STATIONARY SETS FOR THE WAVE EQUATION IN CRYSTALLOGRAPHIC DOMAINS

MARK L. AGRANOVSKY AND ERIC TODD QUINTO

Abstract. Let W be a crystallographic group in ℝ^{n} generated by reflections and let Ω be the fundamental domain of W. We characterize stationary sets for the wave equation in Ω when the initial data is supported in the interior of Ω. The stationary sets are the sets of time-invariant zeros of nontrivial solutions which are identically zero at t = 0.

We show that, for these initial data, the (n − 1)-dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group W̃, W < W̃ . This part comes from a corresponding odd symmetry of the initial data.

In physical language, the result is that if the initial source is localized strictly inside of the crystalline Ω then unmovable interference hypersurfaces can be only faces of a crystalline substructure of the original one.

1. Formulation of the problem and main results

Let Ω be a convex bounded polyhedron in ℝ^{n}, which is the fundamental domain of a crystallographic group, W, that is generated by orthogonal reflections around finite number of hyperplanes.

Let us consider the following initial boundary value problem for the wave equation in Ω with Dirichlet boundary conditions:

(1.1)

(∂_{t}^{2} − Δ)u = 0 in Ω, u = u(x, t), x ∈ Ω, t > 0,
u(x, t) = 0, x ∈ ∂Ω, t > 0,
u(x, 0) = 0, x ∈ Ω,
∂_{t}u(x, 0) = f(x), x ∈ Ω.

Define the stationary set S_{Ω}[f] as

(1.2)  S_{Ω}[f] = {x ∈ Ω : u(x, t) = 0, ∀ t > 0}, u ≢ 0.

Thus, S_{Ω}[f] is the set of all points in Ω where the solution u(x, t) to (1.1) vanishes for all time t > 0 assuming u is identically zero at t = 0.

For a separable solution to (1.1), u(x, t) = sin λt · φ_{λ}(t), where φ_{λ} is the eigenfunction of the Laplacian, Δφ_{λ} = −λ^{2}φ_{λ}, the stationary set S_{Ω}[f] coincides with the nodal set φ_{λ}^{−1}(0). Characterizing nodal sets is a well-known and difficult problem. There is a rich literature on this and we will mention here only some results relating to our subject. The analytic structure of nodal sets is quite well understood: they consist of (n − 1)-dimensional smooth (real-analytic) submanifolds and closed analytic sets of lower dimension (see, e.g. [6, 11, 13, 14, 7, 15] and others).

2000 Mathematics Subject Classification. Primary: 35L05, 44A12 Secondary: 35B05, 35S30.

The first author was supported by the Israel Science Foundation (grant No.279/02-1).

The second author was partially supported by NSF grants 9877155 and 0200788.

On the other hand, understanding global geometry of a single nodal set is a very difficult problem (see, e.g. [16] where the question about existence of closed nodal lines for the second eigenvalue is answered). Nevertheless, and this is the main point of our paper, joint nodal sets in certain situations can be completely described geometrically.

More precisely, we study the case when the stationary sets are intersections of infinitely many nodal sets. This happens, in particular, when the initial velocity f in the wave equation has support strictly in the interior of the domain. It might be expected that in this case the (n − 1)-dimensional component, if it is nonempty, has a precisely determined geometry.

Indeed, our main result states that the stationary hypersurfaces in crystallographic domains Ω = ℝ^{n}/W, where the W is a crystallographic group generated by reflections, are hyperplanes constituting a crystallographic substructure in Ω. Here is a more precise formulation. We will say A ⋐ B if and only if A is a compact subset of the interior of B, int B. Theorem 1.1. Let Ω be a crystallographic domain in ℝ^{n} generated by the reflection group, W, and assume f ∈ C(Ω), f ≠ 0, and supp f is a compact subset of int Ω, supp f ⋐ Ω. Then the stationary set S_{Ω}[f] (1.2) is the union

S_{Ω}[f] = S_{0} ∪ V,

where S_{0}, assuming S_{0} ≠ ∅, is the union of cross-sections of Ω by hyperplanes and V is an analytic set in Ω, dim V < n − 1.

Moreover, there exists a crystallographic group W̃ generated by reflections, W is a subgroup (of finite index) of W̃ , such that S_{0} defines a partition of Ω into a finite number of fundamental polyhedra of the group W̃ .

In this case, the initial data f must be skew-symmetric with respect to reflections through the hyperplanes in S_{0}.

Let us briefly comment on the result. Brünning [9] characterized the compact sets in the plane with piecewise smooth boundary with the following property: there are an infinite number of eigenfunctions of the Laplacian on the set with common nodal curves that are line segments. His result shows that the only planar polygons with this property are rectangles, equilateral triangles and triangles with the angles π/4, π/4, π/2 and π/6, π/3, π/2 (and sets made up from them). Note that the listed polygons are just fundamental domains of crystallographic groups generated by reflections. Our result for n = 2 states that common nodal lines for infinite families of eigenfunctions in these polygons are always straight lines, provided the eigenfunctions appear as projections on eigenspaces of a function f with supp f ⋐ int Ω. Note that, in general, for these polygons, nodal curves for individual eigenfunctions can be very complicated with many nonlinear curves, including closed curves.

