PERTURBATION OF DOMAINS IN THE POMPEIU PROBLEM

Toshiyuki KOBAYASHI* Department of Mathematical Sciences University of Tokyo Komaba, Meguro, Tokyo, 153, Japan toshi@tansei.cc.u-tokyo.ac.jp

Abstract. An old problem in integral geometry called the Pompeiu problem is closely related to the existence of a solution of the over-determined Neumann problem:

(N)_{λ} \{ \begin{matrix}Δu + λu = 0 & in Ω, \\ \frac{∂u}{∂ν} = 0, u ≡ constant & on ∂Ω. \end{matrix}

It is easy to see (N)_{λ} holds if Ω is a ball. In this paper we shall give a quantitative estimate of the following statement in terms of one parameter family of domains and some special values of Bessel functions: If Ω is sufficiently ‘close to’ a ball and if (N)_{λ} holds for a bounded λ, then Ω must be a ball.

§1. Introduction and Statements

The study of the relations between the geometry of a given domain and the spectrum of the Laplace operator is very old and has been an area of active research. One of these problems is a free boundary problem called Schiffer’s conjecture ([20], Problem 80), related to the Pompeiu problem which has originated in harmonic analysis ([15], [16]).

Suppose Ω is a bounded domain in ℝ^{n} whose boundary ∂Ω is C^{1} diffeomorphic to S^{n−1}. We associate the following three objects to Ω:

i) The null variety 𝒩(Ω) := \{ζ ∈ ℂ^{n} : Fχ_{Ω}(ζ) = 0\}. Here Fχ_{Ω}(ζ) = ∫_{Ω} e^{\sqrt{−1}(x_{1}ζ_{1}+...+x_{n}ζ_{n})} dx_{1} … dx_{n} is the Fourier transform of the characteristic function χ_{Ω}, which is a holomorphic function of the variables ζ = (ζ_{1}, …, ζ_{n}) ∈ ℂ^{n}.

Typeset by AMS-TEX

*Supported by the NSF grant DMS-9100383.

ii) An integral transform T_{Ω} : C(ℝ^{n}) ⟶ C(M(n)) defined by (T_{Ω}f)(g) = ∫_{Ω} f(gx)dx.

Here M(n) = O(n) ⋉ ℝ^{n} is the Euclidean motion group. iii) An overdetermined problem:

(N)_{λ} \{ \begin{matrix}Δu + λu = 0 & in Ω, \\ \frac{∂u}{∂ν} = 0, u ≡ constant & on ∂Ω. \end{matrix}

Here \frac{∂}{∂ν} stands for the outward normal vector field on ∂Ω. Then it is a well known result using an argument of spectral synthesis that these three

objects are related with one another:

Fact 1.1 ([7], [18]). In the above setting, the following three conditions on Ω are equivalent: 1) There exists r > 0 such that 𝒩(Ω) ⊃ S_{ℂ}(0 : r).

2) Ker T_{Ω} ≠ {0}.

3) There exists λ > 0 and a nontrivial solution u of (N)_{λ}.

Here, S_{ℂ}(a : r) := {ζ ∈ ℂ^{n} : Σ_{j=1}^{n}(ζ_{j} − a_{j})^{2} = r^{2}} for a = (a_{1}, …, a_{n}) ∈ ℝ^{n} and r ∈ ℝ. (1)

and (3) are related by λ = r^{2}.

A ball in ℝ^{n} satisfies the above three conditions. In fact, if Ω is the unit ball, then

(1.2)  Fχ_{Ω}(ζ) = f\frac{n}{2} (\sqrt{ζ_{1}^{2} + … + ζ_{n}^{2}}) ,  (ζ = (ζ_{1}, …, ζ_{n}) ∈ ℂ^{n}),

where f_{ν}(z) := (2π)^{\frac{n}{2}} \frac{J_{ν}(z)}{z^{ν}} is a holomorphic function of z ∈ ℂ (Lemma (2.2)(1)). Hence,

(1.3.1)  𝒩(Ω) ⊃ S_{ℂ}(0 : r),
(1.3.2)  Ker T_{Ω} ∋ f_{\frac{n}{2} −1} (r\sqrt{x_{1}^{2} + … + x_{n}^{2}}) ,
(1.3.3)  f_{\frac{n}{2} −1} (r\sqrt{x_{1}^{2} + … + x_{n}^{2}}) is a solution to (N)_{r^{2}},

whenever r is a positive zero of the Bessel function J\frac{n}{2} (z) (there exist countably many).

Conversely, it has been a long standing conjecture (the Pompeiu problem, Schiffer’s conjecture) that a ball would be the only domain satisfying one of (any of) (1) - (3) in Fact (1.1).

Berenstein [3], Aviles [2] (see also [4] Proposition 1), and Brown and Kahane [6] have made progress regarding (N)_{λ} which might be summarized as follows: For simplicity, suppose Ω is convex in ℝ^{2} (this assumption can be weakened in (1.4.1) and (1.4.2)). (1.4.1) If (N)_{λ} holds for infinitely many λ_{j}, then Ω is a ball. (1.4.2) If (N)_{λ} holds and λ ≤ ν_{7}, the seventh Neumann eigenvalue, then also Ω is a ball. (1.4.3) If (N)_{λ} holds and n = 2, then 2 min_{θ∈S^{1}} H_{Ω}(θ) > max_{θ∈S^{1}} H_{Ω}(θ), where H_{Ω}(θ) := sup_{x∈Ω}(x, θ)−

inf_{x∈Ω}(x, θ).

For further progress and some survey related to this conjecture, see [5], [9], and [14] and the references therein.

Loosely speaking, the result (1.4.3) of Brown and Kahane asserts that a long thin convex domain in ℝ^{2} (‘far from’ being a ball) never satisfies the conditions (1) - (3) in Fact (1.1). In contrast to this, our concern in this paper will be with the case where Ω is sufficiently ‘close to’ a ball. In order to define the ‘closeness’ and to give a quantitative estimate on how perturbations of a ball affect the properties in Fact (1.1), we need to take parallel translations and similarity transforms into account because they do not affect the properties in Fact (1.1). So, we first define unessential perturbations as follows.

Given a continuous function g : S^{n−1} ⟶ ℝ_{+} = {x ∈ ℝ : x > 0}, we define a star-shaped region Ω(g) with respect to the origin by

(1.5)  Ω(g) := {ρ · η ∈ ℝ^{n} : η ∈ S^{n−1}, 0 ≤ ρ < g(η)}.

Suppose g : [0, T] × S^{n−1} ⟶ ℝ_{+} is a C^{1}-map such that g(0, η) ≡ 1 (η ∈ S^{n−1}). Then Ω_{t} := Ω(g(t, ·)) forms a family of domains parametrized by t ∈ [0, T] with Ω_{0} = B(0 : 1), the unit ball.

We call {Ω_{t}} unessential if there exist a ∈ ℝ and b ∈ ℝ^{n} such that g_{t}(0, η) = a + (b, η) (g_{t} := \frac{∂g}{∂t}) (i.e. g_{t} ∈ E_{0} + E_{1} with the notation of Lemma (2.5) on spherical harmonics). This means that {Ω_{t}} is degenerate at t = 0 up to similarity transformations and parallel translations (see Proposition (5.7)).

Fix 0 < T, 0 < α ≤ 1 and 0 ≪ R. For a C^{1,α} function g : [0, T] × S^{n−1} ⟶ ℝ_{+} we shall define (see (2.6.4), (3.3.3)):

[g]_{R,α} := \frac{|g_{t}(0, ·)|_{R}}{∥g∥_{1+α}} (≥ 0),

by using spherical harmonics and Bessel functions with the following property (see (2.6.5)):

[g]_{R,α} = 0 ⇔ {Ω(g(t, ·))} is unessential.

