GENERIC SIMPLICITY OF THE EIGENVALUES
FOR A SUPPORTED PLATE EQUATION
MARCONE C. PEREIRA
Abstract. In this work, we show that the eigenvalues of the problem

(∆2 + V (x) + λ)u(x) = 0 x ∈ Ω
u(x) = ∆u(x) = 0
x ∈ ∂Ω
are generically simple in the set of C 4 regular regions of Rn , n ≥ 2. In fact, we
prove that there exists a residual set of regions C 4 diffeomorphic to a given Ω
such that all the eigenvalues for a supported plate equation are simple.
1. Introduction
n
Let Ω ⊂ R , n ≥ 2, be an open bounded region with smooth boundary, V (·) a
C α function defined in Ω, and consider the eigenvalue problem
(∆2 + V (x) + λ)u(x) = 0 x ∈ Ω
(1.1)
u(x) = ∆u(x) = 0
x ∈ ∂Ω.
In this note, we show the generic simplicity of all the eigenvalues of (1.1). More
precisely, we prove that there exists a residual set of regions (in a suitable topology)
such that all the eigenvalues of (1.1) are simple.
As was pointed to me by J. M. Arrieta, if the potential V ≡ 0 in Ω, the generic
simplicity of the eigenvalues of (1.1) follows from the generic simplicity of the eigenvalues of the Dirichlet problem for the Laplacian (which was proved by Micheletti
[3] and Uhlenbeck [15]). In fact, if V ≡ 0, the problem (1.1) is the square of the
Dirichlet problem for the Laplacian. Therefore, they must have the same eigenfunctions.
Generic properties for boundary value problems in PDE’s arise in many contexts
and have been investigated by several authors from various points of view. We refer
to C. Rocha [12], Saut and Teman [13] and A. L. Pereira [5], [7] for perturbations
of the nonlinearity of parabolic equations and [8], [9] and [10] for perturbation of
the domain of semilinear elliptic problems.
Several works on boundary perturbations appeared in the literature using the
concept of “shape differentiation” or “shape analysis”. Among others, we mention Ortega and Zuazua [4] for a example of a stabilization problem where generic
simplicity of the eigenvalues was successfully applied (see also [14] and [16]).
One of the difficulties here is that the functional spaces change as we change the
region. Our first task is then to find a way to deal of problem (1.1) in different
regions. One possible approach is the one taken by Henry in [1] where a kind of
differential calculus with the domain as the independent variable was developed.
2000 Mathematics Subject Classification. 35J40,35B30,58C40.
Key words and phrases. biharmonic equations; bilaplacian operator; generic properties; boundary perturbations.
1
2
M. C. PEREIRA
This approach allows the utilization of standard analytic tools such as Implicit
Function Theorem and the Lyapunov-Schmidt method. In his work, Henry also
formulated and proved a generalized form of the Transversality Theorem, which is
the main tool we use in our arguments. Following Henry’s approach, A. L. Pereira
[6] obtained results on the eigenvalues of the Dirichlet problem for the Laplacian
operator on symmetric regions. It is likely that analogous results can be proved for
plate equations. This ought to be the subject of a future study.
This paper is organized as follows: in section 2, we collect some background
results that we need from [1], and in the section 3, we show the main result of the
paper.
2. Preliminaries
The results in this section were taken from the monograph of Henry [1], where
full proofs can be found.
2.1. Differential Calculus. Given an open bounded, C m region Ω ⊂ Rn , consider
the following open subset of C m (Ω, Rn )
Diff m (Ω)
= {h ∈ C m (Ω, Rn ) | h is injective and 1/| det h0 (x)| is bounded in Ω}.
We introduce a topology in the collection of all regions {h(Ω) | h ∈ Diff m (Ω)}
by defining (a sub-basis of) the neighborhoods of a given Ω̃ by
{h(Ω̃) | kh − iΩ̃ kC m (Ω̃,Rn ) < , sufficiently small}.