This paper continues a series of works by the authors on stationary sets for the wave equation with compactly supported initial data. For the problem (1.1) with Ω = ℝ^{2} and f ∈ C_{c}(ℝ^{2}) (continuous functions with compact support), the stationary sets were completely described in [2]. They have reflection symmetry and consist, up to finite sets, of straight lines constituting a Coxeter system (equiangular configuration of straight lines through one point). In higher dimensions, n > 2, the stationary sets S[f] in ℝ^{n} were described for the case of finitely supported distributions f [4], and the general case of compactly supported initial data f is still open, but we have conjectured [3] that the result is analogous to the plane case, namely, the (n − 1)−dimensional parts of stationary sets are cones. Articles such as [1, 5] provide insight into the general case.

The general idea of the proof of Theorem 1.1 is related to results from [2, 3, 4] and is the following. The first step is to extend the problem to the whole space ℝ^{n} by reflections around the faces of the fundamental polyhedron Ω which results in a tiling of ℝ^{n}. Then we relate the stationary sets to the kernel of a spherical Radon type transform. A crucial step is a local symmetry principle that we derive from the microlocal analysis of the spherical transform. This and arguments of irrational wrapping type, help us analyze W -orbits of tangent planes to stationary surfaces and prove that the hypersurfaces are hyperplanes. The final conclusion follows from the reflection principle for the wave equation.

2. Stationary sets and the spherical means

Consider the tiling of ℝ^{n} resulting from reflections of the fundamental polyhedron Ω through its faces and extend f to ℝ^{n} by odd symmetry: f(σ_{1} … σ_{k}x) = (−1)^{k}f(x), x ∈ Ω, where each element σ_{i} ∈ W is a reflection. It easily follows from the definition of fundamental domain that the extended function is well defined. Let us call the extended function the W -extension of f.

Using this extension, from now on, we assume that f is defined in ℝ^{n} and is skew-symmetric with respect to any reflection σ ∈ W, f ∘ σ = −f. Let us consider the following initial value problem in ℝ^{n} :

(2.1)  \begin{matrix}(∂_{t}^{2} − Δ)u = 0, u = u(x, t), x ∈ ℝ^{n}, t > 0, \\ u(x, 0) = 0, x ∈ ℝ^{n}, \\ ∂_{t}u(x, 0) = f(x), x ∈ ℝ^{n}, \end{matrix}

where f ∈ C(ℝ^{n}) and f has the properties:

1) C ∩ supp f ⋐ int C for any open cell C of the group W (i.e., C = w(Ω) for some w ∈ W).

2) f ∘ σ = −f for any reflection σ ∈ W.

By the symmetry principle for the wave equation, the solution u(x, t) to (2.1) possesses the same symmetry u(σx, t) = −u(x, t), x ∈ ℝ^{n}, t > 0, for any reflection σ ∈ W. Moreover by 1) and 2), the restriction of u(x, t) to the original polyhedron Ω solves the problem (1.1) with the initial data f |_{Ω} .

Now introduce the spherical transform

(2.2)  Rf(x, r) = ∫_{S(x,r)} f(y)dA(y),

where S(x, r) = {y ∈ ℝ^{n} : ∥y − x∥ = r} and dA(y) is the normalized area measure on the sphere S(x, r). Lemma 2.1. The following identity holds:

S[f] = {x ∈ ℝ^{n} : Rf(x, r) = 0, ∀ r > 0}.

The proof of the lemma is given, for instance, in [3] and directly follows from the Kirchoff-Poisson formula for the solution u(x, t) and from the invertibility of the resulting generalized Volterra-Abel integral equation.

Lemma 2.2. Let f = ∑_{k=1}^{∞} c_{k}φ_{k} be the Fourier decomposition of f|_{Ω} in the space

L^{2}(Ω) into the orthonormal basis of Dirichlet eigenfunctions φ_{k}, (Δ + λ_{k}^{2})φ_{k} = 0,

λ_{k} ≠ 0. Then S[f] = ⋂_{k=0,c_{k}≠0}^{∞} φ̃_{k}^{−1}(0), where φ̃_{k} is the W -extension of φ_{k} to ℝ^{n}.

Furthermore, each φ̃_{k} is real-analytic.

Proof. Note that the extended functions φ̃_{k} are (smooth) eigenfunctions for the Laplace operator in ℝ^{n}, because they vanish on the faces of the polyhedra wΩ, w ∈ W and the normal derivatives in adjacent polyhedra agree on the common faces.