We put B(a : r) = {x ∈ ℝ^{n} : ∑_{j=1}^{n}(x_{j} − a_{j})^{2} < r^{2}}, S(a : r) = {x ∈ ℝ^{n} : ∑_{j=1}^{n}(x_{j} − a_{j})^{2} = r^{2}} for

a = (a_{1}, …, a_{n}) ∈ ℝ^{n}, r > 0. Now we are ready to state our main result: Theorem. Let R ≫ 0. There exists a constant C(n, R) > 0 with the following property: Suppose 0 < T, 0 < α ≤ 1 and that Ω_{t} ≡ Ω(g(t, ·)) (0 ≤ t ≤ T) is a family of domains in ℝ^{n} given by a C^{1,α} map g : [0, T] × S^{n−1} ⟶ ℝ_{+} satisfying g(0, ·) ≡ 1 and |g_{t}(0, η)| ≤ 1 (η ∈ S^{n−1}). If there exist t_{0} ∈ ℝ, x ∈ ℝ^{n} and r > 0 such that

0 ≤ t_{0} < min (T, (C(n, R)[g]_{R,α})^{\frac{1}{α}}),
𝒩(Ω_{t_{0}})_{ℝ} ∩ B(0 : R) ⊃ S(x : r),

then t_{0} = 0 and so Ω_{t_{0}} is a ball. Remark. Actually it suffices to assume R ≥ j(\frac{n}{2}, 1) the first positive zero of J\frac{n}{2} (z), when the constant C(n, R) is given in (4.5.2). In Proposition (4.4) we drop the technical assumption |g_{t}(0, η)| ≤ 1 (η ∈ S^{n−1}). A small price to pay is the condition (4.4.1)(b) on t_{0} is somewhat complicated (see also Lemma (4.5)). Corollary 1. Let R ≫ 0 and retain the setting as in the theorem above. Assume that there exist λ_{0}, t_{0} ∈ ℝ and u ∈ C^{2}(Ω_{t_{0}}) ∩ C^{1}(\overline{Ω_{t_{0}}}) such that

0 < λ_{0} < R^{2},

0 ≤ t_{0} < min (T, (C(n, R)[g]_{R,α})^{\frac{1}{α}}),
u ≠ 0 is a solution of (N)_{λ_{0}}.

Then t_{0} = 0 and Ω_{t_{0}} is the unit ball.

In a slightly different formulation we may define a perturbation of domains {Ω_{t}} by giving the boundaries by means of a continuous map G : [0, T]×S^{n−1} ⟶ ℝ^{n} with G(0, η) = η (η ∈ S^{n−1}) such that

∂Ω_{t} = G(t, S^{n−1}) := Image (G(t, ·) : S^{n−1} ⟶ ℝ^{n}).

If G is of the C^{1} class, then we find a uniquely defined function g : [0, T′] × S^{n−1} ⟶ ℝ_{+}(0 < T′ ≤ T) such that Ω_{t} = Ω(g(t, ·)) (see Lemma (5.4)). In §5 we discuss a geometric aspect of unessential perturbations. In particular, {Ω_{t}} is unessential if and only if there exist 0 < T′ ≤ T and C^{1}-maps

a: [0, T′] → ℝ,  a(0) = 1,
b: [0, T′] → ℝ^{n},  b(0) = ^{t}(0, …, 0),
φ: [0, T′] × S^{n−1} → S^{n−1}, φ(0, ω) ≡ ω, φ(t, ·) is a C^{1}-diffeomorphism of S^{n−1},

such that G̃(t, ω) := a(t)G(t, φ(t, ω))+b(t) is degenerate at t = 0 in the sense that \frac{∂}{∂t} |_{t=0} G̃(t, ω) ≡ 0. Because of the ambiguity of the map G up to Map([0, T], Homeo(S^{n−1})), we present only a weaker version of Corollary 1 by means of a sequence t_{j} in this formulation, instead of giving an explicit estimate of a single t_{0} as in Corollary 1.

Corollary 2. Let 0 < T and 0 < α ≤ 1. Suppose {Ω_{t}} (0 ≤ t ≤ T) is not an unessential family of domains in ℝ^{n} given by a C^{1,α} map G : [0, T] × S^{n−1} ⟶ ℝ^{n} with G(0, η) ≡ η (η ∈ S^{n−1}). Assume that there exists a sequence t_{j} ∈ [0, T], λ_{j} ∈ ℝ (j ∈ ℕ) and u_{j} ∈ C^{2}(Ω_{t_{j}}) ∩ C^{1}(\overline{Ω_{t_{j}}}) such that

lim_{j→∞} t_{j} = 0,
sup_{j∈ℕ} |λ_{j}| < ∞,
u_{j} is a solution of (N)_{λ_{j}} in Ω_{t_{j}}.

Then there exists n_{0} ∈ ℕ such that Ω_{t_{j}} is a ball for any j ≥ n_{0}.

We finally mention that the results here were announced in [11] Theorem 3 in 1986 (see also [12] Chapter 3 and [14] Theorem 5.12) with the C^{2} assumption on the parameter. Recently C. Berenstein brought to our attention a preprint of Agranovsky [1]. There Agranovsky gives a similar result to Corollary 2 assuming that the dimension n = 2 and assuming the existence of a solution to (N)_{λ_{t}} for all t with the condition that both the boundary ∂Ω_{t} and the eigenvalues λ_{t} depend analytically on the parameter t. His approach using Riemann’s mapping theorem for ℂ ≃ ℝ^{2} is quite different from ours.

§2 Preliminary results on Bessel functions and spherical harmonics

In this section we give a short survey on Bessel functions and spherical harmonics on S^{n−1} that will be used in later sections.

For ν ∈ ℂ and z ∈ ℂ \ (−∞, 0], the ν-th Bessel function is defined by

J_{ν}(z) : = (\frac{z}{2})^{ν} ∑_{n=0}^{∞} \frac{(−1)^{n}(\frac{z}{2})^{2n}}{n!Γ(ν + n + 1)}

= \frac{(\frac{z}{2})^{ν}}{\sqrt{π} Γ(ν + \frac{1}{2})} ∫_{−1}^{1} e^{\sqrt{−1}zt}(1 − t^{2})^{ν− \frac{1}{2}} dt. (2.1)

If we put θ = z \frac{d}{dz}, then both J_{ν}(z) and J_{−ν}(z) are the solutions of the ordinary differential equation (θ^{2} + (z^{2} − ν^{2}))f(z) = 0.

Here is a collection of some basic properties of the Bessel function:

Lemma 2.2.

1) \frac{J_{ν}(z)}{z^{ν}} is an entire function of z ∈ ℂ with \frac{J_{ν}(z)}{z^{ν}}|_{z=0} = \frac{1}{2^{ν}Γ(ν+1)}.

2) (see [8], (7.5.38), (7.15.42)) Let z ∈ ℂ. We have

∑_{n=−∞}^{∞} J_{n}(z)^{2} = 1. (2.2.2)(a)

∑_{n=0}^{∞}(2n + 1)J_{n+ \frac{1}{2}} (z)^{2} = \frac{2z}{π} . (2.2.2)(b)

3) For real ν > −1, J_{ν}(z) has only real zeros (Lommel, see [17], pp.482-483). Let denote by j(ν, 1) the first positive zero of J_{ν}(z)(ν > 0). Then j(ν, 1) > ν ([17], p.485).

4) (Siegel, see [17], p.484) J_{ν}(z) and J_{ν+m}(z)(m ∈ ℕ_{+}) have no common zeros other than zero (Bourget’s conjecture).

The first and the last statements in Lemma (2.2) imply the following lemma, in which the constants ε(n, R) and δ(n, R) will be used in a quantitative estimate of our Theorem. Lemma 2.3. Let j(\frac{n}{2}, k) (k ∈ ℕ_{+}) be the k-th positive zero of the Bessel function J\frac{n}{2} (x). We fix R ≥ j(\frac{n}{2}, 1) and define k ∈ ℕ_{+} by

0 = j(\frac{n}{2}, 0) < j(\frac{n}{2}, 1) < … < j(\frac{n}{2}, k_{R}) ≤ R < j(\frac{n}{2}, k_{R} + 1).

We define an entire function of z ∈ ℂ by

(2.3.1)  f(z) := (2π)^{\frac{n}{2}} \frac{J\frac{n}{2} (z)}{z^{\frac{n}{2}}}.

Put

W (R) := {z ∈ ℂ : 0 ≤ Re z ≤ R, | Im z| ≤ 1}.

Then there exist ε ≡ ε(n, R) > 0 and δ ≡ δ(n, R) > 0 such that the following three conditions hold.

(2.3.2)  δ(n, R) < \frac{1}{2} min_{1≤k≤n} (j(\frac{n}{2}, k) − j(\frac{n}{2}, k − 1)) .

(2.3.3) ζ ∈ W (R) ∩ B(j(\frac{n}{2}, k) : δ) for some k ∈ \{1, …, k_{R}\}, then

|f(z)| ≥ ε(n, R) |z − j(\frac{n}{2}, k)| .

(2.3.4) If z ∈ W (R) \ ∪_{k=1}^{k_{R}} B(j(\frac{n}{2}, k) : δ), then

|f(z)| ≥ ε(n, R)δ(n, R).

Remark 2.4. We might take δ(n, R) = 1, however, we do not pursue this kind of numerical estimates of the Bessel function in this paper. On the other hand, we can and do assume

(2.4.1)  ε(n, R) ≤ δ(n, R) ≤ 1,
(2.4.2)  ε(n, R) ≤ ω_{n−1}.