When kh − iΩ kC m (Ω,Rn ) is small, h is a C m imbedding of Ω in Rn , a C m diffeomorphism to its image h(Ω). Micheletti shows in [3] that this topology is metrizable, and the set of regions C m diffeomorphic to Ω may be considered a separable metric space which we denote by Mm (Ω), or simply Mm . We say that a
function F defined in the space Mm with values in a Banach space is C m or analytic if h 7→ F (h(Ω)) is C m or analytic as a map of Banach spaces (h near iΩ in
C m (Ω, Rn )). In this sense, we may express problems of perturbation of the boundary of a boundary value problem as problems of differential calculus in Banach
spaces. More specifically,
consider
a formal non-linear differential operator u 7→ v
given by v(x) = f x, u(x), Lu(x) , x ∈ Rn where
∂u
∂2u
∂2u
∂u
Lu(x) = u(x),
(x), ...,
(x), 2 (x),
(x), ... , x ∈ Rn .
∂x1
∂xn
∂x1
∂x1 ∂x2
More precisely, suppose Lu(·) has values in Rp and f (x, λ) is defined for (x, λ)
in some open set O ⊂ Rn × Rp . For subsets Ω ⊂ Rn define FΩ by
FΩ (u)(x) = f (x, Lu(x)), x ∈ Ω
(2.1)
for sufficiently smooth functions u in Ω such that (x, Lu(x)) ∈ O for any x ∈ Ω̄.
Let h : Ω 7→ Rn be C m imbedding. We define the composition map (or pull-back ) h∗
of h by h∗ u(x) = (u ◦ h)(x) = u(h(x)), x ∈ Ω where u is a function defined in h(Ω).
Then h∗ is an isomorphism from C m (h(Ω)) to C m (Ω) with inverse h∗ −1 = (h−1 )∗ .
The same is true in other function spaces.
The differential operator Fh(Ω) : DFh(Ω) ⊂ C m (h(Ω)) 7→ C 0 (h(Ω)) given by
(2.1) is called the Eulerian form of the formal operator v 7→ f (·, Lv(·)), whereas
h∗ Fh(Ω) h∗ −1 : h∗ DFh(Ω) ⊂ C m (Ω) 7→ C 0 (Ω) is called the Lagrangian form of the
same operator.
GENERIC SIMPLICITY OF THE EIGENVALUES
3
The Eulerian form is often simpler for computations, while the Lagrangian form
is usually more convenient to prove theorems, since it acts in spaces of functions
that do not depend on h, facilitating the use of standard tools such as the Implicit Function or the Transversality Theorem. However, a new variable, h is
introduced. We then need to study the differentiability properties of the map
(u, h) 7→ h∗ Fh(Ω) h∗ −1 u.
This has been done in [1] where it is shown that, if (y, λ) 7→ f (y, λ) is C k or
analytic then so is the map above, considered as a map from Diff m (Ω) × C m (Ω) to
C 0 (Ω) (other function spaces can be used instead of C m ). To compute the derivative we then need only compute the Gateaux derivative that is, the t-derivative
along a smooth curve t 7→ (h(t, .), u(t, .)) ∈ Diff m (Ω) × C m (Ω). For this purpose, it
∂
∂
−U (x, t) ∂x
, U (x, t) =
is convenient to introduce the differential operator Dt = ∂t
−1
∂h
∂h
∂x
∂t which is is called the anti-convective derivative. The results below (Theorems 2.1 and 2.4) are the main tools we use to compute derivatives.
Theorem 2.1. Suppose f (t, y, λ) is C 1 in an open set in R × Rn × Rp , L is a
constant-coefficient differential operator of order ≤ m with Lv(y) ∈ Rp (where
defined). For open sets Q ⊂ Rn and C m functions v on Q, let FQ (t)v be the
function
y 7→ f (t, y, Lv(y)), y ∈ Q.
where defined.