The solution u(x, t) to (2.1) is given by the series

(2.3)  u(x, t) = ∑_{k=0}^{∞} c_{k} \frac{sin λ_{k}t}{λ_{k}} φ̃_{k}(x)

which converges in L^{2} on compact subsets of ℝ^{n}. It follows that u(x, t) = 0 for all t > 0 is equivalent to φ̃_{k}(x) = 0, k ∈ ℕ, c_{k} ≠ 0. □

3. Local symmetry principle (the support theorem)

The symmetry principle for the wave equation says that if a solution u(x, t) vanishes on a hyperplane S for all t > 0, then u(x, t) and, in particular, the initial data f(x) = ∂_{t}u(x, 0) are odd with respect to the reflection through S.

If the zero hypersurface S is not a hyperplane, this strong symmetry principle does not hold. Nevertheless, we will show that a weaker symmetry principle holds in the nonlinear case as well. Definition 3.1. Let Π be a hyperplane in ℝ^{n} and a ∈ Π. We say that a set K ⊂ ℝ^{n} is locally symmetric with respect to the pair (Π, a) (or (Π, a)-locally symmetric) if whichever a point x_{0} ∈ K gives a local minimum of the distance function K ∋ x → dist(x, a) = ∥x − a∥, the Π-mirror point x_{0}^{∗} belongs to K.

Note that this definition of local symmetry does not imply K is symmetric about Π even near x_{0}, only that if x_{0} is a boundary point of K and K is locally to one side of S(a, ∥x_{0} − a∥) near x_{0}, then the reflection of x_{0} in Π is also in K.

The following theorem, which is equivalent to Theorem 3.2 in [4], is the key microlocal result. Throughout the article, if S is a manifold and a ∈ S then T_{a}(S) and N_{a}(S) will denote tangent and normal spaces, correspondingly to S at a, and T_{a}^{∘}(S) = a + T_{a}(S) will denote the affine tangent plane. Theorem 3.2 (Theorem 3.2 [4]). Let S be a real-analytic hypersurface in ℝ^{n} and f ∈ D′(ℝ^{n}) be a distribution in ℝ^{n} such that Rf(x, r) = 0 for all x ∈ S and r > 0. Then, for each a ∈ S supp f is (T_{a}^{∘}(S), a)-locally symmetric.

This theorem is a generalization of a theorem of Courant and Hilbert ([12], p. 699 ff.) which is equivalent, due to Lemma 2.1, to the reflection principle for the wave equation and states that if S = Π is a hyperplane and Rf(a, r) = 0 for all a ∈ Π and r > 0, then f is odd about Π and so supp f is globally symmetric about Π. For Theorem 3.2, S can be curved, and in this case supp f is only locally symmetric about the tangent planes to S in the weak sense of Definition 3.1.

Since the statement of this theorem is slightly different in [4], let us outline the main ideas of the proof.

The key point is the regularity property of the operator R which implies cancellation of analytic wave front sets at points where a sphere, centered at a point a ∈ S, touches the support of f. In turn, the cancellation is possible only at mirror points with respect to the tangent plane T_{a}^{∘}(S) and this implies that the point which is mirror to the touching point must be in the support as well, and the local symmetry follows.

More precisely, let x_{0} give a local minimum for the distance from a to supp f and let r_{0} = ∥x_{0} − a∥. Let WF_{A}(f) be the analytic wavefront set of f [17] and let ξ_{0} be a conormal covector to S(a, r_{0}) at x_{0}. Then, locally near x_{0}, supp f lies outside S(a, r_{0}). By a theorem of Kawai, Kashiwara, and Hörmander [17] about analytic wavefront at boundary points of supp f, (x_{0}, ξ_{0}) ∈ WF_{A}(f). Let x_{0}^{∗} be the reflection of x_{0} in T_{a}^{∘}(S). By Theorem 3.3 in [4], x_{0}^{∗} must be in supp f (if not, (x_{0}, ξ_{0}) could not be in WF_{A}(f)). Therefore, supp f is (T_{a}^{∘}(S), a)-locally symmetric.

4. W -orbits of the tangent bundle T (S[f])

In the sequel, S will denote a connected real-analytic open (n−1)−submanifold of S[f]. Our aim in this section is to analyze the behavior of the pairs (T_{a}^{∘}(S), a), a ∈ S, under the W − action.

4.1. Preparations. First, we use the Bieberbach theorem [10] (actually the definition in some books) that states that the crystallographic group W contains a lattice subgroup ℤ[h_{1}, ⋯, h_{n}] generated by n linearly independent vectors h_{i}:

ℤ[h_{1}, ..., h_{n}] = {∑_{i=1}^{n} k_{i}h_{i} : k_{i} ∈ ℤ}.

From now on, the basis h_{1}, ⋯, h_{n} is fixed. Lemma 4.1. Assume that S is not contained in a hyperplane. Then there exists a point a ∈ S and a vector τ ∈ T_{a}(S),

τ = τ_{1}h_{1} + ⋯ + τ_{n}h_{n}, τ_{i} ∈ ℝ,

such that the coordinates τ_{i} are linearly independent over the field ℚ of rational numbers.