Here we define

(2.4.3)  ω_{n−1} := Vol(S^{n−1}) = \frac{2π^{\frac{n}{2}}}{Γ(\frac{n}{2})}

Next, we review spherical harmonics. We induce a Riemannian metric g on S^{n−1} from the standard metric on ℝ^{n}. Denote by Δ_{S^{n−1}} the Laplace-Beltrami operator on S^{n−1} with respect to g. Then the Laplace operator Δ ≡ Δ_{ℝ^{n}} on ℝ^{n} is represented as Δ = \frac{∂^{2}}{∂r^{2}} + \frac{n−1}{r} \frac{∂}{∂r} + \frac{1}{r^{2}} Δ_{S^{n−1}} in the spherical coordinate x = rω (r > 0, ω ∈ S^{n−1}). We write L^{2}(S^{n−1}) for the Hilbert space of square integrable functions with respect to the measure given by the metric g. For each non-negative integer k ∈ ℕ we define an eigenspace

E_{k} := {f ∈ C^{∞}(S^{n−1}) : Δ_{S^{n−1}}f = −k(k + n − 2)f} ⊂ L^{2}(S^{n−1}).

The orthogonal group O(n) naturally acts on the unit sphere S^{n−1}. This action is transitive and we have O(n)/O(n − 1) ≃ S^{n−1}. Since this action is isometric, Δ_{S^{n−1}} belongs to the algebra of O(n)-invariant differential operators and so E_{k} is an O(n)-submodule of L^{2}(S^{n−1}), on which O(n) acts unitarily by f(x) ↦ f(g^{−1}x) for f ∈ L^{2}(S^{n−1}), x ∈ S^{n−1} and g ∈ O(n).

In fact, it turns out that each E_{k} is an irreducible O(n)-module and the eigenvalue of Δ_{S^{n−1}} is given as an evaluation of the Casimir operator. Here are classical results on spherical harmonics: Lemma 2.5 (see [10], Introduction, Theorem 3.1 and Lemma 3.6).

1) L^{2}(S^{n−1}) = ∑_{k=0}^{∞} ⊕E_{k}, (orthogonal Hilbert space decomposition).

2) E_{k} is irreducible as an O(n)-module.

3) E_{k} is the ℂ-span of {(a_{1}x_{1} + ⋯ + a_{n}x_{n})^{k} : a_{j} ∈ ℂ, ∑_{j=1}^{n} a_{j}^{2} = 0, (x_{1}, …, x_{n}) ∈ S^{n−1}} . In

particular,

(2.5.1)  E_{0} = ℂ · 1,
(2.5.2)  E_{1} = ℂx_{1} ⊕ … ⊕ ℂx_{n}.

4) (see also [8], p. 247,(11.4.22)) For h ∈ L^{2}(S^{n−1}), r ∈ ℝ and ω ∈ S^{n−1} we put

(2.5.3)  ĥ(r, ω) := ∫_{S^{n−1}} e^{\sqrt{−1}r(η,ω)}h(η)dη.

If h ∈ E_{k}(k ∈ ℕ), then we have for ω ∈ S^{n−1} and r ∈ ℂ

(2.5.4)  ĥ(r, ω) = (2π)^{\frac{n}{2}} \sqrt{−1}^{k}h(ω)\frac{J_{k+ \frac{n}{2} −1}(r)}{r^{\frac{n}{2} −1}}.

We introduce a family of seminorms | · |′_{r} on L^{2}(S^{n−1}) parametrized by r > 0 by

(2.6.1)  |h|′_{r} := \{∑_{k=1}^{∞} ∥h_{k}∥_{L^{2}(S^{n−1})}^{2}J_{k+ \frac{n}{2} −1}(r)^{2}\}^{\frac{1}{2}}

for h = ∑_{k=0}^{∞} h_{k} ∈ L^{2}(S^{n−1}) = ∑_{k=0}^{∞} ⊕E_{k}. If h ∈ ∑_{k=1}^{∞} ⊕E_{k} (i.e. h_{0} = 0), then it follows from (2.5.4) that

(2.6.2)  ∥ĥ(r, ω)∥_{L^{2}(S^{n−1})} = (2π)^{\frac{n}{2}} r^{1− \frac{n}{2}} |h|′_{r}

Since ∑_{k=0}^{∞} J_{k+ \frac{n}{2} −1}(r)^{2} < max(1, r) (see Lemma (2.2)(2)) if ℕ ∋ n ≥ 2, we have

(2.6.3)  |h|′_{r} ≤ ∥h∥_{L^{2}(S^{n−1})} max(1, r^{\frac{1}{2}} ).

Let j(ν, k)(k ∈ ℕ_{+}) be the positive zeros of J_{ν}(z) arranged in ascending order. For R ≥ j(\frac{n}{2}, 1) let k_{R} ∈ ℕ_{+} be the integer such that

0 < j (\frac{n}{2}, 1) < j (\frac{n}{2}, 2) < … < j (\frac{n}{2}, k_{R}) ≤ R < j (\frac{n}{2}, k_{R} + 1) .

For h ∈ L^{2}(S^{n−1}), we define

(2.6.4)  |h|_{R} := min_{1≤k≤k_{R}} |h|′_{j( \frac{n}{2} ,k)}.

It follows from Lemma (2.2)(4) that

(2.6.5)  |h|_{R} = 0 ⇔ h ∈ E_{0} + E_{1}.

§3. Some estimates of error terms.

In this section we give the first and second terms of Fχ_{Ω_{t}}(ζ · ω), the Fourier transform of a family of domains {Ω_{t}} in ℝ^{n}, and also give an estimate of the error term.

We define a function F : ℂ × ℂ → ℂ by

(3.1.1)  F (s, u) := \frac{(−1)^{n}(n − 1)!}{(\sqrt{−1}u)^{n−1}} \{1 − e^{\sqrt{−1}su} ∑_{k=0}^{n−1} \frac{(−\sqrt{−1}su)^{k}}{k!} \} .

By a simple calculation we have

(3.1.2)  \frac{∂F}{∂s} (s, u) = s^{n−1}e^{\sqrt{−1}su},

and thus

(3.1.3)  ∫_{0}^{s} e^{\sqrt{−1}ρu}ρ^{n−1}dρ = F (s, u).

By means of this function F we have a formula of Fχ_{Ω(g)}(ζ):

Lemma 3.2. Suppose g : S^{n−1} → ℝ_{+} is a continuous function and that Ω(g) is a star-shaped region with respect to the origin as in (1.5). Then we have a formula

Fχ_{Ω(g)}(ζ) = ∫_{S^{n−1}} F (g(η), (η, ζ))dη, ζ ∈ ℂ^{n}.

Proof. From definition we have

Fχ_{Ω(g)}(ζ) = ∫_{Ω(g)} e^{\sqrt{−1}(x,ζ)}dx
= ∫_{S^{n−1}} ∫_{0}^{g(η)} e^{\sqrt{−1}r(η,ζ)}r^{n−1}drdη
= ∫_{S^{n−1}} F (g(η), (η, ζ))dη.

□ We fix 0 < T and 0 < α ≤ 1. For a C^{0,α}-map f : [0, T] → ℝ, we put

∥f∥_{C^{0,α}([0,T])} := sup_{0≤t≤T} |f(t)| + sup_{0≤s<t≤T} \frac{|f(t) − f(s)|}{|t − s|^{α}}(< ∞).

Next, suppose g : [0, T] × S^{n−1} → ℝ is a C^{1,α}-map. We put

(3.3.1)  ∥g∥_{∞} : = sup_{\begin{matrix}0≤t≤T \\ η∈S^{n−1} \end{matrix}} |g(t, η)|,
(3.3.2)  ∥g∥_{0+α} : = sup_{η∈S^{n−1}} ∥g(·, η)∥_{C^{0,α}([0,T])},
(3.3.3)  ∥g∥_{1+α} : = ∥g∥_{∞} + ∥g_{t}∥_{0+α},

where g_{t}(t, η) := \frac{∂}{∂t}g(t, η). Lemma 3.4. Let 0 < T and 0 < α ≤ 1. Suppose g : [0, T] × S^{n−1} → ℝ_{+} is a C^{1,α} map such that

(3.4.1)  g(0, η) ≡ 1 for η ∈ S^{n−1}.

We write Ω_{t} = Ω(g(t, ·)) (see (1.5)). In particular, Ω_{0} is the unit ball. We put

(3.4.2)  Y (g : r, T, α) := \frac{1}{1 + α} sup_{\begin{matrix}η∈S^{n−1} \\ ω∈S^{n−1} \end{matrix}} ∥g(t, η)^{n−1}e^{\sqrt{−1}rg(t,η)(η,ω)}g_{t}(t, η)∥_{C^{0,α}([0,T])}.

