EIGENVALUES PROBLEM FOR THE BIHARMONIC OPERATOR ON Z 2 -SYMMETRIC REGIONS A. L. PEREIRA AND M. C. PEREIRA Abstract. In this work we show that the eigenvalues of the Dirichlet problem for the Biharmonic operator are generic simple in the set of Z 2 -symmetric regions of R n with a suitable topology. To establish it, we combine the Baire’s Lemma, a generalized version of the Transversality Theorem, due to Henry[6], and the method of rapidly oscillating functions developed in [15]. 1. Introduction Generic properties for boundary value problems in partial differential equations arise in many contexts and have been investigated by several authors from various points of view. Among others, we may mention the pioneering works of Micheletti [7] and Uhlenbeck [8], who studied the generic simplicity of the eigenvalues of the Dirichlet problem for self-adjoint operator with respect to variations of the domain; Rocha [18] and Pereira [13], who showed the generic hyperbolicity of equilibria of a reaction-diffusion equations by perturbation of the nonlinear term and diffu- sion coefficient; Brunovsk´ y-Pol´ acik [17] and Joly [20], who proved that the generic Kupka-Smale property holds for a class of dynamical system defined for partial differential equations; and Saut and Teman [19], who considered the generic hyper- bolicity of equilibria of a semi linear second order elliptic problem with respect to perturbation of the boundary. These results are interesting because they play an important role in the qualitative study of partial differential equations. In some sense, they can ensure the stability of the qualitative behavior of the solutions. There are also many works on perturbation of the boundary in the literature using the concept of shape differentiation (see e.g. [3]). We mention Ortega and Zuazua [4] and [5] where generic simplicity of the eigenvalues was successfully ap- plied to study a stabilization problem. Many problems of the same type were considered in the monograph of Henry [6], where the author developed a general theory on perturbation of domains and proved several results on boundary perturbations for second order elliptic operators. In particular, he proved, using Thom-Abraham’s Transversality Theorem, the generic simplicity of the eigenvalues of the Laplacian with Dirichlet boundary conditions even if one restrict attention to Z 2 -symmetric regions. Following his approach, Pereira [11] obtained generic results on the eigenvalues of the Dirichlet problem for the Laplace operator in regions satisfying rather general symmetry properties. In this work, it is shown that, except for the very special case G = Z 2 ⊕··· Z 2 , 2000 Mathematics Subject Classification. 35J40,35B30,58C40. Key words and phrases. biharmonic equations; bilaplacian operator; generic properties; bound- ary perturbations. Research partially supported by Fapesp 2006/06278-7. 1
Transcript
EIGENVALUES PROBLEM FOR THE BIHARMONIC
OPERATOR ON Z2-SYMMETRIC REGIONS
A. L. PEREIRA AND M. C. PEREIRA
Abstract. In this work we show that the eigenvalues of the Dirichlet problem
for the Biharmonic operator are generic simple in the set of Z2-symmetric
regions of Rn with a suitable topology. To establish it, we combine the Baire’sLemma, a generalized version of the Transversality Theorem, due to Henry[6],
and the method of rapidly oscillating functions developed in [15].
1. Introduction
Generic properties for boundary value problems in partial differential equationsarise in many contexts and have been investigated by several authors from variouspoints of view. Among others, we may mention the pioneering works of Micheletti[7] and Uhlenbeck [8], who studied the generic simplicity of the eigenvalues of theDirichlet problem for self-adjoint operator with respect to variations of the domain;Rocha [18] and Pereira [13], who showed the generic hyperbolicity of equilibriaof a reaction-diffusion equations by perturbation of the nonlinear term and diffu-sion coefficient; Brunovsky-Polacik [17] and Joly [20], who proved that the genericKupka-Smale property holds for a class of dynamical system defined for partialdifferential equations; and Saut and Teman [19], who considered the generic hyper-bolicity of equilibria of a semi linear second order elliptic problem with respect toperturbation of the boundary. These results are interesting because they play animportant role in the qualitative study of partial differential equations. In somesense, they can ensure the stability of the qualitative behavior of the solutions.
There are also many works on perturbation of the boundary in the literatureusing the concept of shape differentiation (see e.g. [3]). We mention Ortega andZuazua [4] and [5] where generic simplicity of the eigenvalues was successfully ap-plied to study a stabilization problem.
Many problems of the same type were considered in the monograph of Henry [6],where the author developed a general theory on perturbation of domains and provedseveral results on boundary perturbations for second order elliptic operators. Inparticular, he proved, using Thom-Abraham’s Transversality Theorem, the genericsimplicity of the eigenvalues of the Laplacian with Dirichlet boundary conditionseven if one restrict attention to Z2-symmetric regions. Following his approach,Pereira [11] obtained generic results on the eigenvalues of the Dirichlet problemfor the Laplace operator in regions satisfying rather general symmetry properties.In this work, it is shown that, except for the very special case G = Z2 ⊕ · · ·Z2,
2000 Mathematics Subject Classification. 35J40,35B30,58C40.Key words and phrases. biharmonic equations; bilaplacian operator; generic properties; bound-
ary perturbations.Research partially supported by Fapesp 2006/06278-7.
1
2 PEREIRA AND PEREIRA
multiple eigenvalues must appear in any G-symmetric region. This result extendsin a straightforward way to any elliptic operator commuting with the orthogonalgroup as we indicate below for our special case.
In this paper, we consider the following eigenvalue problem for the Biharmonicoperator
(∆2 + λ)u = 0 in Ωu = ∂u
∂N = 0 on ∂Ω.(1.1)
It is well known that the Biharmonic operator with the boundary conditions aboveis self-adjoint and possesses a countable number of negative eigenvalues 0 > λ0 >λ1 > ...→ −∞. Let now Γ be a subgroup of the orthogonal group O(n) isomorphicto Z2 that is,
Γ = 1, g where g 6= 1 and g2 = 1
where g is either a rotation of angle π or a reflection with respect to some hyper-plane.
We show that, generically in the set of C4-regular regions, Ω ⊂ Rn, n ≥ 2,satisfying gΩ = Ω, all eigenvalues of (1.1) are simple. As a consequence we obtainthat all eigenfunctions u of (1.1) are either even (u g = u) or odd (u g = −u)in Ω. The results can be easily extend to the case where G = Z2 ⊕ · · ·Z2, butwe have chosen to stick to the case of just one copy of Z2 for simplicity. Westress the fact that, unlike the case of the Laplacian, this result does not followfrom Thom-Abraham’s version of the Transversality Theorem. To establish it, wewill need a generalized version, due to Henry[6], and the method developed in[15] to compute approximate solutions of the inverse (modulo finite rank) of somedifferential operators.
In [16], the authors employed these methods with success to show a similar resultfor the Dirichlet problem of a nonlinear plate equation. The goal of this approach isto reduce the problem to the investigation of a suitable unique continuation genericproperty of the eigenfunctions. Another questions were considered in [9] and [14],where the method described in [15] was not necessary.