Proof. Pick an arbitrary point a ∈ S. If the conclusion of the lemma is not true

then for each vector τ ∈ T_{a}(S), τ = ∑_{i=1}^{n} τ_{i}h_{i}, there are integers r_{i} such that

∑_{i=1}^{n} r_{i}τ_{i} = 0.

Thus, the linear space T_{a}(S) is covered by the countable family of subspaces

T_{a}(S) ∩ { ∑_{i=1}^{n} τ_{i}h_{i} ∈ ℝ^{n} : ∑_{i=1}^{n} r_{i}τ_{i} = 0}, r_{i} ∈ ℤ.

and this is possible only if T_{a}(S) coincides with one of these subspaces, i.e.

T_{a}(S) = { ∑_{i=1}^{n} τ_{i}h_{i} ∈ ℝ^{n} : ∑_{i=1}^{n} r_{i}τ_{i} = 0},

for some (r_{1}, ⋯, r_{n}) ∈ ℤ^{n}.

There are only countable many such linear spaces, and since the tangent space T_{a}(S) depends on a ∈ S continuously and S is connected, then T_{a}(S) is a fixed hyperplane for all a ∈ S and S is contained in a hyperplane. □

Proposition 4.2. [Kronecker, cf. [8],Ch.7,1.3, Prop.7] Let θ_{1}, ⋯, θ_{n−1} ∈ ℝ. Then the vectors

(4.1)  (k_{1} + k_{n}θ_{1}, ⋯, k_{n−1} + k_{n}θ_{n−1}), k_{i} ∈ ℤ

are dense in ℝ^{n−1} if and only if the numbers θ_{1}, ⋯ θ_{n−1}, 1 are linearly independent over ℚ.

In fact, we will need a bit stronger version of Kronecker’s theorem: Corollary 4.3. If θ_{1}, ⋯, θ_{n−1}, 1 are linearly independent over ℚ then the vectors (4.1) are dense in ℝ^{n−1} when (k_{1}, ⋯, k_{n}) ∈ ℤ^{n} and k_{n} ≥ 0.

Proof. . Denote V_{+} the semigroup of all vectors (4.1) with k_{n} ≥ 0 and let V_{−} = −V_{+}.

If V_{+} is not dense in ℝ^{n−1} then V_{−} is not dense, either, and there exists an open ball B = B(b, R), R > 0 that contains no element from the closure cl V_{−}. By Proposition 4.2, cl V_{+} ∪ cl V_{−} = ℝ^{n−1} and therefore B ⊂ cl V_{+}.

From the definition (4.1) we have ℤ^{n−1}+cl V_{+} ⊂ cl V_{+}, and in particular, ℤ^{n−1} ⊂ cl V_{+}. Also B+cl V_{+} ⊂ cl V_{+}+cl V_{+} ⊂ cl V_{+} because cl V_{+} is a semigroup. Therefore, for any s ∈ ℕ:

ℤ^{n−1} + sb + B(0, sR) = ℤ^{n−1} + sB = ℤ^{n−1} + \underbrace{B + ⋯ + B}_{s times} ⊂ cl V_{+}

Write sb = [sb] + {sb}, where [sb] and {sb} are vectors of integer and fractional parts of the coordinates, respectively. Since ℤ^{n−1} + [sb] = ℤ^{n−1}, it follows from the above inclusion that

{sb} + B(0, sR) ⊂ cl V_{+}

and, since {sb} ∈ [0, 1]^{n−1} and s ∈ ℕ is arbitrary, we see that cl V_{+} = ℝ^{n−1}. □

Lemma 4.4. Let X, Y be linear normed vector spaces, dim X = n, dim Y = n−1, and

P : X → Y

be a linear operator with one-dimensional kernel

ker P = ℝτ, τ ∈ X \ {0}.

Fix a basis h_{1}, ⋯, h_{n} ∈ X and define the half-lattice

ℤ_{+}[h] = {∑_{j=1}^{n} k_{j}h_{j} : k_{j} ∈ ℤ, k_{n} ≥ 0}.

If the coordinates τ_{1}, ⋯, τ_{n} of τ in the basis {h_{i}} are linearly independent over ℚ then the image P (ℤ_{+}[h]) is dense in Y.

Proof. Let v_{i} = P (h_{i}), i = 1, ⋯, n. Then

∑_{j=1}^{n} τ_{j}v_{j} = P (τ) = 0

and since all τ_{i} ≠ 0 we can write

v_{n} = θ_{1}v_{1} + ⋯ + θ_{n−1}v_{n−1},

where θ_{i} = −τ_{i}/τ_{n}.