If ζ ∈ ℂ, r ≥ 0 and ω ∈ S^{n−1}, then we have (see (2.4.3) for the definition of ω_{n−1}) |Fχ_{Ω_{t}}(ζ · ω) − Fχ_{Ω_{0}}(ζ · ω)| ≤ t∥g∥_{∞}^{n−1}∥g_{t}∥_{∞}ω_{n−1}e^{| Im ζ|∥g∥_{∞}} (3.4.3)

|Fχ_{Ω_{t}}(r · ω) − Fχ_{Ω_{0}}(r · ω) − t \frac{∂}{∂t}|_{t=0} Fχ_{Ω_{t}}(r · ω)| (3.4.4)

≤ t^{1+α}ω_{n−1}Y (g : r, T, α).

Proof. Let ζ ∈ ℂ and ω ∈ S^{n−1}. By Lemma (3.2) we have

Fχ_{Ω_{t}}(ζ · ω) = ∫_{S^{n−1}} F (g(t, η), ζ(η, ω))dη.

Then, from (3.1.2) we have

(3.4.5)  \frac{∂}{∂t}Fχ_{Ω_{t}}(ζ · ω) = ∫_{S^{n−1}} g(t, η)^{n−1}g_{t}(t, η)e^{\sqrt{−1}g(t,η)ζ(η,ω)}dη.

Thus,

|Fχ_{Ω_{t}}(ζ · ω) − Fχ_{Ω_{0}}(ζ · ω)| ≤ t sup_{0≤s≤t} |\frac{∂}{∂s}Fχ_{Ω_{s}}(ζ · ω)|
≤ tω_{n−1}∥g∥_{∞}^{n−1}∥g_{t}∥_{∞}e^{| Im ζ|∥g∥_{∞}},

which shows the first inequality (3.4.4).

It follows from the definition of Y (g : r, T, α) and (3.4.5) that

|\frac{∂}{∂t}Fχ_{Ω_{t}}(r · ω) − \frac{∂}{∂t}|_{t=0} Fχ_{Ω_{t}}(r · ω)| ≤ (1 + α)Y (g : r, T, α)ω_{n−1}t^{α},

for 0 ≤ t ≤ T, ω ∈ S^{n−1} and r > 0. Therefore, we have

|Fχ_{Ω_{t}}(r · ω) − Fχ_{Ω_{0}}(r · ω) − t \frac{∂}{∂s}|_{s=0} Fχ_{Ω_{s}}(r · ω)|

= |∫_{0}^{t}\{\frac{∂}{∂s}Fχ_{Ω_{s}}(r · ω) − \frac{∂}{∂s}|_{s=0} Fχ_{Ω_{s}}(r · ω)\}ds|

≤ t^{1+α}Y (g : r, T, α)ω_{n−1}.

□

We shall need the following estimate of Y in §4.

Lemma 3.5. Retain the setting of Lemma (3.4). We define g̃ : [0, T] × S^{n−1} × ℝ → ℝ by

(3.5.1)  g̃(t, η, a) := (1 + at)g(t, η).

Assume

(3.5.2)  0 < T ≤ 1 and |a|T ≤ 1.

Then

(3.5.3)  Y (g̃(·, ·, a) : r, T, α) ≤ 2^{n+1}(n + r)∥g∥_{∞}^{n−1}∥g∥_{0+α}∥g∥_{1+α}(1 + |a|)^{2}.

Proof. First we treat the case a = 0. Fix r > 0 and η, ω ∈ S^{n−1}. We put h(x) := x^{n−1}e^{\sqrt{−1}xr(η,ω)}. Then

∥h(g(·, η))∥_{C^{0,α}([0,T])}

≤ sup_{0≤t≤T} |h(g(t, η))| + sup_{0≤t≤T} |h_{t}(g(t, η))| sup_{0≤s<t≤T} \frac{|g(t, η) − g(s, η)|}{|t − s|^{α}}

≤ ∥g∥_{∞}^{n−1} + {(n − 1)∥g∥_{∞}^{n−2} + ∥g∥_{∞}^{n−1}r}∥g∥_{0+α}

≤ (n + r)∥g∥_{∞}^{n−1}∥g∥_{0+α}.

Here we have used ∥g∥_{∞} ≥ 1 in the last inequality. Hence

Y (g : r, T, α) = \frac{1}{1 + α} sup_{η,ω∈S^{n−1}} ∥h(g(·, η))g_{t}(·, η)∥_{C^{0,α}([0,T])}

≤ ∥g_{t}∥_{0+α} sup_{η,ω∈S^{n−1}} ∥h(g(·, η))∥_{C^{0,α}([0,T])}

≤ (n + r)∥g∥_{∞}^{n−1}∥g∥_{0+α}∥g_{t}∥_{0+α} (3.5.4)

Next, let us estimate Y for g̃(t, η, a) = (1 + at)g(t, η) in terms of g. We assume T ≤ 1 and |a|T ≤ 1. Since

g̃_{t}(t, η, a) = ag(t, η) + (1 + at)g_{t}(t, η),
∥1 + at∥_{0+α} ≤ ∥1 + at∥_{0+1} ≤ 2 + |a|,

we have

∥g̃∥_{∞} ≤ 2∥g∥_{∞},
∥g̃(·, ·, a)∥_{0+α} ≤ ∥1 + at∥_{0+α}∥g∥_{0+α}
≤ (2 + |a|)∥g∥_{0+α},
∥g̃_{t}(·, ·, a)∥_{0+α} ≤ |a|∥g∥_{0+α} + ∥1 + at∥_{0+α}∥g_{t}∥_{0+α}
≤ 2(1 + |a|)∥g∥_{1+α}.

Applying (3.5.4) to g̃, we have

Y (g̃(·, ·, a) : r, T, α) ≤ (n + r)2^{n−1}∥g∥_{∞}^{n−1}(2 + |a|)∥g∥_{0+α}2(1 + |a|)∥g∥_{1+α}
≤ 2^{n+1}(n + r)(1 + |a|)^{2}∥g∥_{∞}^{n−1}∥g∥_{0+α}∥g∥_{1+α}.
□

§4. Proof of Theorem

Now we are ready to prove our main Theorem. We start with the following: Lemma 4.1. In the setting of Lemma (3.4), let ρ be a positive zero of J\frac{n}{2} (x). Assume

(4.1.1)  0 ≤ t < min (T, (\frac{(2π)^{\frac{n}{2}} |g_{t}(0, ·)|′_{ρ}}{ρ^{\frac{n}{2} −1}ω_{n−1}^{\frac{3}{2}} Y (g : ρ, T, α)})^{\frac{1}{α}} ) ,

(see (2.6.1) and (3.4.2) for notation). Then 𝒩(Ω_{t}) ⊃ S(0 : ρ) if and only if t = 0. Here we write Ω_{t} = Ω(g(t, ·)). Proof. If t = 0, then Fχ_{Ω_{0}}(r · ω) = (2π)^{\frac{n}{2}} \frac{J_{\frac{n}{2}} (r)}{r^{\frac{n}{2}}} and so we have Fχ_{Ω_{0}}(ρ · ω) = 0 for any ω ∈ S^{n−1}, that is, 𝒩(Ω_{0}) ⊃ S(0 : ρ).

Conversely, if t ≥ 0 satisfies 𝒩(Ω_{t}) ⊃ S(0 : ρ). Then we have Fχ_{Ω_{t}}(ρ · ω) = Fχ_{Ω_{0}}(ρ · ω) = 0 for ω ∈ S^{n−1}. Using the inequality (3.4.4), we have

t | \frac{∂}{∂t}|_{t=0} Fχ_{Ω_{t}}(ρ · ω)| ≤ t^{1+α}ω_{n−1}Y (g : ρ, T, α).

Taking the L^{2}-norm on S^{n−1}, we have from (2.6.2) the following inequality:

t\frac{(2π)^{\frac{n}{2}}}{ρ^{\frac{n}{2} −1}} |g_{t}(0, ·)|′_{ρ} ≤ t^{1+α}ω_{n−1}^{\frac{3}{2}} Y (g : ρ, T, α).

This implies t = 0 or \frac{(2π)^{\frac{n}{2}} |g_{t}(0,·)|′_{ρ}}{ρ^{\frac{n}{2} −1}ω_{n−1} ^{\frac{3}{2}} Y (g:ρ,T,α)} ≤ t^{α}. Thus we have completed the proof. □

Lemma 4.2. Let 0 < T, 0 < α ≤ 1 and g : [0, T] × S^{n−1} → ℝ_{+} be a C^{1,α} map such that g(0, η) ≡ 1 for η ∈ S^{n−1}. We put

(4.2.1)  B ≡ B(g) := ∥g∥_{∞}^{n−1}∥g∥_{0+1}ω_{n−1}.