Suppose t 7→ 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) 7→ ∂t ∂xj h(t, x), ∂xk h(t, x), ∂xk u(t, x) are
−1
continuous on R × Ω near t = 0, and h(t, ·)∗ u(t, ·) is in the domain of FΩ(t) .
Then, at points of Ω
0
Dt (h∗ FΩ(t) (t)h∗ −1 )(u) = (h∗ ḞΩ(t) (t)h∗ −1 )(u) + (h∗ FΩ(t)
(t)h∗ −1 )(u) · Dt u
where Dt is the anti-convective derivative defined above,
ḞQ (t)v(y) =
∂f
(t, y, Lv(y))
∂t
and
FQ0 (t)v · w(y) =
∂f
(t, y, Lv(y)) · Lw(y), y ∈ Q
∂λ
is the linearisation of v 7→ FQ (t)v.
α
P
∂
Example 2.2. Suppose we deal with a linear operator A = |α|≤m aα (y) ∂y
not explicitly dependent on t, and h(t, x) = x + tV (x) + o(t) as t → 0 and x ∈ Ω.
Then at t = 0
∂ ∗ ∗ −1 ∗
∗ −1 (h Ah
u)
= Dt (h∗ Ah∗ −1 u)
+ h−1
u)
x ht ∇(h Ah
∂t
t=0
t=0
t=0
∂u
− V · ∇u + V · ∇(Au)
= A
∂t
∂u
= A
+ [V · ∇, A]u
∂t
since
∂
∂t A
= 0.
4
M. C. PEREIRA
We also need to be able to differentiate boundary conditions, and a quite general
form is
b(t, y, Lv(y), M NΩ(t) (y)) = 0 for y ∈ ∂Ω(t),
where L, M are constant-coefficient differential operators and NΩ(t) (y) is the outward unit normal for y ∈ ∂Ω(t), extended smoothly as a unit vector field on a
neighborhood of ∂Ω(t). We choose some extension of NΩ in the reference region
and then define NΩ(t) = Nh(t,Ω) by
h∗ Nh(t,Ω) (x) = Nh(t,Ω) (h(x)) =
T
(h−1
x ) NΩ (x)
−1 T
k(hx ) NΩ (x)k
(2.2)
T
for x near ∂Ω, where (h−1
x ) is the inverse-transpose of the Jacobian matrix hx and
k · k is the Euclidean norm. This is the extension understood in the above boundary
condition: b(t, y, Lv(y), M NΩ(t) (y)) is defined for y ∈ Ω near ∂Ω and has limit zero
(in some sense, depending on the functional space employed) as y → ∂Ω.
Lemma 2.3. Let Ω be a C 2 -regular region, NΩ(·) a C 1 unit-vector field defined on
a neighborhood of ∂Ω which is the outward normal on ∂Ω, and for a C 2 function
h : Ω 7→ Rn define Nh(Ω) on a neighborhood of h(∂Ω) = ∂h(Ω) by (2.2) above.
Suppose h(t, ·) is an imbedding for each t, defined by
∂
h(t, x) = V (t, h(t, x)) for x ∈ Ω, h(0, x) = x,
∂t
(t, y) 7→ V (t, y) is C 2 and Ω(t) = h(t, Ω), NΩ(t) = Nh(t,Ω) . Then for x near
∂
∂Ω, y = h(t, x) near ∂Ω(t), we may compute the derivative ( ∂t
)y=constant and, if
y ∈ ∂Ω,
∂
NΩ(t) (y)
∂t
= Dt (h∗ Nh(t,Ω) )(x)
∂NΩ(t) = − ∇∂Ω(t) σ + σ
(y)
∂NΩ(t)
where σ = V · NΩ(t) is the normal velocity and ∇∂Ω(t) σ is the component of the
gradient tangent to ∂Ω.