This paper is organized as follows: in section 2.1, we collect some backgroundresults that we need from [6]. In section 3, we show that symmetries other thanZ2 force the appearance of multiple eigenvalues and in the section 4, we show thegeneric simplicity in the Z2 case.
2. Preliminaries
The results in this section (except Lemma 2.3) were taken from the monographof Henry [6], where more details and proofs can be found.
2.1. Some notation and geometrical preliminaries. Given a real function fdefined in a neighborhood of x ∈ Rn, its m-derivative at x can be consideredas a m-linear symmetric form h −→ Dmf(x)hm in Rn, with norm |Dmf(x)| =max|h|≤1|Dmf(x)hm|. If Ω is an open subset of Rn and E is a normed vectorspace, Cm(Ω, E) is the space of m-times continuously and bounded differentiablefunctions on Ω whose derivatives extend continuously to the closure Ω, with theusual norm ‖f‖Cm(Ω,E) = max0≤j≤msupx∈Ω|Dmf(x)|. If E = R, we write simply
Cm(Ω). Cmunif (Ω, E) is the closed subspace of Cm(Ω, E) of functions whose mth
derivative is uniformly continuous.
EIGENVALUES ON SYMMETRIC REGIONS 3
We say that an open set Ω ⊂ Rn is Cm- regular if there exists φ ∈ Cm(Rn,R),which is at least in C1
unif (Rn,R), such that Ω = x ∈ Rn;φ(x) > 0 and φ(x) = 0
implies |∇φ| ≥ 1.We sometimes need to use differential operators (gradient, divergence and Lapla-
cian) in a hypersurface S ⊂ Rn. These operators can be defined via the inducedRiemannian metric of S ⊂ Rn but, for our purposes the following (equivalent) defi-nitions are more convenient. Let S be a C1 hypersurface in Rn and let φ : S −→ Rbe C1 (so it can be extended to be C1 on a neighborhood of S), then ∇Sφ isthe tangent vector field in S such that, for each C1 curve t −→ x(t) ⊂ S, wehave d
dtφ(x(t)) = ∇Sφ(x(t)) · x(t). Now, let S be a C2 hypersurface in Rn and
~a : S −→ Rn a C1 vector field tangent to S. Then divS~a : S −→ Rn is thecontinuous function such that, for every C1 φ : S −→ R with compact sup-port in S,
∫S
(divS~a)φ = −∫S~a · ∇Sφ. Finally, if u : S −→ R is C2, then
∆Su = divS(∇Su) or, equivalently, for all C1 φ : S −→ R with compact support∫Sφ∆Su = −
∫S∇Sφ · ∇Su.
Theorem 2.1. (1) If S is a C1 hypersurface and φ : Rn −→ R is C1 in aneighborhood of S, then, on S, ∇Sφ(x) is the component of ∇φ(x) tangent
S at x, that is ∇Sφ(x) = ∇φ(x)− ∂φ∂N (x)N(x) where N is an unit normal
field on S.(2) If S is a C2 hypersurface in Rn, ~a : S −→ Rn is C1 in a neighborhood of
S, N : Rn −→ Rn is a C1 unit normal field in a neighborhood of S andH = divN is the mean curvature of S, then
divS~a = div~a−H~a ·N − ∂
∂N(a ·N) on S.
(3) If S is a C2 hypersurface, u : Rn −→ R is C2 in a neighborhood of S andN is a normal vector field for S, then
∆Su = ∆u−H ∂u
∂N− ∂2u
∂N2+∇Su ·
∂N
∂Non S.
We may choose N so that ∂N∂N = 0 on S and then the final term vanishes.
Remark 2.2. As an immediate consequence of the Theorem 2.1, we have that
∆u = ∂2u∂N2 on ∂Ω for all u ∈ H2
0 (Ω) ∩ C2,α(Ω).
We often need the following uniqueness theorem
Lemma 2.3. Let Ω ⊂ Rn be an open, connected, bounded, C4-regular region andJ ⊂ ∂Ω a nonempty open set in the boundary. Then, if u is solution of
(∆2 + λ)u = 0 in Ω
u = ∂u∂N = ∂2u
∂N2 = ∂3u∂N3 = 0 on J,
(2.1)
u ≡ 0 in Ω.
Proof. This result follows from Lemma 3.6, page 276 of [2].
2.2. Boundary perturbations. Given an open, bounded, Cm region Ω0 ⊂ Rn,consider the following open subset of Cm(Ω,Rn)
Diffm(Ω) = h ∈ Cm(Ω,Rn) | h is injective and1
|deth′(x)|is bounded in Ω.
4 PEREIRA AND PEREIRA
and the collection of regions h(Ω0) | h ∈ Diffm(Ω0). We introduce a topology inthis set by defining a (sub-basis of) the neighborhoods of a given Ω by
where iΩ : Ω 7→ Rn is the inclusion. When ‖h − iΩ‖Cm(Ω,Rn) is small, h is a Cm
imbedding of Ω in Rn, a Cm diffeomorphism to its image h(Ω). Michelleti [7]shows this topology is metrizable, and the set of regions Cm-diffeomorphic to Ωmay be considered a separable metric space which we denote by Mm(Ω) = Mm.We say that a function F defined in the space Mm with values in a Banach spaceis Cm or analytic if h 7→ F (h(Ω)) is Cm or analytic as a map of Banach spaces (hnear iΩ in Cm(Ω,Rn)). In this sense, we may express problems of perturbation ofthe boundary of a boundary value problem as problems of differential calculus inBanach spaces.
More specifically, consider a formal non-linear differential operator u 7→ v
v(y) = f(y, u(y),
∂u
∂y1(y), ...,
∂u
∂yn(y),
∂2u
∂y21
(x),∂2u
∂y1∂y2(y), ...
), y ∈ Rn
To simplify the notation, we define a constant matrix coefficient differential operator
Lu(y) =(u(y),
∂u
∂y1(y), ...,
∂u
∂yn(y),
∂2u
∂y21
(y),∂2u
∂y1∂y2(y), ...
), y ∈ Rn
with as many terms as needed, so our nonlinear operator becomes u 7→ v(·) =f(·, Lu(·)). More precisely, suppose Lu(·) has values in Rp and f(y, λ) is definedfor (y, λ) in some open set O ⊂ Rn × Rp. For subsets Ω ⊂ Rn define FΩ byFΩ(u)(y) = f(y, Lu(y)), y ∈ Ω for sufficiently smooth functions u in Ω such that(y, Lu(y)) ∈ O for any y ∈ Ω. For example, if f is continuous, Ω is bounded andL involves derivatives of order ≤ m, the domain of FΩ is an open subset (perhapsempty) of Cm(Ω), and the values of FΩ are in C0(Ω). (Other function spaces couldbe used with obvious modifications).