We recall notations of Lemma (2.3) such as ε(n, R), δ(n, R), k_{R} and j(\frac{n}{2}, k). Let R > j(\frac{n}{2}, 1). Assume

(4.2.2)(a)  0 ≤ r_{0} ≤ R,
(4.2.2)(b)  0 ≤ t_{0} < min (\frac{ε(n, R)δ(n, R)}{B(g)}, T ) ,
(4.2.2)(c)  ω_{0} ∈ S^{n−1},
(4.2.2)(d)  Fχ_{Ω_{t_{0}}} (r_{0} · ω_{0}) = 0.

Then there exist k ∈ {1, 2, …, k_{R}} and a ∈ ℝ such that

(4.2.3)(a)  r_{0} = (1 + at_{0})j(\frac{n}{2}, k),
(4.2.3)(b)  |a| ≤ \frac{B(g)}{ε(n, R)},
(4.2.3)(c)  |r_{0} − j(\frac{n}{2}, k)| < δ(n, R).

Proof. If t_{0} = 0 then we can take a = 0. Hence we assume t_{0} > 0. It follows from Lemma (3.4) that

|Fχ_{Ω_{t_{0}}} (r_{0} · ω_{0}) − Fχ_{Ω_{0}}(r_{0} · ω_{0})| ≤ B(g)t_{0}.

Since Fχ_{Ω_{t_{0}}} (r_{0} ·ω_{0}) = 0 and Fχ_{Ω_{0}}(r_{0} ·ω_{0}) = f(r_{0}) (see (2.3.1) for the definition of f), we have

(4.2.4)  |f(r_{0})| ≤ B(g)t_{0} < ε(n, R)δ(n, R).

Therefore, from Lemma (2.3), we find k ∈ {1, …, k_{R}} such that

(4.2.5)  ε(n, R) |r_{0} − j(\frac{n}{2}, k)| ≤ |f(r_{0})|.

Now, we put a := \frac{r_{0}−j( \frac{n}{2} ,k)}{t_{0}j( \frac{n}{2} ,k)}. Then we have

|a| ≤ \frac{|r_{0} − j(\frac{n}{2}, k)|}{t_{0}} ≤ \frac{B(g)}{ε(n, R)}.

Here, the second inequality follows from (4.2.4) and (4.2.5), which proves (4.2.3)(b). The inequality (4.2.3)(c) is direct from (4.2.4) and (4.2.5). □ Lemma 4.3. Retain the setting of Lemma (4.2). Assume

(4.3.1)(a)  0 ≤ r_{0} ≤ R, 1 ≤ R,
(4.3.1)(b)  0 ≤ t_{0} < min (\frac{ε(n, R)δ(n, R)}{B(g)}, (\frac{(2π)^{\frac{n}{2}} ε(n, R)^{2}[g]_{R,α}}{2^{n+4}nω_{n−1}^{\frac{1}{2}} R^{\frac{n}{2}} B(g)^{3}} )^{\frac{1}{α}} , T, 1) ,
(4.3.1)(c) 𝒩(Ω_{t_{0}}) ⊃ S(0 : r_{0}).

Then t_{0} = 0. Proof. It follows from Lemma (4.2) that we find a ∈ ℝ and k ∈ {1, …, k_{R}} such that

(4.3.2)(a)  r_{0} = (1 + at_{0})j(\frac{n}{2}, k),
(4.3.2)(b)  |a| ≤ \frac{B(g)}{ε(n, R)}.

We put g̃(t, η, a) := (1 + at_{0})g(t, η). Then we have

(4.3.3)  𝒩(Ω(g̃(t_{0}, ·, a))) ⊃ S(0 : j(\frac{n}{2}, k)),

because

Fχ_{Ω((1+at_{0})g(t_{0},·))}(j(\frac{n}{2}, k) · ω) = (1 + at_{0})^{n}Fχ_{Ω(g(t_{0},·))}((1 + at_{0})j(\frac{n}{2}, k) · ω)
= (1 + at_{0})^{n}Fχ_{Ω(g(t_{0},·))}(r_{0} · ω).

We want to apply Lemma (4.1) to g̃(·, ·, a) in order to show (4.3.3) implies t_{0} = 0. To do this, it suffices to show (4.3.1)(b) implies (4.1.1) with g replaced by g̃, that is,

(4.3.4)  \frac{(2π)^{\frac{n}{2}} ε(n, R)^{2}[g]_{R,α}}{2^{n+4}nω_{n−1}^{\frac{1}{2}} R^{\frac{n}{2}} B(g)^{3}} ≤ \frac{(2π)^{\frac{n}{2}} |g̃_{t}(0, ·, a)|′_{ρ}}{ρ^{\frac{n}{2} −1}ω_{n−1}^{\frac{3}{2}} Y (g̃, ·ρ, T, α)},

where ρ = j(\frac{n}{2}, k)(≤ R). Since |a| ≤ \frac{B(g)}{ε(n,R)}, t_{0} ≤ 1 and t_{0} ≤ \frac{ε(n,R)δ(n,R)}{B(g)} ≤ \frac{ε(n,R)}{B(g)} (see (2.4.1)), we can apply Lemma (3.5) and thus we have for 0 < ρ ≤ R,

Y (g̃(·, ·, a) : ρ, T, α) ≤ 2^{n+1}(n + R)∥g∥_{∞}^{n−1}∥g∥_{0+α}∥g∥_{1+α} (1 + \frac{B(g)}{ε(n, R)})^{2}

≤ 2^{n+2}nR\frac{B(g)}{ω_{n−1}} ∥g∥_{1+α}2^{2} \frac{B(g)^{2}}{ε(n, R)^{2}}

= \frac{2^{n+4}nRB(g)^{3}∥g∥_{1+α}}{ω_{n−1}ε(n, R)^{2}}

Here we have used R ≥ 1 and ε(n, R) ≤ ω_{n−1} ≤ B(g) (see (2.4.2)). Therefore,

\frac{(2π)^{\frac{n}{2}} ε(n, R)^{2}[g]_{R,α}}{2^{n+4}nω_{n−1}^{\frac{1}{2}} R^{\frac{n}{2}} B(g)^{3}} ≤ \frac{(2π)^{\frac{n}{2}} |g_{t}(0, ·)|_{R}}{R^{\frac{n}{2} −1}ω_{n−1}^{\frac{3}{2}} Y (g̃(·, ·, a) : ρ, T, α)}

= \frac{(2π)^{\frac{n}{2}} |g̃_{t}(0, ·, a)|_{R}}{R^{\frac{n}{2} −1}ω_{n−1}^{\frac{3}{2}} Y (g̃(·, ·, a) : ρ, T, α)}.

Hence we have shown (4.3.4) and thus Lemma (4.3). □ Proposition 4.4. Let 0 < α ≤ 1, 0 < T and 0 < R. Suppose g : [0, T] × S^{n−1} ⟶ ℝ_{+} is a C^{1,α} map such that g(0, ·) ≡ 1. Retain the notation as in Lemma (4.3). Assume

0 < r, x ∈ ℝ^{n}, (4.4.1)(a)

0 ≤ t_{0} < min (\frac{ε(n, R)δ(n, R)}{B(g)e^{∥g∥_{∞}}}, (\frac{(2π)^{\frac{n}{2}} ε(n, R)^{2}[g]_{R,α}}{2^{n+4}nω_{n−1}^{\frac{1}{2}} R^{\frac{n}{2}} B(g)^{3}} )^{\frac{1}{α}} , T, 1) , (4.4.1)(b)

𝒩(Ω_{t_{0}})_{ℝ} ∩ B(0 : R) ⊃ S(x : r). (4.4.1)(c)

Then x = 0 and t_{0} = 0. Proof. It suffices to show x = 0 by virtue of Lemma (4.3). Because t_{0} satisfies (4.4.1)(b), the inequality (3.4.3) gives

(4.4.2)  |Fχ_{Ω_{t_{0}}} (ζ · ω) − Fχ_{Ω_{0}}(ζ · ω)| ≤ t_{0}B(g)e^{| Im ζ|∥g∥_{∞}} < ε(n, R)δ(n, R),

for ζ ∈ ℂ with | Im ζ| ≤ 1 and ω ∈ S^{n−1}. On the other hand, if ζ ∈ ℂ satisfies |ζ − j(\frac{n}{2}, k)| = δ(n, R) for some k ∈ {1, …, k_{R}}, then

|Fχ_{Ω_{0}}(ζ · ω)| ≥ ε(n, R)δ(n, R)

from Lemma (2.3). Therefore by using Rouche’s theorem Fχ_{Ω_{t_{0}}} (ζ · ω) has a unique zero in \{ζ ∈ ℂ : |ζ − j(\frac{n}{2}, k)| < δ(n, R)\} for each ω ∈ S^{n−1}.