The condition for τ_{i} implies that θ_{1}, ⋯ θ_{n−1}, 1 are linearly independent over ℚ and by Corollary 4.3 the vectors

x = P (∑_{j=1}^{n} k_{j}h_{j}) = ∑_{j=1}^{n} k_{j}v_{j} = ∑_{j=1}^{n−1}(k_{j} + k_{n}θ_{j})v_{j},

are dense in Y, where k_{j} runs over ℤ and k_{n} ≥ 0. More precisely, Corollary 4.3 gives denseness in ℝ^{n−1} of vectors (k_{1} + k_{n}θ_{1}, …, k_{n−1} + k_{n}θ_{n−1}) ∈ R^{n−1} and then the denseness of vectors x in Y follows because the system v_{1}, ⋯, v_{n−1} is

a basis in Y (indeed, if ∑_{i=1}^{n−1} αv_{i} = 0 for some real α_{i} then ∑_{i=1}^{n−1} α_{i}h_{i} ∈ ker P, so

(α_{1}, ⋯, α_{n−1}, 0) = t(τ_{1}, ⋯, τ_{n}) for some t ∈ ℝ and τ_{n} ≠ 0 implies all α_{i} = 0). □

4.2. A geometric construction. In what follows K is a fixed compact set in ℝ^{n}. Fix the point a ∈ S and the vector τ ∈ T_{a}(S) in accordance with Lemma 4.1. Possibly using a rotation, we can normalize τ by

∥τ∥ = 1, τ_{n} > 0.

Choose a unit normal vector e ∈ N_{a}(S).

We can assume that

(4.2)  max_{x∈K}(x, τ) = 0.

Denote

Q = {x ∈ ℝ^{n} : (x, τ) = 0.}

the supporting hyperplane. Let K_{0} = K ∩ Q be the portion of K in Q. Choose a real number d_{0} so that

(4.3)  max_{x∈K_{0}}(x, e) < d_{0}

and define the hyperplane

(4.4)  Π_{0} = {x ∈ ℝ^{n} : (x, e) = d_{0}}.

Finally, chose two closest points x_{0} ∈ K_{0} and p_{0} ∈ Π_{0}:

(4.5)  ∥x_{0} − p_{0}∥ = dist(K_{0}, Π_{0})

Since the hyperplanes Q and Π_{0} are orthogonal and x_{0} ∈ Q the point p_{0} belongs to the both hyperplanes, p_{0} ∈ Q ∩ Π_{0}. Also

dist(K_{0}, Π_{0}) = d_{0} − max_{x∈K_{0}}(x, e) > 0 .

4.3. The supporting plane Q as a limit of supporting spheres. Let P_{Q} : ℝ^{n} → Q be the orthogonal projection. In this subsection we construct, using the notation and preparations made in subsections 4.1 and 4.2, a sequence of spheres having the following properties:

1) they support (touch) the prescribed compact set K,

2) they converge to the supporting hyperplane Q,

3) the P_{Q}−projections of centers of the spheres converge to the point p_{0} ∈ Q, defined in (4.5). Because ker P_{Q} = ℝτ and since the vector τ is chosen in accordance with Lemma

4.1, i.e., its coordinates τ_{i} with respect to the basis h_{i} are linearly independent over ℚ, Lemma 4.4 applies to the operator P_{Q} and yields that P_{Q} takes the half-lattice Z_{+}[h] onto a dense set in the hyperplane Q.

In particular, if one fixes a natural number M,

(4.6)  M > |(a)_{n}|

where (a)_{n} is the n^{th} coordinate of the point of tangency, a, in the basis h_{j} , then there exists an infinite sequence t_{m} ∈ ℤ_{+}[h] such that

P_{Q}(t_{m}) → p_{0} − P_{Q}(a + Mh_{n}), m → ∞

where p_{0} is defined in (4.5).

If we let

(4.7)  a_{m} = a + t_{m} + Mh_{n}

then

lim_{m→∞} P_{Q}(a_{m}) = p_{0} .

The formula (4.7) defines a translation w_{m} : a → a_{m} that belongs to the group W, so a_{m} is in W − orbit of the point a.

The sequence a_{m} is unbounded because it is infinite and discrete, because it belongs to the lattice Z_{+}[h], so passing to a subsequence we can assume that ∥a_{m}∥ → ∞ as m → ∞.

Now represent a_{m} as the direct sum

(4.8)  a_{m} = P_{Q}(a_{m}) ⊕ (a_{m}, τ)τ

The first summand converges and hence is bounded, therefore ∥(a_{m}, τ)∥ → ∞, m → ∞. If a_{m} = ∥a_{m}∥ξ_{m}, ∥ξ_{m}∥ = 1, then dividing (4.8) by ∥a_{m}∥ = (∥P_{Q}(a_{m})∥^{2} + |(a_{m}, τ)|^{2})^{1/2} and letting m → ∞ we obtain

ξ_{m} → ±τ, m → ∞.

However, ξ_{m} → −τ is impossible because the n^{th} coordinates, in the basis h_{i}, of the vectors τ and ξ_{m} have opposite signs: τ_{n} > 0 by construction, and ∥a_{m}∥(ξ_{m})_{n} = (t_{m})_{n} + (a)_{n} + M > (t_{m})_{n} ≥ 0 because t_{m} ∈ ℤ_{+}[h] and by the choice of M, (4.6).

