z^{k} will suffice to satisfy the hypotheses of Proposition 3.1. Let Γ_{k}^{T} be the inverse image of Γ_{1}^{T} under z ↦ z^{k}. We have proved the following

Theorem 3.1. Suppose that A and B > max(A, 0) are real numbers. Let Ω_{k} = {the domain inside Γ_{k}} and Ω_{k}^{T} = {the domain inside Γ_{k}^{T} }, where Γ_{k} and Γ_{k}^{T} are as defined above, for a positive integer k. If a, b, c in (3.2) and f are in the Vekua class of some neighborhood of \overline{Ω_{k}^{T}} and if f(z_{1}, w_{1}) ≠ 0, where (z_{1}, w_{1}) = ((z_{∗})^{1/k}, (F_{∗})^{1/k}) for some choice of branch of the k th root and ∗ ∈ {−, +}, then the overdetermined Cauchy problem

(3.18)  \{ \begin{matrix}ℒ u = f & in Ω_{k} \\ u = |∇u| = 0 & on Γ_{k} \end{matrix}

has no solution.

We could choose a ≡ b ≡ 0, c ≡ α for some α ∈ C^{∗} and f ≡ 1. By the result of Williams mentioned in the introduction we have the following

Corollary 3.1. The domain Ω_{k} has the Pompeiu property.

We can use Proposition 3.1 to prove a result in which the assumptions are made on the conformal map of the domain onto the unit disc D instead of on the Schwarz function of the boundary. Using this result we construct another class of domains in which the overdetermined Cauchy problems corresponding to (3.18) have no solution.

Again, suppose that Ω is a bounded simply connected domain. Let φ be a conformal map of Ω onto D. Let us also assume that Γ = ∂Ω is a nonsingular analytic curve so that φ extends holomorphically to some neighborhood Ω′ of \overline{Ω}.

Theorem 3.2. Suppose that φ takes in Ω′ \ \overline{Ω} no value in \overline{D}. If there is a point z_{1} in Ω′ such that φ′(z_{1}) = 0 then the overdetermined Cauchy problem

(3.19)  \{ \begin{matrix}ℒ u = f & in Ω \\ u = |∇u| = 0 & on Γ, \end{matrix}

where a, b, c and f are in the Vekua class of some neighborhood of Ω′ and f does not vanish in the Vekua hull of this neighborhood (in fact, it suffices to assume that f does not vanish at a certain point; cf. Proposition 3.1), has no solution.

Proof. We assert that the Schwarz function s of Γ extends holomorphically to Ω′ \V , where V is some closed set contained in Ω, maps Ω′ \\overline{Ω} into \{z : z̄ ∈ Ω\} and that s′(z_{1}) = 0. The theorem then follows from Proposition 3.1. To show the assertion we observe that the following identity

(3.20)  s(z) = \overline{φ^{−1}(1/\overline{φ(z)} )},

where φ^{−1} is the inverse of φ defined in some neighborhood U of \overline{D}, holds on Γ (both sides are equal to z̄). Also, since φ maps Ω′ \ \overline{Ω} to the outside of \overline{D} the right side of (3.20) is well defined and holomorphic in Ω′ \ V , where V is the image under φ^{−1} of the closed set inside D which is reflected (w ↦ 1/w̄) to the outside of U, and, hence, s is holomorpic in Ω′ \ V and the identity (3.20) holds there. Clearly, V is contained in Ω and s̄ maps Ω′ \ \overline{Ω} into Ω. We just need to check that s′(z_{1}) = 0, but this follows immediately from (3.20) by differentiating both sides and recalling that φ′(z_{1}) = 0.

As above this theorem has the corollary:

Corollary 3.2. If Ω satisfies the hypotheses of Theorem 3.2 then Ω has the Pompeiu property.

Next, we give some examples of domains that satisfy the hypotheses of Theorem 3.2.

Example 3.2. An immediate example is obtained by choosing a function f which is holomorphic in some neighborhood of the closed unit disc \overline{D}, univalent in D and which has a vanishing derivative at z = 1, e.g. f(z) = (z − 1)^{2} . Then, any domain Ω with \overline{Ω} ⊂ D for which f(Ω) is a disc satisfies the hypothesis of Theorem 3.2. Using f(z) = (z − 1)^{2} we deduce, from Corollary 3.2, that the inverse image (under f) of any disc D with \overline{D} contained in the conchoid f(D) has the Pompeiu property.