If h : Ω 7→ Rn is a Ck imbedding, we can also consider Fh(Ω) : Cm(h(Ω)) 7→C0(h(Ω)). But then the problem will be posed in different spaces. To bring it backto the original spaces we consider the ‘pull-back’ of h
h? : Ck(h(Ω)) 7→ Ck(Ω) (0 ≤ k ≤ m)
defined by h?(ϕ) = ϕh (which is a diffeomorphism) and then h?Fh(Ω)h?−1 is again
a map from Cm(Ω) into C0(Ω). This is more convenient to apply tools like theImplicit Function or Transversality theorems. On the other hand, a new variable his introduced. We then need to study the differentiability properties of the function(h, u) 7→ h?Fh(Ω)h
?−1u. This has been done in [6] where it is shown that, if (y, λ) 7→f(y, λ) is Ck or analytic then so is the map above, considered as a map fromDiffm(Ω) × Cm(Ω) to C0(Ω) (other function spaces can be used instead of Cm).To compute the derivative we then need only compute the Gateaux derivative thatis, the t-derivative along a smooth curve t 7→ (h(t, .), u(t, .)) ∈ Diffm(Ω) × Cm(Ω).For this purpose, it is convenient to introduce the differential operator Dt = ∂
∂t −
U(x, t) ∂∂x , U(x, t) =
(∂h∂x
)−1∂h∂t which is is called the anti-convective derivative.
The result below (Theorems 2.4) is the main tools used to compute derivatives.
Theorem 2.4. Suppose f(t, y, λ) is C1 in an open set in R × Rn × Rp, L is aconstant-coefficient differential operator of order ≤ m with Lv(y) ∈ Rp (where
EIGENVALUES ON SYMMETRIC REGIONS 5
defined). For open sets Q ⊂ Rn and Cm functions v on Q, let FQ(t)v be thefunction
y −→ f(t, y, Lv(y)), y ∈ Q.where defined.
Suppose t −→ h(t, ·) is a curve of imbeddings of an open set Ω ⊂ Rn, Ω(t) =h(t,Ω) and for |j| ≤ m, |k| ≤ m + 1 (t, x) −→ ∂t∂
jxh(t, x), ∂kxh(t, x), ∂kxu(t, x) are
continuous on R × Ω near t = 0, and h(t, ·)∗−1u(t, ·) is in the domain of FΩ(t).
Then, at points of Ω
Dt(h∗FΩ(t)(t)h
∗−1)(u) = (h∗FΩ(t)(t)h∗−1)(u) + (h∗F ′Ω(t)(t)h
∗−1)(u) ·Dtu
where Dt is the anti-convective derivative defined above,
FQ(t)v(y) =∂f
∂t(t, y, Lv(y))
and
F ′Q(t)v · w(y) =∂f
∂λ(t, y, Lv(y)) · Lw(y), y ∈ Q
is the linearization of v −→ FQ(t)v.
Remark 2.5. Suppose we deal with a linear operator A =∑|α|≤m aα(y)
(∂∂y
)αnot
explicitly dependent on t, and h(t, x) = x+ tV (x) + o(t) as t→ 0 and x ∈ Ω. Thenat t = 0
∂
∂t(h∗Ah∗−1u)
∣∣∣t=0
= Dt(h∗Ah∗−1u)
∣∣∣t=0
+ h−1x ht∇(h∗Ah∗−1u)
∣∣∣t=0
= A∂u
∂t+ [V · ∇, A]u since
∂
∂tA = 0.
2.3. The Transversality Theorem. A basic tool for our results is the Transver-sality Theorem in the version below, due to Henry [6]. We first recall some defini-tions.
A map T ∈ L(X,Y ) where X and Y are Banach spaces is a semi-Fredholm mapif the range of T is closed and at least one (or both, for Fredholm) of dim N (T ),codim R(T ) is finite; the index of T is then ind(T ) = dim N (T ) − codim R(T ).We say that a subset F of a topological space X is rare if its closure has emptyinterior and meager if it is contained in a countable union of rare subsets of X. Wesay that F is residual if its complement in X is meager. We also say that X is aBaire space if any residual subset of X is dense. Let P be a property dependingof a parameter x ∈ X, where X is a Baire topological space. We say that P isgeneric (in x) if it holds for all x in a residual set of X. Suppose f is a Ck mapbetween Banach spaces. We say that x is a regular point of f if the derivative f ′(x)is surjective and its kernel is finite-dimensional. Otherwise, x is called a criticalpoint of f . A point is critical if it is the image of some critical point of f . Letnow X be a Baire space and I = [0, 1]. For any closed or σ-closed F ⊂ X and anynonnegative integer m we say that the codimension of F is greater or equal to m(codim F≥ m) if the subset φ ∈ C(Im, X) | φ(Im)∩F is non-empty is meager inC(Im, X). We say codim F = k if k is the largest integer satisfying codim F ≥ m.
Theorem 2.6. Suppose given positive numbers k and m; Banach manifolds X,Y, Zof class Ck; an open set A ⊂ X × Y ; a Ck map f : A → Z and a point ξ ∈ Z.Assume for each (x,y) ∈ f−1(ξ) that:
6 PEREIRA AND PEREIRA
(1) ∂f∂x (x, y) : TxX 7→ TξZ is semi-Fredholm with index < k.
(3) (x, y) 7→ y : f−1(ξ) 7→ Y is σ-proper, that is f−1(ξ) is a countable union ofsets Mj such that (x, y) 7→ y : Mj 7→ Y is a proper map for each j.[Given(xn, yn) ∈ Mj such that yn converges in Y , there exists a subsequence(or subnet) with limit in Mj.]
We note that (3) holds if f−1(ξ) is Lindelof [every open cover has a countablesubcover] or, more specifically, if f−1(ξ) is a separable metric space, or if X,Y areseparable metric spaces.
Let Ay = x | (x, y) ∈ A and Ycrit = y | ξ is a critical value of f(·, y) : Ay 7→Z. Then Ycrit is a meager set in Y and, if (x,y) 7→ y : f−1(ξ) 7→ Y is proper,
Ycrit is also closed. If ind(∂f∂x
)≤ −m < 0 on f−1(ξ), then 2(α) implies 2(β) and
Ycrit = y | ξ ∈ f(Ay, y) has codimension ≥ m in Y. [Note Ycrit is meager iffcodim Ycrit ≥ 1].
3. Multiplicity of the eigenvalues on symmetric regions
We first recall some definitions from the representation theory of compact groups.We refer to [1] for details and proofs. Let G be a compact group. A representationof G in a Hilbert space H is a group homomorphism V : G → GL(H), whereGL(H) is the group (under composition) of invertible continuous linear operatorsin H. We say that V is finite dimensional if H is a finite dimensional space. If His a complex (resp. real) Hilbert space the representation V is called unitary (resp.orthogonal) if the image V (g), which we denote in the sequel by Vg, is an unitary(resp. orthogonal) operator, for any g ∈ G.
We say that V is (strongly) continuous if limg→e Vgξ = ξ for any ξ ∈ H. Twocontinuous representations V : G→ GL(H) and V ′ : G→ GL(H ′) are equivalent ifthere exists a linear isometry T : H → H ′ such that V ′g T = T Vg for any g ∈ G.