Next, the condition (4.4.1)(c) means Fχ_{Ω_{t_{0}}} (x + r · ω) = 0 for any ω ∈ S^{n−1}. Applying Lemma (4.2) with r_{0} = ∥x + r · ω∥(≤ R) (with the notation there), we find k ∈ {1, …, k_{R}} for each ω ∈ S^{n−1} such that

|∥x + r · ω∥ − j(\frac{n}{2}, k)| < δ(n, R).

By (2.3.2) and because S^{n−1} is connected, the above integer k does not depend on ω ∈ S^{n−1}. This means that there exists k ∈ {1, …, k_{R}} such that

(4.4.4)  |∥x + r · ω∥ − j(\frac{n}{2}, k)| < δ(n, R)

for any ω ∈ S^{n−1}.

Now, let us show x = 0. Suppose x ≠ 0. We put

z_{+} := (1 + \frac{r}{∥x∥}) x, z_{−} := |1 − \frac{r}{∥x∥}| x.

Then, from (4.4.4) we have

(4.4.5)  |∥z_{±}∥ − j(\frac{n}{2}, k)| < δ(n, R).

On the other hand, since \overline{Fχ_{Ω_{t_{0}}} (ζ · ω)} = Fχ_{Ω_{t_{0}}} (−ζ · ω) for ζ ∈ ℝ, 𝒩(Ω_{t_{0}}) ⊃ S(x : r) if and only if 𝒩(Ω_{t_{0}}) ⊃ S(−x : r). Therefore Fχ_{Ω_{t_{0}}} (±x ± r · ω) = 0 for any ω ∈ S^{n−1}. In particular, we have

(4.4.6)  Fχ_{Ω_{t_{0}}} (z_{±}) = 0.

Because Fχ_{Ω_{t_{0}}} (ζ · \frac{x}{∥x∥}) has a unique zero in \{ζ ∈ ℂ : |ζ − j(\frac{n}{2}, k)| < δ(n, R)\}, we conclude ∥z_{+}∥ = ∥z_{−}∥. This implies r = 0, which contradicts to (4.4.1)(a). Therefore x must be the origin. Now Lemma (4.4) is reduced to Lemma (4.3). □

In the normalized case where ∥g_{t}(0, ·)∥_{L^{∞}(S^{n−1})} ≤ 1, we can replace the assumption (4.4.1)(b) by a simpler one: Lemma 4.5. We put

(4.5.1)  C_{n} := \frac{1}{ω_{n−1}^{\frac{1}{2}}} min(1, \frac{1}{2^{n+3}ω_{n−1}} , \frac{(2π)^{\frac{n}{2}}}{2^{4n+7}nω_{n−1}^{3}} ).

For (1 <)j(\frac{n}{2}, 1) ≤ R, we put

(4.5.2)  C(n, R) := \frac{C_{n}ε(n, R)^{2}}{R^{\frac{n}{2}}} ,

where ε(n, R) is the constant in Lemma (2.3). Let 0 < T and 0 < α ≤ 1. Suppose g : [0, T] × S^{n−1} ⟶ ℝ_{+} is a C^{1,α} map such that

(4.5.3)(a)  g(0, η) ≡ 1,  for any η ∈ S^{n−1},
(4.5.3)(b)  sup_{η∈S^{n−1}} |g_{t}(0, η)| ≦ 1.

If t satisfies

(4.5.4)  0 ≤ t < min(T, (C(n, R)[g]_{R,α})^{\frac{1}{α}}),

then t satisfies (4.4.1)(b). Proof. If r ≥ 1, then it follows from (2.6.3) that

|g_{t}(0, ·)|′_{r} ≤ ∥g_{t}(0, ·)∥_{L^{2}(S^{n−1})}r^{\frac{1}{2}} ≤ ∥g_{t}(0, ·)∥_{L^{∞}(S^{n−1})}ω_{n−1}^{\frac{1}{2}} r^{\frac{1}{2}} .

Therefore, from the definition (2.6.4), we have

(4.5.5)  [g]_{R,α} = \frac{|g_{t}(0, ·)|_{R}}{∥g∥_{1+α}} ≤ \frac{ω_{n−1}^{\frac{1}{2}} R^{\frac{1}{2}}}{∥g∥_{1+α}} .

Suppose t satisfies (4.5.4). Then

t^{α} < \frac{C_{n}ε(n, R)^{2}}{R^{\frac{n}{2}}} [g]_{R,α} ≤ \frac{ω_{n−1}^{− \frac{1}{2}} · 1}{R^{\frac{n}{2}}} \frac{ω_{n−1}^{\frac{1}{2}} R^{\frac{1}{2}}}{∥g∥_{1+α}} ≤ \frac{1}{∥g∥_{1+α}} ≤ \frac{1}{2}.

Therefore we have

t ≤ \frac{1}{2},
|g_{t}(t, η)| ≤ |g_{t}(0, η)| + t^{α}∥g∥_{1+α} ≤ 1 + \frac{1}{2} · 2 = 2,
|g(t, η)| ≤ |g(0, η)| + t∥g_{t}∥_{∞} ≤ 1 + \frac{1}{2} · 2 = 2,
B(g) = ∥g∥_{∞}^{n−1}∥g∥_{0+1}ω_{n−1} ≤ 2^{n+1}ω_{n−1},
B(g)e^{∥g∥_{∞}} ≤ 2^{n+1}ω_{n−1} · e^{2}  < 2^{n+4}ω_{n−1}.

Thus from (4.5.2), (4.5.5) and (2.4.1) we have

t ≤ t^{α} < C_{n} \frac{ε(n, R)^{2}}{R^{\frac{n}{2}}} [g]_{R,α}

≤ \frac{1}{2^{n+3}ω_{n−1}^{\frac{3}{2}}} \frac{ε(n, R)δ(n, R)}{R^{\frac{n}{2}}} \frac{ω_{n−1}^{\frac{1}{2}} R^{\frac{1}{2}}}{∥g∥_{1+α}}

≤ \frac{ε(n, R)δ(n, R)}{2^{n+4}ω_{n−1}}

≤ \frac{ε(n, R)δ(n, R)}{B(g)e^{∥g∥_{∞}}}.

On the other hand, since C_{n} ≤ \frac{(2π)^{\frac{n}{2}}}{2^{4n+7}nω_{n−1} ^{\frac{7}{2}}} ≤ \frac{(2π)^{\frac{n}{2}}}{B(g)^{3}2^{n+4}nω_{n−1} ^{\frac{1}{2}}} , we have

t^{α} < C_{n} \frac{ε(n, R)^{2}[g]_{R,α}}{R^{\frac{n}{2}}}

≤ \frac{(2π)^{\frac{n}{2}} ε(n, R)^{2}[g]_{R,α}}{2^{n+4}nω_{n−1}^{\frac{1}{2}} B(g)^{3}R^{\frac{n}{2}}} .

Hence t satisfies (4.4.1)(b). □

Now, Theorem follows from Proposition (4.4) and Lemma (4.5). Corollary 1 is an immediate consequence of Theorem (with x = 0) and Fact (1.1). Corollary 2 is a direct consequence of Corollary 1.

§5. Appendix (geometric aspect of unessential perturbation)

In Definition (1.5), we have defined a star-shaped region (with respect to the origin) by using the polar coordinate, that is, Ω(g) ⊂ ℝ^{n} for g ∈ C(S^{n−1}, ℝ_{+}). In this section, we also consider a star-shaped region Ω by defining the boundary G: S^{n−1} → ∂Ω, and discuss a geometric aspect of “unessential perturbation” that we have treated. This is done in Proposition (5.7). Throughout this section we use the notation C^{k} where k stands for l (l ∈ ℕ = {0, 1, … }), (l, α) (l ∈ ℕ, 0 < α ≤ 1), ∞ or ω (real analytic). The condition k ≥ 1 means that we exclude the cases k = 0 and k = (0, α) from the above possibilities. We define a standard norm in ℝ^{n} by ∥x∥ := (∑_{j=1}^{n} x_{j}^{2})^{\frac{1}{2}} for x = ^{t}(x_{1}, …, x_{n}) ∈ ℝ^{n}. Definition 5.1. A C^{k}-map G: S^{n−1} → ℝ^{n} is called to bound a C^{k}star-shaped region (with respect to the origin) if the following two conditions are satisfied:

Image(G) ∌ 0 (5.1.1)

(5.1.2) \frac{G}{∥G∥}: S^{n−1} → S^{n−1}, ω ↦ \frac{G(ω)}{∥G(ω)∥} defines a C^{k}-diffeomorphism of S^{n−1}.

Then Image(G) is the boundary of a star-shaped region (see Lemma (5.2.1)). We write C_{⋆}^{k}(S^{n−1}, ℝ^{n}) for the totality of such maps.

Clearly, if h ∈ C^{k}-Diffeo(S^{n−1}) and G ∈ C_{⋆}^{k}(S^{n−1}, ℝ^{n}), then both G and G ∘ h define the same boundary Image(G) = Image(G ∘ h). The following lemma relates Definition (1.5) with Definition (5.1).