Example 3.3. If we specialize Theorem 2.1 to simply connected domains it can be reformulated as follows: Any domain which is the conformal image of the unit disc under a rational function, other than a Möbius transformation, has the Pompeiu property. In this example we show that if the rational function “goes the other way” (see Theorem 3.3) then Ω satisfies the hypotheses of Theorem 3.2 and, consequently, it has the Pompeiu property.

Let Ω be a bounded simply connected domain with nonsingular analytic boundary and suppose that f, the conformal map of Ω onto D, is a rational function, but not a Möbius transformation. Since the boundary of Ω is regular f is univalent in some neighborhood V of \overline{Ω}. Consider f as a holomorphic map of the Riemann sphere (or complex projective line) P onto itself and let Ω′ be the component of f^{−1}(D′), where D′ = P \ \overline{D}, that meets V . Since Ω′ is maximal each point in D′ has a neighborhood the inverse image (under f) of which is relatively compact in Ω′, i.e. the closure is a compact subset of Ω′. This means that f makes Ω′ into a complete covering surface of D′ (see [AS], p. 42). Since f is no Möbius transformation and f is univalent in V it follows that f^{−1}(D) consists of two disjoint open sets (not necessarily connected, though, since f^{−1}(D) may have several components) the closures of which are also disjoint. Consequently, the boundary of Ω′ is disconnected and, thus, Ω′ is not simply connected. The fact that D′ is simply connected implies (the monodromy theorem in [AS], p. 31, and the argument on p. 41 in [AS]) that the covering f must be ramified in at least two points. This means that there is at least one finite point z_{1} in Ω′ such that f′(z_{1}) = 0. We state this as a lemma:

Lemma 3.1. Let f be a rational function, other than a Möbius transformation, and suppose that f maps a domain V conformally onto a neighborhood of the unit disc D. Then there is a neighborhood U of \overline{V} such that f takes in U \\overline{V} no value in D and a point z_{1} in U \ \overline{V} such that f′(z_{1}) = 0.

This lemma shows that the following is true:

Theorem 3.3. Let Ω be a bounded simply connected domain with nonsingular analytic boundary and suppose that f, the conformal map of Ω onto D, is a rational function other than a Möbius transformation. Then Ω satisfies the hypotheses of Theorem 3.2 and, in particular, Ω has the Pompeiu property.

We remark that the arguments in this example carry through even if we just assume that f is meromorphic in some domain U ⊂ P with \overline{Ω} ⊂ U. In this case, however, we need to postulate that Ω′ is relatively compact in U.

4. Concluding remarks

At the time when the work in this paper was carried out the author was unaware of the paper [GS2] and, consequently, some results may overlap. The main result, concerning the Pompeiu problem, in that paper states that if a holomorphic parameterization of ∂Ω, e.g. the conformal map φ of D onto Ω extended across and restricted to ∂D (by assumption, Ω has a nonsingular analytic boundary), extends holomorphically to a neighborhood of ∂D with a nondegenerate critical point then, under certain additional hypotheses (we refer the reader to [GS2] for the exact formulation), Ω has the Pompeiu property. This result seems to complement our Theorem 3.2, although the approach taken in [GS2] is the same as in Section 2 of this paper, in an interesting way. Recall that one hypothesis on the domain in that theorem is that φ^{−1} extends holomorphically with a critical point outside Ω. However, note that Theorem 3.2 concerns a larger class of overdetermined Cauchy problems than the eigenvalue problems of Williams (a ≡ b ≡ 0, c ≡ α and f ≡ 1 in (3.19)). An interesting observation is that if φ(z) = e^{z} then neither result applies. Also, Garofalo and Segala claim to have strong evidence that Ω has the Pompeiu property if ∂Ω is parameterized by a rational function unless, of course, Ω is a disc. This should be compared to Theorem 2.1 in this paper which contains the slightly weaker statement that any simply connected domain which is the conformal image of the unit disc under a rational function, other than a Möbius transformation, has the Pompeiu property.

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Received 9 April 1992