We say that a closed subspace H1 ⊂ H is invariant for V if VgH1 ⊂ H1 forany g ∈ G. In this case the representation of G in H1 defined by restriction iscalled a subrepresentation of V . The representation V is irreducible if it admits nonontrivial subrepresentation. Any irreducible unitary (orthogonal) representaion ofa compact group G is finite dimensional and G is commutative if and only if allits unitary irreducible representations are one dimensional. If H = H1 ⊕H2 ⊕ · · · ,with H1 invariant for V , we write V = V|H1
⊕ V|H2⊕ · · · and say that V is a direct
sum of the representations V|Hi . If V is a finite dimensional representation of G thefunction XV
g ∈ G 7→ trVg,
where tr denotes the trace of the operator Vg, is called the character of V . Clearly,two equivalent representations have the same character.
Let G be a compact subgroup of the orthogonal group O(n). The set of allequivalent classes of continuous irreducible representations of G is called the dualobject of G and denoted by G. We denote by Xσ the character of any representationin the class σ ∈ G and by dσ its dimension. If H is a Hilbert space and V : G 7→L(H) is a continuous orthogonal representation of G, we can define, for each σ ∈ G,
EIGENVALUES ON SYMMETRIC REGIONS 7
the operator Pσ in H by
〈Pσξ , η〉 =
∫G
〈Vxξ , η〉dσXσ(x) dx
Pσ is a continuous projection (see [1]). We set Mσ := PσH.The following decomposition theorem will be important in the sequel. A proof
for unitary representations can be found in [1]. For real spaces it can be obtainedfrom this result by complexification (see [12]).
Theorem 3.1. Let G be a compact subgroup of O(n) and V a continuous (unitary)
orthogonal representation of G in H. For every σ ∈ G, let Pσ be the operator in Hdefined by
〈Pσξ , η〉 =
∫G
〈Vxξ , η〉dσXσ(x) dx.
Then Pσ is a projection operator in H.If σ 6= σ′ then Mσ and Mσ′ are orthogonal subspaces of H, H =
⊕σ∈GMσ.
For each σ ∈ G,Mσ is either 0 or a direct sum of mσ pairwise orthogonal, dσ-invariant subspaces Lσ,j, on each of which V|Lσ,j ∈ σ.The cardinal number mσ may be finite or infinite.The subspace Mσ is the smallest closed subspace of H containing all invariantsubspaces of H on which V is in the class σ.This direct sum decomposition of V is unique in the following sense. If
H =⊕λ∈Λ
Nλ,
where each Nλ is an invariant subspace on which V is irreducible, then
⊕Nλ | V|Nλ ∈ σ = Mσ
and there are mσ subspaces Nλ on each of which V|Nλ ∈ σ.
We have a natural action of G in Rn: (g, x) 7→ gx. The subgroup Gx := g ∈G | gx = x is called the isotropy group of x ∈ Rn, and G(x) := gx | g ∈ G isthe orbit of x ∈ Rn under this action.
Let now Ω ⊂ Rn be a domain satisfying gΩ ⊂ Ω for every g ∈ G. Such a domainis called G-symmetric or G-invariant. In this case we can define an orthogonalrepresentation of G in H = L2(Ω) by
Γg(u) = u g−1, ∀g ∈ Gwhich we call the quasi-regular representation of G in Ω. This representation en-joys an important property: it contains any irreducible representation of G. Moreprecisely, we have
Theorem 3.2. Let Ω ⊂ Rn be a G-invariant domain and suppose there exists x ∈ Ωsuch that Gx = Id. Then, any orthogonal representation of G is equivalent to asubrepresentation of the quasi-regular representation Γ of G in Ω.
Proof. See [11].
As an immediate consequence of this result we see that the spaces Mσ ⊂ L2(Ω)obtained in the decomposition of the quasi-regular representation accordingly toTheorem 3.1 are nonempty, for every σ ∈ G (and, in fact, infinite dimensional ascan be seen from the proof of the theorem).
8 PEREIRA AND PEREIRA
Lemma 3.3. For any σ ∈ G the spaces Mσ are invariant by the Biharmonicoperator. More precisely, the Biharmonic can be considered as an operator in Mσ
with domain Mσ ∩H4 ∩H20 (Ω).
Proof. If u belongs to Mσ ∩H4 ∩H20 (Ω) and g ∈ G then, clearly u g belongs to
H4(Ω) and u g ≡ 0 in ∂Ω. Also, if N is the unit outward normal field in ∂Ω,N(g · x) = g ·N(x), for any x ∈ ∂Ω. Therefore
Since H4 ∩H20 (Ω) is dense in L2(Ω), it follows that Pσ∆2u = ∆2Pσu, that is, Pσ
and ∆2 commute, from which the result follows immediately.
Theorem 3.4. Let V : G 7→ L(V ) be an irreducible representation of G withdimension d. Then, there exists an eigenvalue λ of (1.1) and a subspace W ofthe corresponding eigenspace Uλ such that the representation Γ restricted to W isequivalent to V . In particular, dimUλ ≥ d.
Proof. Suppose V is in the class σ. By the previous lemma, we can restrict ∆2 toMσ. Let λ be an eigenvalue of this (self-adjoint) operator. Since ∆2(u g−1) =(∆2u
) g−1 for any g ∈ G, it follows that the eigenspace Uλ ⊂Mσ is an invariant
subspace for the representation Γ and so, by Theorem 3.1 a direct sum of irreduciblespaces for the representation Γ restricted to which Γ is in the class σ.
Remark 3.5. We remember that any noncommutative group has complex irre-ducible unitary representations of dimension greater than one. Also, even in thecase of commutative groups, there always exist (real) orthogonal irreducible repre-sentations if G 6= Z2 ⊕ Z2 ⊕ · · · ,⊕Z2. Thus, except in this exceptional case, thereare multiple eigenvalues in any G-symmetric region.
Remark 3.6. The results above extend in a straightforward way to a differentialoperator given by a polinomial of the Laplacian, with suitable boundary conditions.We have chosen to treat here only the case of the Biharmonic operator mainly fornotational simplicity.
EIGENVALUES ON SYMMETRIC REGIONS 9
4. Generic simplicity of the eigenvalues on symmetric regions
Let Γ be a subgroup of the orthogonal group O(n) isomorphic to Z2:
Γ = 1, g where g 6= 1 and g2 = 1,
g is either a rotation of angle π around some axis or a reflection with respect toa hyperplan. We introduce a topology in the set of Γ-symmetric regions given byrestriction of the one described by section 2.2. In this section, we show the genericsimplicity of the eigenvalues of (1.1) with respect to this topology.
Let Ω ⊂ Rn be an open, connected, bounded, Ck-regular, Γ-symmetric region,k ≥ 2, and consider
Diffkg(Ω) = h ∈ Diffk(Ω) | h(gx) = gh(x) for all x ∈ Ω.