Lemma 5.2. We have a bijection:

(5.2.1)  C_{⋆}^{k}(S^{n−1}, ℝ^{n}) ≃ C^{k}(S^{n−1}, ℝ_{+}) × C^{k}-Diffeo(S^{n−1}), G ↦ (g_{G}, h_{G}),

characterized by the following property:

(5.2.2)  G ∘ h_{G}(η) = g_{G}(η)η  (η ∈ S^{n−1}).

In particular,

1) Image(G) = ∂Ω(g_{G}), the boundary of the star-shaped region Ω(g_{G}) (see (1.5)).

2) If G_{1}, G_{2} ∈ C_{⋆}^{k}(S^{n−1}, ℝ^{n}) define the same boundary Image(G_{1}) = Image(G_{2}), then there exists an h ∈ C^{k}-Diffeo(S^{n−1}) such that G_{1} ∘ h = G_{2}.

Proof. Given G ∈ C_{⋆}^{k}(S^{n−1}, ℝ^{n}), we put h_{G} := (\frac{G}{∥G∥})^{−1} ∈ C^{k}-Diffeo(S^{n−1}) and g_{G}(η) := ∥G ∘ h_{G}(η)∥ (η ∈ S^{n−1}). It follows from definition of h_{G} that \frac{G(h_{G}(η))}{∥G(h_{G}(η))∥} = η and so G ∘ h_{G}(η) = g_{G}(η)η. Conversely, given (g, h) ∈ C^{k}(S^{n−1}, ℝ_{+})×C^{k}-Diffeo(S^{n−1}), we define G(ω) := g(h^{−1}(ω)) h^{−1}(ω). Then it follows from definition that G ∈ C_{⋆}^{k}(S^{n−1}, ℝ^{n}) and that (g_{G}, h_{G}) = (g, h). This establishes the bijection (5.2.1). The remaining part (1) is followed from (5.2.2), and (2) holds if we put h_{G} := h_{G_{1}} ∘ h_{G_{2}}^{−1}. □

Next, we consider a deformation of domains by Map([0, T], C_{⋆}^{k}(S^{n−1}, ℝ^{n})). We prepare the following lemma asserting that it is an open condition for a domain to be star-shaped under a C^{1}-deformation of domains.

Lemma 5.3. Let k ≥ 1. Suppose G: [0, T] × S^{n−1} → ℝ^{n} is a C^{k}-map such that G(0, ·) ∈ C_{⋆}^{k}(S^{n−1}, ℝ^{n}) (see Definition (5.1)). Then there exists a positive number 0 < T′ ≤ T such that G(t, ·) ∈ C_{⋆}^{k}(S^{n−1}, ℝ^{n}) for any t ∈ [0, T′]. Moreover, if we define g_{G}(t, η) = g_{G(t,·)}(η) and h_{G}(t, η) = h_{G(t,·)}(η) with the notation in Lemma (5.2), then g_{G} : [0, T′] × S^{n−1} → ℝ_{+} and h_{G} : [0, T′] × S^{n−1} → S^{n−1} are C^{k}-maps satisfying

(5.3.1)  h_{G}(t, ·) is a C^{k}-diffeomorphism of S^{n−1} for t ∈ [0, T′].
(5.3.2)  G(t, h_{G}(t, η)) = g_{G}(t, η)η  for t ∈ [0, T′], η ∈ S^{n−1}.

Proof. We fix T′′ > 0 such that G([0, T′′]×S^{n−1}) ⊂ ℝ^{n} \{0}. We define a C^{k}-map H: [0, T′′]× S^{n−1} → S^{n−1} by H(t, ω) := \frac{G(t,ω)}{∥G(t,ω)∥} for (t, ω) ∈ [0, T′′] × S^{n−1}. Let us show that the map H(t, ·): S^{n−1} → S^{n−1} is a C^{k}-diffeomorphism for any small t.

Since G(0, ·) bounds a star-shaped region, we find C^{k}-maps h: S^{n−1} → S^{n−1} and g: S^{n−1} → ℝ_{+} such that G(0, h(η)) = g(η)η by Lemma (5.2). Then the Jacobian JH(0, ·) of the map H(0, ·) = h^{−1} : S^{n−1} → S^{n−1} is nowhere vanishing. Because H(t, ω) is in the C^{k} class and because S^{n−1} is compact, there exists (T′′ ≥) T′ > 0 such that the Jacobian JH(t, ·) of the map H(t, ·): S^{n−1} → S^{n−1} is nowhere vanishing for all t ∈ [0, T′]. This means that the map H(t, ·): S^{n−1} → S^{n−1} is locally C^{k}-diffeomorphic. Because a local homeomorphism from a compact manifold to a connected one is automatically a covering map, the map H(t, ·): S^{n−1} → S^{n−1} is a covering map. On the other hand, consider the following commutative diagram of homology groups:

H(t, ·)_{∗}: H_{n−1}(S^{n−1}, ℤ) →H_{n−1}(S^{n−1}, ℤ)
∥  ↓
G(t, ·)_{∗}: H_{n−1}(S^{n−1}, ℤ) →H_{n−1}(ℝ^{n} \ {0}, ℤ).

Since G(t, w) is continuous, the mapping degree of H(t, ·) equals to that of H(0, ·), that is, ±1. Therefore H(t, ·) is a C^{k}-diffeomorphism from S^{n−1} to S^{n−1} for any t ∈ [0, T′]. Hence G(t, ·) ∈ C_{⋆}^{k}(S^{n−1}, ℝ^{n}) for any t ∈ [0, T′] by definition. Now the last statement is proved in the same way as Lemma (5.2). □

A deformation of domains starting from a unit ball is a special case of the previous lemma and so we have:

Lemma 5.4. Let k ≥ 1. Suppose G: [0, T] × S^{n−1} → ℝ^{n} is a C^{k}-map such that G(0, ω) ≡ ω (ω ∈ S^{n−1}). Then there exist a positive number 0 < T′ ≤ T, and C^{k}-maps g_{G} : [0, T′]×S^{n−1} → ℝ_{+} and h_{G} : [0, T′] × S^{n−1} → S^{n−1} satisfying (5.3.1), (5.3.2) and

(5.4.1)  g_{G}(0, η) ≡ 1, h_{G}(0, η) ≡ η for η ∈ S^{n−1}.

Thus the map G defines a family of star-shaped regions

(5.4.2)  Ω_{t} := Ω(g_{G}(t, ·))  (see (1.5))

parametrized by t ∈ [0, T′] with Ω_{0} = B(0 : 1), the unit ball.

Let T ℝ^{n} → ℝ^{n} be the tangent bundle, and T ℝ^{n}_{|S^{n−1}} → S^{n−1} its restriction to S^{n−1}. We write Γ^{k}(T ℝ^{n}_{|S^{n−1}}) for the space of sections in the C^{k} class.

Definition 5.5. Suppose G: [0, T] × S^{n−1} → ℝ^{n} is a C^{1}-map such that G(0, ω) ≡ ω (ω ∈ S^{n−1}). We define the velocity X_{G} ∈ Γ^{1}(T ℝ^{n}|_{S^{n−1}}) of a deformation Ω_{t} = Ω(g_{G}(t, ·)) (see (5.4.2)) by

X_{G}(ω) := \frac{∂}{∂t}|_{t=0} G(t, ω).

The map G(t, ω) is called degenerate at t = 0 if X_{G}(ω) ≡ 0 (ω ∈ S^{n−1}).

The groups C^{k}-Diffeo(S^{n−1}) and AF (ℝ^{n}) := GL(n, ℝ) ⋉ ℝ^{n} act naturally on C^{k}(S^{n−1}, ℝ^{n}) by G ↦ π ∘ G ∘ φ, where φ ∈ C^{k}-Diffeo(S^{n−1}), π ∈ AF (ℝ^{n}) and G ∈ C^{k}(S^{n−1}, ℝ^{n}). In particular, if we define π(a, b) ∈ AF (ℝ^{n}) (a > 0, b ∈ ℝ^{n}) by π(a, b)x := ax + b (x ∈ ℝ^{n}) describing a parallel translation and similarity transform, then (π(a, b) ∘ G ∘ φ) (ω) = aG(φ(ω)) + b.