Thus,

(4.9)  a_{m}/∥a_{m}∥ = ξ_{m} → τ, m → ∞ .

Lemma 4.5. Denote r_{m} = dist(a_{m}, K) and let x_{m} ∈ K be such that ∥x_{m} − a_{m}∥ = r_{m}. Then the sequence x_{m} converges to the point x_{0} defined in subsection 4.2.

Proof. It suffices to proof that x_{0} is the only limit point of the sequence x_{m}. Let x̃_{0} be a limit point, x̃_{0} = lim_{k→∞} x_{m_{k}}. We can assume, by renumbering, that x_{m} → x̃_{0} as m → ∞.

Clearly, x̃_{0} ∈ K because x_{m_{k}} ∈ K and K is closed. We now show that x̃_{0} ∈ K ∩ Q = K_{0}. Indeed, for any x ∈ K,

∥x_{m} − a_{m}∥^{2} = r_{m}^{2} ≤ ∥x − a_{m}∥^{2}

or, equivalently,

(x, a_{m}) ≤ (x_{m}, a_{m}) + (1/2)(∥x∥^{2} − ∥x_{m}∥^{2}).

Dividing both sides by ∥a_{m}∥ and letting m → ∞, so ∥a_{m}∥ → ∞, we obtain from (4.9):

(x, τ) ≤ (x̃_{0}, τ).

Thus,

(x̃_{0}, τ) = max_{x∈K}(x, τ) = 0

and hence x̃_{0} ∈ Q by the definition of the hyperplane Q. Geometrically, what we have checked is that the spheres S(a_{m}, r_{m}), touching the compact K tend to the supporting hyperplane Q when the centers a_{m} go to infinity.

Note that (x_{m}, τ) ≤ 0 by (4.2) because x_{m} ∈ K. On the other hand, (a_{m}, τ) → +∞, m → ∞, and therefore ∃m_{0} ∈ ℕ such that (a_{m}, τ) > 0 for m ≥ m_{0}. Thus, for such m, x_{m} and a_{m} are on opposite sides of the hyperplane Q, and therefore the open ball B(a_{m}, r_{m}) meets Q. Hence its intersection with Q coincides with nonempty intersection of Q and another open ball centered on Q:

B(a_{m}, r_{m}) ∩ Q = B(p_{m}, s_{m}) ∩ Q,

where p_{m} = P_{Q}(a_{m}).

By minimality of r_{m}, no open ball B(a_{m}, r_{m}) contains points from K, and since B(p_{m}, s_{m}) ∩ Q ⊂ B(a_{m}, r_{m}), the ball B(p_{m}, s_{m}) ∩ Q is disjoint from K. Therefore ∥x_{0} − p_{m}∥ ≥ s_{m}, ∥x̃_{0} − p_{m}∥ ≥ s_{m} because x_{0}, x̃_{0} ∈ K_{0} ⊂ K. In particular, the sequence s_{m} is bounded (because p_{m} converges to p_{0}) and passing to a subsequence we can assume that s_{m} → s_{0}, m → ∞. Taking limits we obtain

(4.10)  ∥x_{0} − p_{0}∥ ≥ s_{0}, ∥x̃_{0} − p_{0}∥ ≥ s_{0}

On the other hand, we have for m ≥ m_{0}:

∥P_{Q}(x_{m})−a_{m}∥^{2} = ∥x_{m}−(x_{m}, τ)τ−a_{m}∥^{2} = ∥x_{m}−a_{m}∥^{2}+(x_{m}, τ)(2(a_{m}, τ)−(x_{m}, τ)) ≤ r_{m}^{2},

because ∥x_{m} − a_{m}∥ = r_{m}, (x_{m}, τ) ≤ 0 and (a_{m}, τ) > 0.

So, P_{Q}(x_{m}) belongs to the closed ball \overline{B}(a_{m}, r_{m}) and since it belongs to Q we have P_{Q}(x_{m}) ∈ \overline{B}(p_{m}, s_{m}), i.e. ∥P_{Q}(x_{m})−p_{m}∥ ≤ s_{m}. Letting m → ∞ we conclude

∥x̃_{0} − p_{0}∥ ≤ s_{0} .

Here we have taken into account that x̃_{0} ∈ Q and therefore P_{Q}(x̃_{0}) = x̃_{0}. Together with (4.10) it gives

(4.11)  ∥x̃_{0} − p_{0}∥ = s_{0}.

To complete the proof, recall that, by the construction in 4.3, the vector p_{0} − x_{0} is proportional to the vector e (normal to Π_{0}): p_{0} − x_{0} = αe, where the coefficient α = dist(K_{0}, Π_{0}). Write, based on (4.11), p_{0} − x̃_{0} = s_{0}η, ∥η∥ = 1.

From the first inequality in (4.10), we have s_{0} ≤ α. Then using the fact that x̃_{0} ∈ K_{0} and p_{o} ∈ Π_{0} we obtain

s_{0} ≤ α = dist(K_{0}, Π_{0}) ≤ dist(x̃_{0}, Π_{0}) = (p_{0} − x̃_{0}, e) = s_{0}(η, e) ≤ s_{0}.