Observe that, if Ω is Γ-invariant, then h(Ω) is also Γ-invariant.From Theorem 3.1 (applied to the special case of the quasi-regular representation
of a finite commutative group) it follows that L2(Ω) can be decomposed as a directsum of a finite number of orthogonal subspaces MX , where
MX = u ∈ L2(Ω) | u g−1 = X (g)u, ∀g ∈ G.
If G = Z2, this gives L2(Ω) = L2e(Ω) ⊕ L2
o(Ω) (as an orthogonal sum), whereL2e(Ω) = u ∈ L2(Ω) | u g = u in Ω (the even subspace) and L2
o(Ω) = u ∈L2(Ω) | u g = −u in Ω (the odd subspace). Let He(Ω) = H4 ∩H2
0 (Ω) ∩L2e(Ω)
and Ho(Ω) = H4 ∩H20 (Ω) ∩ L2
o(Ω).We apply the Transversality Theorem to show that there exists a residual set of
h ∈ Diff4g(Ω) such that all the eigenvalues of the Dirichlet problem for the ∆2 are
simple in h(Ω). As a consequence we obtain that all eigenfunctions u of (1.1) areeither even (u g = u) or odd (u g = −u) in Ω.
To obtain our genericity result, we first restrict attention to the even and oddsubspaces, and apply transversality arguments to prove, within this spaces, that alleigenvalues are simple in an open and dense subset of Diff4
g(Ω). At a later stage,we show how to split an eigenvalue which happens to occur simultaneously in bothspaces, that is, it has even and odd eigenfunctions.
In order to apply transversality arguments, we first show that our generic prop-erty is equivalent of zero being a regular value for an appropriate mapping.
Proposition 4.1. Let Ω ⊂ Rn be an open, connected, bounded, C4-regular regionand h ∈ Diff4(Ω). Then, all eigenvalues of (1.1) are simple in h(Ω) if and only ifzero is a regular value of the mapping φh : H4 ∩H2
0 (Ω)× R 7→ L2(Ω) defined by
φh(u, λ) = h∗(∆2 + λ)h∗−1u.
Moreover, if Ω and h(Ω) are G-symmetric regions, and σ ∈ G for some compactsubgroup G of O(n), then zero is a regular value of the mapping φσh : H4 ∩H2
0 (Ω)∩Mσ × R 7→ Mσ, the restriction of φh to Mσ if only if, all eigenvalues of (1.1)restricted to the space Mσ are simple in h(Ω).
Proof. Since L0 : H4 ∩H20 (Ω) 7→ L2(Ω) : u 7→ (∆2 + λ)u is Fredholm with index
zero and h∗ and h∗−1 are isomorphisms, we have that 0 is a regular value of φh ifand only if the application
Dφh(u, λ)(u, λ) = h∗(∆2 + λ)h∗−1u+ λu
10 PEREIRA AND PEREIRA
is onto for all (u, λ) ∈ H4 ∩H20 (Ω)×R with φh(u, λ) = 0. Now, Dφ(u, λ) is onto if
and only if
R(∆2 + λ)⊕ [u] = L2(Ω),
that is, if and only if λ is simple eigenvalue ofh∗(∆2 + λ)h∗−1u(x) = 0 x ∈ Ωu(x) = ∂
∂N u(x) = 0 x ∈ ∂Ω.(4.1)
It remains only to prove that v is solution of the problem (1.1) in h(Ω) if and onlyif u = h∗−1v satisfies (4.1) for all h ∈ Diff4(Ω). Let u ∈ H4 ∩H2
0 (Ω). Since h∗ andh∗−1 are isomorphisms, we have
h∗(∆2 + λ)h∗−1u = 0 ⇐⇒ (∆2 + λ)h∗−1u = 0.
It is clear that u = 0 in ∂Ω if and only if h∗−1u = 0 in ∂h(Ω). Observe that v isan eigenfunction of the problem (1.1) in h(Ω) if only if u = h∗v is a solution of
h∗(∆2 + λ)h∗−1u = 0 in Ω
u = h∗ ∂∂Nh
h∗−1u = 0 on ∂Ω(4.2)
where Nh is the normal of the region h(Ω). Now,(h∗
∂
∂Nhh∗−1u
)(x) =
n∑i=1
(h∗
∂
∂yih∗−1u
)(x)(Nh)i(h(x))
=
n∑i,j=1
bij(x)∂u
∂xj(x)(Nh)i(h(x))
= Nh(h(x))b(x)∇u(x)
where bij(x) = (h−1x )ji(x) [the i,j-th entry in the transposed inverse of the Jacobian
Matrix of h] and b(x) = (bij)(x) with x ∈ Ω. Since u = 0 on ∂Ω we have ∇u =∂u∂NN on ∂Ω. Observe that for all x ∈ Ω ∪ ∂Ω, b(x) is a non singular matrix and
b(x)N(x) is in same direction of Nh(h(x)). Thus h∗ ∂∂Nh
h∗−1u = 0 on ∂Ω ⇐⇒∂u∂N = 0 on ∂Ω, that is, v is solution of (1.1) in h(Ω) if and only if u = h∗v is
solution of (4.1) (note that the pull-back map h∗ : H20 (h(Ω)) → H2
0 (Ω) is welldefined).
Now, since the restriction of (∆2 + λ) to the space Mσ is still a self-adjointoperator and, therefore, Fredholm of index 0, second part follows in the sameway.
It follows from the Proposition 4.1 that, to show generic simplicity of the eigen-values of (1.1) in the even and odd subspaces, it is enough to show that 0 is aregular value of φh, generically in h ∈ Diff4
g(Ω). Using the Transversality Theorem,we show that 0 is a regular value of
F : BiM × [−M, 0]× UgM 7→ L2(Ω)
(u, λ, h) 7→ h∗(∆2 + λ)h∗−1u (4.3)
for i = 1, 2, where B1M = u ∈ He(Ω)\0 | ‖u‖ ≤M, B2
M = u ∈ Ho(Ω)\0 |‖u‖ ≤ M and UgM is an open dense set in Diff4
g(Ω) for all M ∈ N. Taking the
intersection of UgM for M , we obtain by Baire Theorem the desired residual set.
EIGENVALUES ON SYMMETRIC REGIONS 11
Remark 4.2. The set of h ∈ Diff4g(Ω) such that all eigenvalues λ ∈ (−M, 0) of
(1.1) are simple in h(Ω) is an open set in Diff4g(Ω). There are only finitely many
such eigenvalues and each depends continuously on h (see [10, section 3 and 4]).To prove density, we may work with more regular (for example C∞) regions.
The two lemmas below are a kind of ‘generic unique continuation results’ thatwill be needed in the sequel.