As we are interested in the perturbation of domains up to parallel translations and similarity transforms, we consider:

Definition 5.6. Suppose G_{i}: [0, T_{i}] × S^{n−1} → ℝ^{n} is a C^{k}-map such that G(0, ω) ≡ ω (i = 1, 2). We call G_{1} is C^{k}-equivalent to G_{2} if there exist 0 < T′′ ≤ T and C^{k}-maps

a: [0, T′′] → ℝ,  a(0) = 1,
b: [0, T′′] → ℝ^{n},  b(0) = ^{t}(0, …, 0),
φ: [0, T′′] × S^{n−1} → S^{n−1}, φ(0, ω) ≡ ω, φ(t, ·) is a C^{k}-diffeomorphism of S^{n−1},

such that G_{2}(t, ·) = π(a(t), b(t)) ∘ G_{1}(t, ·) ∘ φ(t, ·), that is, G_{2}(t, ω) = a(t)G_{1}(t, φ(t, ω)) + b(t) for t ∈ [0, T′], ω ∈ S^{n−1}. We write [G]^{k} for the C^{k}-equivalence class containing G.

Proposition 5.7. Let k ≥ 1. Suppose G: [0, T] × S^{n−1} → ℝ^{n} is a C^{k}-map such that G(0, ω) ≡ ω. Let g_{G}: [0, T′] × S^{n−1} → ℝ_{+} be the corresponding function in Lemma (5.3). Then the following conditions on the map G are equivalent:

1) Ω(g_{G}(t, ·)) is an unessential family of perturbations (see §1).

2) Among the C^{k}-equivalence class [G]^{k} (Definition (5.6)), there exists G̃ which is degenerate at t = 0 (see Definition (5.5)).

The rest of this section is devoted to the proof of Proposition (5.7).

Let T_{S^{n−1}}ℝ^{n} → S^{n−1} be the normal bundle associated to the embedding of Riemannian manifolds S^{n−1} ⊂ ℝ^{n}. According to a Whitney sum of vector bundles over S^{n−1}

T ℝ^{n}_{|S^{n−1}} = T S^{n−1} ⊕ T_{S^{n−1}}ℝ^{n},

we have a direct sum of vector spaces:

Γ^{k}(T ℝ^{n}_{|S^{n−1}}) = 𝒳^{k}(S^{n−1}) ⊕ C^{k}(S^{n−1}),

where we denote by 𝒳^{k}(S^{n−1}) the space of vector fields over S^{n−1} in the C^{k} class, and identify C^{k}(S^{n−1}) with Γ^{k}(T_{S^{n−1}}ℝ^{n}) by the outer normal vector field. We recall E_{0}, E_{1} are the subspace of C^{ω}(S^{n−1}) which are the eigenspaces of the Laplace-Beltrami operator Δ_{S^{n−1}} with eigenvalues 0, 1 − n, respectively (see Lemma (2.5)). Thus we regard 𝒳^{k}(S^{n−1}) ⊕ E_{0} ⊕ E_{1} ⊂ Γ^{k}(T ℝ^{n}_{|S^{n−1}}). We write a quotient map as

(5.8) p: Γ^{k}(T ℝ^{n}_{|S^{n−1}}) → Γ^{k}(T ℝ^{n}_{|S^{n−1}})/ (𝒳^{k}(S^{n−1}) ⊕ E_{0} ⊕ E_{1}) ≃ C^{k}(S^{n−1}) ∩ (∑_{k=2}^{∞} ⊕E_{k}) .

Formula 5.9. Suppose T > 0 and we are given C^{1}-maps

G: [0, T] × S^{n−1} → ℝ^{n}, G(0, ω) ≡ ω (ω ∈ S^{n−1})
a: [0, T] → ℝ,  a(0) = 1,
b: [0, T] → ℝ^{n},  b(0) = ^{t}(0, …, 0),
φ: [0, T] × S^{n−1} → S^{n−1}, φ(0, ω) ≡ ω.

If we put G̃(t, ω) := a(t)G(t, φ(t, ω)) + b(t), then with the notation in Definition (5.5)

(5.9.1)  XG̃(ω) = X_{G}(ω) + X_{φ}(ω) + a′(0)ω + b′(0).

In particular, p(X_{G}) ∈ C^{k}(S^{n−1}) ∩ ∑_{k=2}^{∞} ⊕E_{k} is determined only by the equivalent class [G]^{k}.

Proof. The formula (5.9.1) follows by a direct calculation. □

Proof of Proposition (5.7). In the setting of Proposition (5.7), we consider another condition on the map G:

(5.7.3)  p(X_{G}) = 0, that is, X_{G} ∈ 𝒳^{k}(S^{n−1}) ⊕ E_{0} ⊕ E_{1}.

i) (5.7.1) ⇔ (5.7.3): Because G(t, h_{G}(t, η)) = g_{G}(t, η)η, we have \frac{∂}{∂t} |_{t=0} G(t, h_{G}(t, η)) = \frac{∂}{∂t} |_{t=0} g_{G}(t, η)η, that is, X_{G}(η) + X_{h_{G}}(η) = g_{G}(η)η since G(0, η) = h_{G}(0, η) = η. Because X_{h_{G}} ∈ 𝒳^{k}(S^{n−1}) and because (η ↦ g_{G}(η)η) ∈ Γ^{k}(T_{S^{n−1}}ℝ^{n}) is identified with g_{G} ∈ C^{k}(S^{n−1}) via the outer normal vector field, we have

p(X_{G}) = g_{G} mod E_{0} ⊕ E_{1}.

Therefore p(X_{G}) = 0 if and only if g_{G} ∈ E_{0} ⊕ E_{1}, that is, (5.7.3) is equivalent to (5.7.1). ii) (5.7.2) ⇔ (5.7.3): Assume there exists G̃ ∈ [G]^{k} such that XG̃ = 0. It follows from Formula (5.9) that p(X_{G}) = p(XG̃) = 0. Therefore, (5.7.2) implies (5.7.3). Conversely, assume p(X_{G}) = 0, that is, X_{G} ∈ 𝒳^{k}(S^{n−1}) ⊕ E_{0} ⊕ E_{1}. In view of (2.5.1) and (2.5.2) we find Y ∈ 𝒳^{k}(S^{n−1}) and a ∈ ℝ, b ∈ ℝ^{n} such that

X_{G}(ω) = Y (ω) + (a + (b, ω))ω (ω ∈ S^{n−1}).

Because S^{n−1} is compact, there exist 0 < T′′ ≤ T′ and a one parameter family of diffeomorphisms φ: [0, T′′] × S^{n−1} → S^{n−1} such that

\frac{∂}{∂t}|_{t=0} φ(t, ω) = −Y (ω) + (b − (b, ω)ω),

from fundamental existence and uniqueness theorems of a system of first order differential equations. We put a(t) := 1 − at, b(t) := −bt and G̃(t, ω) := a(t)G(t, φ(t, ω)) + b(t). Then, from Formula (5.9.1), we have

XG̃(ω) = X_{G}(ω) + (−Y (ω) + b − (b, ω)ω) − aω − b = 0.

Hence we have proved (5.7.3) implies (5.7.2) □ Remark 5.10. Given a C^{k}-map G: [0, T] × S^{n−1} → ℝ^{n} such that G(0, η) ≡ η for k = ∞ or ω. If Ω(g_{G}(t, ·)) is not unessential (generic case), we can apply Corollary 2. Otherwise (degenerate case), from the equivalent condition (5.7.2) we can find G̃ ∈ [G]^{k} such that G̃(t, η) − η ≡ 0 mod t, so that G_{1}(t, η) := G̃(\sqrt{t}, η) is in the C^{1, \frac{1}{2}} class. Similarly, either we can apply Corollary 2 to G_{1} or we can find ˜ G̃ ∈ [G]^{k} such that ˜ G̃(t, η)−η ≡ 0 mod t^{2}, so that G_{2}(t, η) := ˜ G̃(^{3}\sqrt{t}, η) is in the C^{1, \frac{1}{3}} class. By an iteration of this argument, we can show either

(5.10.1) there exists l ∈ ℕ and G̃ ∈ [G]^{k} such that G̃(t, η) − η ≡ 0 mod t^{l} and that G̃(t^{\frac{1}{l+1}} , η)

∈ C^{1, \frac{1}{l+1}} ([0, T′] × S^{n−1}, ℝ^{n}) defines a family of domains which is not unessential,

or

(5.10.2) for any l ∈ ℕ there exists G̃_{l} ∈ [G]^{k} such that G̃_{l}(t, η) − η ≡ 0 mod t^{l}.

In the real analytic case k = ω, it is likely that (5.10.2) can be replaced by

(5.10.2)′ there exist G̃ ∈ [G]^{ω} such that G̃(t, η) ≡ η.

We remark that the case (5.10.2)′ means that Ω(g_{G}(t, ·)) is a ball for all t, while in the case

(5.10.1) we can apply Corollary 2.

Acknowledgement

The author would like to thank Professors C. Berenstein, B. Ørsted and T. Oshima for their

comments and interest in this work. He is also grateful to Ms. Carol Warfield for her typing

of the manuscript during his stay at the School of Mathematics of the Institute for Advanced

Study at Princeton.

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