This yields η = e and α = s_{0}, i.e., x̃_{0} = x_{0}. So, x_{m} → x_{0}. □

4.4. Dense W − orbits and local symmetry. Lemma 4.6. Let Ω be a fundamental domain of the group W and a compact set K lies strictly inside Ω, K ⋐ Ω. Define

K̃ = ⋃_{w∈W} w(K).

Let S be a real-analytic connected hypersurface in ℝ^{n}. Suppose that K̃ is locally symmetric with respect to any pair (T_{a}^{∘}(S), a), s ∈ S. Then K̃ = ∅ unless S is a hyperplane.

Proof. Suppose that K̃ ≠ ∅ (and therefore K ≠ ∅) and S is not a hyperplane. We apply the constructions and the results of the previous subsections 4.1-4.4 to the compact K. Let a ∈ S and τ ∈ T_{a}(S) be as in Lemma 4.1, and e ∈ N_{a}(S), Q, Π_{0}, x_{m}, a_{m}, x_{0}, p_{0} are as in subsections 4.3 and 4.4.

First of all, applying translation, we can assume that the condition (4.2) is fulfilled for the compact K. As well as the condition (4.3) for the number d_{0}, we require:

(4.12)  max_{x∈K_{0}}(x, e) < d_{0} < ρ/2 + max_{x∈K_{0}}(x, e),

where ρ = dist(K, ∂Ω). Note that ρ > 0 because K ⋐ Ω.

The points a_{m} = w_{m}(a) (4.7) belong to the W −orbit of the point a and since K̃ is W −invariant and (T_{a}^{∘}(S), a)− locally symmetric then K̃ is locally symmetric with respect to any pair (Π_{m}, a_{m}), where Π_{m} = T_{a_{m}}^{∘}(S_{m}) and S_{m} = w_{m}(S).

Because K is open and closed in K̃, the function K̃ ∋ x → ∥x−a_{m}∥ attains local minimum r_{m} at the point x_{m} and, by local symmetry, Theorem 3.2, the (Π_{m}, a_{m})− mirror point x_{m}^{∗} belongs to K̃.

It follows that the Π_{0}− symmetric point x_{0}^{∗} is in K̃. Indeed, since P_{Q} is the orthogonal projection along the vector τ, orthogonal to e, then (a_{m}, e) = (p_{m}, e) (recall that p_{m} = P_{Q}(a_{m})) and therefore the hyperplane Π_{m} is defined by the equation (x, e) = d_{m} where d_{m} = (p_{m}, e).

The Π_{m}-mirror point x_{m}^{∗} = x_{0} + 2(d_{m} − (x_{0}, e))e ∈ K̃. By Lemma 4.5 x_{m} → x_{0}, and also p_{m} → p_{0} so taking limit we have the Π_{0}− mirror point

x_{0}^{∗} = x_{0} + 2(d_{0} − (x_{0}, e))e ∈ K̃, d_{0} = (p_{0}, e).

Furthermore the right inequality in (4.12) implies

∥x_{0}^{∗} − x_{0}∥ = 2|d_{0} − (x_{0}, e)| < ρ

and since x_{0} ∈ K ⊂ Ω then x_{0}^{∗} must be inside Ω by the definition of ρ and so x_{0}^{∗} ∈ Ω ∩ K̃ = K.

We will now show x_{0}^{∗} is not in K by showing x_{0}^{∗} is above K_{0}. This will finish the proof.

Moreover, x_{0}^{∗} ∈ Q∩K = K_{0} because x_{0} ∈ Q, and the hyperplane Q is orthogonal to Π_{0} and hence Q is invariant under reflection in Π_{0}. From this and from the left inequality in (4.12) we have

(x_{0}^{∗}, e) ≤ max_{x∈K_{0}}(x, e) < d_{0}.

However, the same inequality in (4.12) implies

(x_{0}^{∗}, e) = (x_{0}, e) + 2(d_{0} − (x_{0}, e)) = 2d_{0} − (x_{0}, e) ≥ 2d_{0} − max_{x∈K_{0}}(x, e) > d_{0}.

This contradiction proves that if S is not a hyperplane then K̃ = ∅. □

5. Proof of Theorem 1.1

Recall that we deal with the wave equation (2.1) in the whole space ℝ^{n} and the initial data f is obtained by the W −extension in section 1. Correspondingly, the stationary set S[f] is obtained from S_{Ω}[f] (1.2) by applying the group W and adding hyperplanes that bound all the polyhedra w(Ω), w ∈ W. We call these hyperplanes W −mirrors; they are obtained from the system of hyperplanes generating the group W by applying elements from W and each W − mirror consists of fixed points of some reflection in W.

Denote K = Ω ∩ supp f. Since f ≠ 0, we have K ≠ ∅. Because f is skewsymmetric with respect to reflections from the group W, K̃ = supp f = ⋃_{w∈W} wK.