Lemma 4.3. Let Ω ⊂ Rn be an open, connected, bounded, C5-regular region withgΩ = Ω, n ≥ 2 and J ⊂ ∂Ω a nonempty open set. Consider the analytic mapping
Proof. We prove only Lemma 4.4 using the Transversality Theorem, the proof ofthe Lemma 4.3 is similar. The hypotheses (1) and (3) of 2.6 can be verified asin [16, Theorem 5], we prove (2β). Let (u, v, λ, h) ∈ Q−1(0, 0, 0). By ‘changingthe origin, we may assume that h = iΩ. The differentiability of Q is easy toestablish, and its partial derivatives can be computed using Theorem 2.4. Therefore,DQ(u, v, λ, iΩ) : He(Ω) × He(Ω) × R × C5
where C5g (Ω,Rn) = h ∈ C5(Ω,Rn) | h(gx) = gh(x) ∀x ∈ Ω. Suppose that (2β) is
not true for (u, v, λ, iΩ) ∈ Q−1(0, 0, 0), that is,
dim R(DQ(u, v, λ, iΩ))
R( ∂Q∂(u,v,λ) (u, v, λ, iΩ))
=∞.
Then, there exist θ1, ..., θm ∈ L2(Ω)×L2(Ω)×L1(∂Ω) such that for all h ∈ C5g (Ω,Rn)
there exist u, v, λ and c1, ..., cm ∈ R such that
DQ(u, v, λ, iΩ)(u, v, λ, h) =
m∑j=1
cjθj , θj = (θ1j , θ
2j , θ
3j ),
that is,
(∆2 + λ)(u− h · ∇u) + λu =
m∑j=1
cjθ1j (4.4)
(∆2 + λ)(v − h · ∇v) + λv =
m∑j=1
cjθ2j (4.5)
∆u[∆(v−h·∇v)+h·∇(∆v)
]+∆v
[∆(u−h·∇u)+h·∇(∆u)
]∣∣∣∂Ω
=
m∑j=1
cjθ3j . (4.6)
Let u1, ..., up be an orthonormal basis of the eigenspace associated to theeigenvalue λ of (1.1) and consider the linear operators
A∆2+λ : L2(Ω)→ H4 ∩H10 (Ω) and C∆2+λ : H
52 (∂Ω)→ H4 ∩H1
0 (Ω) (4.7)
defined by
w = A∆2+λf + C∆2+λg ∈ H4 ∩H10 (Ω)
if (∆2 + λ)w − f ∈ [u1, ..., up], w⊥[u1, ..., up] and∂w
∂N= g on ∂Ω.
Observe that the even and odd subspaces of L2(Ω) are invariant by (4.7) [see Re-mark 3.3]. Then, by equations (4.4) and (4.5), we have
u− h · ∇u =
p∑j=1
ξjuj +
m∑j=1
cjA∆2+λθ1j − C∆2+λ
(h ·N∆u
)(4.8)
v − h · ∇v =
l∑j=1
ηiui +
m∑j=1
cjA∆2+λθ2j − C∆2+λ
(h ·N∆v
). (4.9)
[Note ∂∂N (u − h · ∇u)|∂Ω = −h ·N∆u|∂Ω and ∂
∂N (v − h · ∇v)|∂Ω = −h ·N∆v|∂Ω.]Substituting (4.8) and (4.9) in (4.6), we obtain that
∆u[h · ∇(∆v)−∆
(C∆2+λ
(h ·N∆v
))]+ ∆v
[h · ∇(∆u)−∆
(C∆2+λ
(h ·N∆u
))]∣∣∣∂Ω
(4.10)
stays in a finite dimensional subspace when h varies in C5g (Ω,Rn). Now, since
iΩ ∈ DgM , it follows from lemma 4.3 that ∆u 6= 0 in a nonempty open set U ⊂ ∂Ω.
Therefore, ∆u∆v ≡ 0 on ∂Ω implies ∆v ≡ 0 on U . It follows then from (4.10) that∆u(h · ∇(∆v)
)−∆v∆
(C∆2+λ
(h ·N∆u
))∣∣∣∂Ω
(4.11)
EIGENVALUES ON SYMMETRIC REGIONS 13
is in a (fixed) finite dimensional subspace for h ∈ C5g (Ω,Rn) with h ≡ 0 on ∂Ω\U∪
gU. In fact, these choices of h imply that C∆2+λ(h ·N∆v) = C∆2+λ(0) belongs to
a finite dimensional subspace and ∆v(h · ∇(∆u)) ≡ 0 on ∂Ω.Now, observe that
∆u(h · ∇(∆v)
)−∆v∆
(C∆2+λ
(h ·N∆u
))∣∣∣U
= ∆u(h · ∇(∆v)
)∣∣∣U.
Therefore, by equation (4.11), the application
Σ : h→ ∆u(h · ∇(∆v)
)∣∣∣U
defined for all h ∈ C5g (Ω,Rn) satisfying h ≡ 0 on ∂Ω\U ∪ gU has finite rank.
Since ∆u 6= 0 on U and dim Ω ≥ 2, this can only occur if ∇(∆v) ≡ 0 on U ⊂ ∂Ω.From this and ∆v ≡ 0 on U it follows that ∂∆v
∂N = ∇(∆v) ≡ 0 on U . Thus, we
obtain that the eigenfunction v of (1.1) satisfies ∆v = ∂∆v∂N = 0 on U, that is, v
satisfies the hypotheses of Theorem 2.3. Therefore, v ≡ 0 on Ω, and we reach acontradiction, proving the result.
Remark 4.5. Results similar to Lemmas 4.3 and 4.4 can be obtained for BoM =u ∈ Ho(Ω) | ‖u‖ ≤M and L2
o(Ω).
Theorem 4.6. There exists a residual set of h ∈ Diff4g(Ω) such that all eigenvalues
of (1.1) restricted to the even [respectively odd] subspace are simple in h(Ω).
Proof. By Remark 4.5, we can suppose that Ω is C5-regular region. We considerhere only the problem for the even space as the other case is completely analo-gous. Consider the analytic mapping F : BeM × [−M, 0] × UgM → L2
e(Ω) defined
by F (u, λ, h) = h∗(∆2 + λ)h∗−1u where UgM = DgM\E
gM . We wish to apply the
Transversality Theorem to conclude that the set
h ∈ UgM | 0 is not regular value of (u, λ)→ F (u, λ, h)is meager and closed in UgM . The result then follows from Baire’s Theorem andProposition (4.1).
It is easy to see that hypotheses (1) and (3) of the Transversality Theorem aresatisfied. We prove (2α). We reason by contradiction. Suppose there exists acritical point (u, λ, iΩ) with F (u, λ, iΩ) = 0. Then, there exists v ∈ L2
R(∆2 +λ)⊥ = N (∆2 +λ). Since ∂Ω is C5, we have that v ∈ C4,α(Ω)∩H5∩H20 (Ω).