According to Lemma 2.2, the stationary set

S[f] = ⋂_{k=0,c_{k}≠0}^{∞} φ̃_{k}^{−1}(0);

and since each nodal set φ̃_{k}(0) is a union of a smooth hypersurface and a lower dimensional analytic set (see Theorem 1.1), the representation

S[f] = S_{0} ∩ V

holds, where S_{0} is a smooth hypersurface and V is an analytic set of dimension less than n − 1.

Theorem 3.2 states that the set K̃ = supp f is (T_{a}^{∘}(S_{0}), a)-locally symmetric for any a ∈ S_{0}. Then Lemma 4.6 implies that any connected smooth hypersurface S ⊂ S_{0} must be a hyperplane. Therefore S_{0} is the union of hyperplanes.

Denote by Σ the system of these hyperplanes, so that

S_{0} = ⋃_{π∈Σ} π.

Since S_{0} ⊂ S[f], then by the reflection principle the initial data f is odd with respect to reflections through any hyperplane in Σ. Also all W −mirrors belong to Σ.

Let W̃ be the group generated by the reflections through the hyperplanes in Σ. Since Σ contains W −mirrors, W ⊂ W̃. By the above property of f, each set S[f], S_{0} and V is W̃ − invariant and for this reason Σ is a W̃ − invariant system of hyperplanes.

We claim that the system Σ is locally finite. Indeed, if it is not then there exists a hyperplane π_{0} in ℝ^{n} that is a limit point of hyperplanes in Σ:

π_{0} = lim_{m→∞} π_{m}, π_{m} ∈ Σ, π_{m} ≠ π_{0}.

By Lemma 2.2 each eigenfunction φ̃_{k} in the Fourier decomposition (2.3) vanishes on S[f] and hence vanishes on each hyperplane π_{m}. Then φ̃_{k} vanishes on π_{0} to infinite order and φ_{k} = 0 due to real analyticity. This would imply f = 0 which is not the case.

Now we want to check that the group W̃ is discrete. It suffices to prove that the unit e, the identity map, is an isolated point of W̃.

Suppose w_{m} → e, where w_{m} ∈ W̃. The convergence is uniform on compact sets. Due to local finiteness of the system Σ, only a finite number of hyperplanes in Σ meet the fundamental polyhedron Ω, therefore Ω \ S_{0} consists of finite number of disjoint open polyhedra. Let Δ be one of them and c ∈ Δ be a generic point (not a fixed point of any nontrivial isometry of Δ). Then w_{m}(c) → c and hence ∃m_{0} ∈ ℕ such that w_{m}(c) ∈ Δ for m ≥ m_{0}. Since for each m > m_{0}, w_{m} is a composition of Σ− reflections and the polyhedron w_{m}(Δ) and Δ have a common point, w_{m}(c), they must coincide: w_{m}(Δ) = Δ, m ≥ m_{0}.

However, the group of isometries of the polyhedron Δ is finite and hence the sequence w_{m} contains only finite number of distinct elements. Then, since c is generic, w_{m}(c) → c implies that w_{m} = e when m is sufficiently large. This means that the group W̃ is discrete.

In addition, the factor-space ℝ^{n}/W̃ is compact because of the imbedding

ℝ^{n}/W̃ ↪ ℝ^{n}/W ⋍ \overline{Ω}/W

and since \overline{Ω} is compact.

Thus the group W̃ is uniform (the factor space is compact) and discrete and therefore is crystallographic (see. e.g. [10], Def. 1.7). Any reflection from W̃ takes the polyhedron Δ to a polyhedron that is either inside Ω or outside \overline{Ω}, the polyhedra obtained by subsequent reflections through the faces are mutually disjoint and cover ℝ^{n} \ S_{0}.

Therefore Δ serves as a fundamental polyhedron of the group W̃. By construction, the group W is a subgroup of the group W̃, of finite index #(W̃ /W), which equals to the number of reflections of the fundamental polyhedron Δ through its faces needed to cover Ω. This completes the proof. Remark 5.1. The proof is valid for more complicated domains, namely domains that tile ℝ^{n} and that are made up of unions of fundamental domains of the same crystallographic group. For example, the hexagon Ω is a union of equilateral triangles, and it tiles the plane. Our theorem is true for the hexagon as long as f is zero near the boundary of Ω and f is oddly symmetric about each reflection in each diameter of Ω. The proof is as given above and the extra condition on f is used to ensure the extension of f to ℝ^{2} by odd symmetry is consistent (see start of §2). Acknowledgments. The authors thank Professors Gregory Soifer and Mikhail Katz for stimulating discussions. The second author would like to thank Dr. Mark Varvares of MEEI and Dr. James McIntyre of MGH for their invaluable help as this research began.

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Bar Ilan University, Ramat Gan, Israel

E-mail address: agranovs@macs.biu.ac.il

Tufts University, Medford, MA, USA

E-mail address: todd.quinto@tufts.edu