Moreover, keeping u = h = 0, we see∫
Ωvu = 0 and allowing h to vary in C5
g (Ω,Rn)we obtain
0 =
∫Ω
(h · ∇u)(∆2 + λ)v − v(∆2 + λ)(h · ∇u)
=
∫∂Ω
(h · ∇u)∂
∂N(∆v)−∆v
∂
∂N(h · ∇u)− v ∂
∂N(∆(h · ∇u)) + ∆(h · ∇u)
∂v
∂N
=
∫∂Ω
(h · ∇u)∂
∂N(∆v)−∆v
∂
∂N(h · ∇u)
14 PEREIRA AND PEREIRA
=
∫∂Ω
h ·N ∂∆v
∂N
∂u
∂N−∆v
∂
∂N
(h ·N ∂u
∂N
)
= −∫∂Ω
h ·N∆v∆u ∀h ∈ C5g (Ω,Rn).
Therefore our hypothesis imply the existence of two even eigenfunctions u andv ∈ C4,α(Ω) ∩H5 ∩H2
0 (Ω) of (1.1) satisfying∫∂Ω
h ·N∆u∆v = 0 (4.13)
for all h ∈ C5(Ω,Rn) with h(gx) = gh(x) in Ω. Now, by Lemma 4.7 below, we can
approximate the even function ∆u∆v by a function of the form h · N in L1(∂Ω).So, (4.13) implies that ∆u∆v ≡ 0 on ∂Ω. Since iΩ ∈ UgM , we reach a contradiction.
Observe that in the case where both eigenfunctions are odd, we also have ∆u∆veven on ∂Ω.
Lemma 4.7. Let g2 = I, g 6= I, g(Ω) = Ω and f : ∂Ω → R be a continuous evenfunction ( f g = f).
Then there exists V : Rn → Rn of class C∞ with V (gx) = gV (x) in Ω such thatV · N |∂Ω is uniformly close to f . If f ∈ Lp(∂Ω), 1 ≤ p < ∞, the approximationcan be done in Lp(∂Ω).
Proof. See [6].
The auxiliary result below will be used in the proof of generic simplicity of theeigenvalues. We show that the existence of eigenfunctions u and v of (1.1), evenand odd respectively, satisfying the additional property (∆u)2 = (∆v)2 on ∂h(Ω)can only happen if h is outside an open dense subset of Diff5
g(Ω).
Lemma 4.8. Let Ω ⊂ Rn be an open, connected, bounded, C5-regular. Considerthe differentiable application
Proof. We proceed as in the proof of the Lemma 4.4. We verify condition (2β) ofthe Theorem 2.4 showing that
dim R(DG(u, v, λ, iΩ))
R( ∂G∂(u,v,λ) (u, v, λ, iΩ))
=∞.
Suppose, by contradiction, this is not true for (u, v, λ, iΩ) ∈ G−1(0, 0, 0). Then,
there exist θ1, ..., θm ∈ L2e(Ω)×L2
o(Ω)×L1(∂Ω) such that ∀h ∈ C5(Ω,Rn) satisfying
EIGENVALUES ON SYMMETRIC REGIONS 15
h(gx) = gh(x) there exist u, v, λ and c1, ..., cm ∈ R such that
DG(u, v, λ, iΩ)(u, v, λ, h) =
m∑i=1
cjθj , θj = (θ1j , θ
2j , θ
3j ),
that is,
(∆2 + λ)(u− h · ∇u) + λu =
m∑i=1
cjθ1j (4.14)
(∆2 + λ)(v − h · ∇v) + λv =
m∑i=1
cjθ2j (4.15)
m∑i=1
cjθ3j =
h · ∇
((∆u)2 − (∆v)2
)+2[∆u∆(u− h · ∇u)−∆v∆(v − h · ∇v)
]∣∣∣∂Ω. (4.16)
Let u1, ..., up, v1, ..., vl be an orthonormal basis for eigenspace associated to theeigenvalue λ th of (1.1) where u1, ..., up are even and v1, ..., vl are odd eigen-functions. Now, observe that the even and odd subspaces of L2(Ω) are invariant bythe operators
A∆2+λ : L2(Ω)→ H4 ∩H10 (Ω) and C∆2+λ : H
52 (∂Ω)→ H4(Ω) ∩H1
0 (Ω)
defined in (4.7). So, we obtain from equations (4.14) and (4.15) that
u− h · ∇u =
p∑i=1
ξiui +
m∑i=1
ciA∆2+λθ1j − C∆2+λ
(h ·N∆u
)(4.17)
v − h · ∇v =
l∑i=1
ξivi +
m∑i=1
ciA∆2+λθ2j − C∆2+λ
(h ·N∆v
). (4.18)
Substituting (4.17) and (4.18) in equation (4.16) we haveh·∇
[(∆u)2−(∆v)2
]+2[∆v∆
(C∆2+λ
(h·N∆v
))−∆u∆
(C∆2+λ
(h·N∆u
))]∣∣∣∂Ω
belongs to a finite dimensional space when h varies in C5g (Ω,Rn). Since
(∆u)2 − (∆v)2 = 0 on ∂Ω (4.19)
we can conclude that operator
Θ(h) =h ·N ∂
∂N
[(∆u)2 − (∆v)2
]+2[∆v∆
(C∆2+λ
(h ·N∆v
))−∆u∆
(C∆2+λ
(h ·N∆u
))]∣∣∣∂Ω
(4.20)
defined for all h ∈ C5(Ω,Rn) with h(gx) = gh(x) in Ω has finite rank.We show in section 4.1 that, if dim Ω ≥ 2, a necessary condition for Θ to be of
finite rank is∂
∂N
[(∆u)2 − (∆v)2
]∣∣∣∂Ω≡ 0. (4.21)
Now, from (4.21), we obtain
(∆u−∆v)∂
∂N(∆u+ ∆v) + (∆u+ ∆v)
∂
∂N(∆u−∆v) = 0 on ∂Ω. (4.22)
16 PEREIRA AND PEREIRA
Since u and v are even and odd eigenfunctions of (1.1) satisfying (4.19) and iΩ ∈ZgM , we have by Lemma 4.3 that there exists a nonempty open set U ⊂ ∂Ω suchthat
∆u−∆v 6= 0 on U and ∆u+ ∆v = 0 on U. (4.23)
Therefore, we may conclude from equations (4.22) and (4.23) that
∆(u+ v) =∂
∂N∆(u+ v) ≡ 0 on U.
Thus, the eigenfunction w = u+v of (1.1), satisfies the hypotheses of Theorem 2.3,that is, w ≡ 0 on Ω. Then u = v = 0 on Ω and we obtain a contradiction, provingthe result.
Theorem 4.9. There exists a residual set of h ∈ Diff4g(Ω) such that all the eigen-
values of (1.1) are simple in h(Ω).
Proof. By Remark 4.5, we may suppose Ω is a C5-regular region. Let λ be aneigenvalue of (1.1) in Ω. We can assume, by Theorem 4.6, that λ is a simpleeigenvalue in each subspace L2
e(Ω) and L2o(Ω). So, if λ is not simple, there exist
eigenfunctions u and v of (1.1) corresponding to λ with u = u g and −v = v gin Ω, and
∫Ωu2 =
∫Ωv2 = 1.
From [6][Example 3.2], we have that a small perturbation h(t, x) = x+ tV (x) ofΩ [V (gx) = gV (x) and t ∈ (−ε, ε) for some ε ∈ R] moves the eigenvalue λ to twobranches of simple eigenvalues
λe(t) = λ+ t
∫Ω
V ·N( ∂2u
∂N2
)2
+O(t2) and (4.24)
λo(t) = λ+ t
∫Ω
V ·N( ∂2v
∂N2
)2
+O(t2). (4.25)
Thus, the eigenvalue λ split into two simple eigenvalues by symmetric perturbationof Ω, unless ( ∂2u
∂N2
)2
=( ∂2v
∂N2
)2
on ∂Ω.
Let w+ = u + v and w− = u − v. Then w+ and w− are eigenfunctions of (1.1) inΩ with
∫Ωw2
+ =∫
Ωw2− = 2 and
∂2w+
∂N2
∂2w−
∂N2 =( ∂2u
∂N2 +∂2v
∂N2
)( ∂2u
∂N2 −∂2v
∂N2
)=
( ∂2u
∂N2
)2
−( ∂2v
∂N2
)2
= (∆u)2 − (∆v)2 = 0 on ∂Ω (4.26)
since for all u, v ∈ H4∩H20 (Ω) we have ∂2
∂N2 |∂Ω = ∆|∂Ω. Now, by Lemma 4.8, (4.26)
can only occur in a closed meager subset of Diff5g(Ω). Therefore, there exists an
open dense set of h ∈ Diff4g(Ω) such that all the eigenvalues of (1.1) with modulus
≤M are simple in h(Ω). Thus, for a residual set of h ∈ Diff4g(Ω) all the eigenvalues
of (1.1) are simple in the Z2-symmetric region h(Ω).
EIGENVALUES ON SYMMETRIC REGIONS 17
4.1. The operator Θ. To conclude the proof of the Theorem 4.9, it is necessaryto show that, if the operator
Θ(h) =h ·N ∂
∂N
[(∆u)2 − (∆v)2
]+2[∆v∆
(C∆2+λ
(h ·N∆v
))−∆u∆
(C∆2+λ
(h ·N∆u
))]∣∣∣∂Ω
defined in (4.20) has finite rank, then (4.21) holds. To this aim, we use the Methodof Rapidly Oscillating Solutions developed in [15]. The argument is very similar tothe example treated in detail there, so we only sketch the proof here.
We show that
Θ(
cos(wθ))
= cos(wθ)∂
∂N
[(∆u)2 − (∆v)2
]∣∣∣∂Ω
+O(w−1) (4.27)
as w → +∞ where θ is a smooth real function on ∂Ω with |∇θ| ≡ 1. So, if Θis assumed to have finite rank, we obtain the identity (4.21) from Lemma 4.10below. To obtain (4.27), we apply the methods of [15] to compute the approximatesolutions of [
∆v∆C∆2+λ(ewiθ∆v)−∆u∆C∆2+λ(ewiθ∆u)]∣∣∣∂Ω
(4.28)
as w → +∞.
Let ewS∑Nk=0
Uk(2w)k
be the approximate value of C∆2+λ
(ewiθ∆u
)given by equa-
tion (13) of [15]. From the [15, Theorem 2], we obtain
C∆2+λ
(ewiθ∆u
)= ewSU0 +O(w−1)
in a neighborhood of ∂Ω ∩ Ω. Therefore, using the ‘normal coordinates’ given byx = y + tN(y), where y ∈ ∂Ω and t ∈ (−r, r) with r > 0 small, we have
∆C∆2+λ
(ewiθ∆u
)∣∣∣∂Ω
=(H
∂
∂N+
∂2
∂N2
)(ewSU0
)∣∣∣∂Ω
+O(w−1)
=(H∂t + ∂tt
)[ewS
(U0
0 + U10 t+ U2
0
t2
2!+ ...
)]∣∣∣t=0
+O(w−1)
= ewiθ(
(2w)U10 +HU1
0 + U20
)+O(w−1).
Choosing
Gk = U1k =
∆u|∂Ω k = 00 k > 0
we obtain
∆C∆2+λ
(ewiθ∆u
)∣∣∣∂Ω
= ewiθ(
∆u(2w) +[H∆u+ U2
0
])∣∣∣∂Ω
+O(w−1).
Analogously we have (following the notation of [15])
∆C∆2+λ
(ewiθ∆v
)∣∣∣∂Ω
= ewiθ(
∆v(2w) +[H∆v + V 2
0
])∣∣∣∂Ω
+O(w−1).
Thus, we may conclude from equation (4.19) that[∆v∆C∆2+λ(ewiθ∆v)−∆u∆C∆2+λ(ewiθ∆u)
]∣∣∣∂Ω
=
= ewiθ[(∆v)2(2w +H) + ∆vV 2
0 − (∆u)2(2w +H)−∆uU20
]∣∣∣∂Ω
+O(w−1)
18 PEREIRA AND PEREIRA
= ewiθ[∆vV 2
0 −∆uU20
]∣∣∣∂Ω
+O(w−1). (4.29)
Moreover, we may compute U20 and V 2
0 as in [15], obtaining
U20 =
(q − α− 2i
∂
∂θ
)∆u∣∣∣∂Ω
and V 20 =
(q − α− 2i
∂
∂θ
)∆v∣∣∣∂Ω.
Therefore, from (4.19), we have(∆vV 2
0 −∆uU20
)∣∣∣∂Ω
=
(q − α)[(∆v)2 − (∆u)2
]−2i
∂
∂θ
[(∆v)2 − (∆u)2
]∣∣∣∂Ω
= −2i∇∂Ωθ · ∇∂Ω
[(∆v)2 − (∆u)2
]∣∣∣∂Ω
= 0 on ∂Ω.
Thus, we obtain[∆v∆C∆2+λ(ewiθ∆v)−∆u∆C∆2+λ(ewiθ∆u)
]∣∣∣∂Ω
= O(w−1) (4.30)
as w → +∞. Since[∆v∆C∆2+λ(cos(wθ)∆v)−∆u∆C∆2+λ(cos(wθ)∆u)
]∣∣∣∂Ω
=
Re[
∆v∆C∆2+λ(ewiθ∆v)−∆u∆C∆2+λ(ewiθ∆u)]∣∣∣∂Ω
(4.27) follows from (4.30).
Lemma 4.10. Suppose S is a C1 manifold; A and B ∈ L2(S) with compact support;θ is C1 on S and real valued with ∇Sθ 6= 0 in supp A ∪ supp B; E is a finitedimensional subspace of L2(S) and u(w) ∈ E for all w ∈ R satisfying
u(w) = A cos(wθ) +B sin(wθ) + o(1) on L2(S)
as w → +∞. Then A = B = 0 on S.
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Antonio L. PereiraInstituto de Matematica e Estatıstica, Universidade de Sao Paulo - Sao Paulo - Brazil
Marcone C. PereiraEscola de Artes, Ciencias e Humanidades, Universidade de Sao Paulo - Sao Paulo -