Theorem 2.4. Let b(t, y, Lv(y), M NΩ(t) (y)) be a C 1 function on an open set of
R×Rn ×Rp ×Rq and let L, M be constant-coefficient differential operators with order
≤ m of appropriate dimensions so b(t, y, Lv(y), M NΩ(t) (y)) makes sense. Assume
that Ω is a C m+1 region, NΩ (x) is a C m unit-vector field near ∂Ω which is the
outward normal on ∂Ω, and define Nh(t,Ω) by (2.2) when h : Ω 7→ Rn is a C m+1
smooth imbedding. Also define Bh(Ω) (t) by
Bh(Ω) v(y) = b(t, y, Lv(y), M Nh(Ω) (y))
for y ∈ h(Ω) near ∂h(Ω).
If t 7→ h(t, ·) is a curve of C m+1 imbeddings of Ω and for |j| ≤ m, |k| ≤ m + 1,
(t, x) 7→ (∂t ∂xj h, ∂xk , ∂t ∂xj u, ∂xk u)(t, x) are continuous on R × Ω near t = 0, then at
points of Ω near ∂Ω
Dt (h∗ Bh(Ω) h∗ −1 )(u)
=
0
(h∗ Ḃh(Ω) h∗ −1 )(u) + (h∗ Bh(Ω)
h∗ −1 )(u) · Dt u
+
(h∗
∂Bh(Ω) ∗ −1
h
)(u) · Dt (h∗ NΩ(t) )
∂N
GENERIC SIMPLICITY OF THE EIGENVALUES
5
0
where h = h(t, ·); Ḃh(Ω) e Bh(Ω)
are defined as in Theorem 2.1,
∂Bh(Ω)
∂b
(v) · n(y) =
(t, y, Lv(y), M Nh(Ω) (y)) · M n(y)
∂N
∂µ
and Dt (h∗ NΩ(t) ) is computed in Lemma 2.3.
∂Ω
2
∂
Example 2.5. Now, consider the following boundary linear operator B = ∂N
2 not
explicitly dependent on t, and h(t, x) = x + tV (x) + o(t) as t → 0 and x ∈ Ω. Then,
by Theorem 2.4, we obtain at t = 0
∂ ∗ ∗ −1 ∗
∗ −1 u)
+ h−1
h
∇(h
Bh
= Dt (h∗ Bh∗ −1 u)
(h Bh
u)
t
x
∂t
t=0
t=0
t=0
∂u
∂u = B
· Ṅ
− V · ∇u + ∇
∂t
∂N
+∇ ∇u · Ṅ · N + V · ∇(Bu)
∂u ∂u
· Ṅ + ∇ ∇u · Ṅ · N
= B
+ [V · ∇, B]u + ∇
∂t
∂N
where Ṅ =
∂
∂t N is given by Lemma 2.3.
t=0
2.2. Change of origin. We can always transfer the ‘origin’ or reference region
from any Ω ⊂ Rn , to another diffeomorphic region. Indeed, if H̃ : Ω → Ω̃ is
a diffeomorphism we define, for any imbedding h : Ω 7→ Rn , another imbedding
h̃ = h ◦ H̃ −1 : Ω̃ 7→ Rn .
H̃ t N (x)
If x̃ = H̃(x), ũ = H̃ −1 ∗ u NΩ̃ (x̃) = NH̃(Ω) (H̃(x)) = kH̃xt NΩ (x)k then h(Ω) = h̃(Ω̃),
x
Ω
h∗ Fh(Ω) h∗ −1 u(x) = h̃∗ Fh̃(Ω̃) (h̃∗ )−1 ũ(x̃), h∗ Bh(Ω) h∗ −1 u(x) = h̃∗ Bh̃(Ω̃) (h̃∗ )−1 ũ(x̃),
using the normal Nh̃(Ω̃) (h̃(x̃)) = Nh(Ω) (h(x)).
This ‘change of origin’ will be frequently used in the sequel, as it allow us to
compute derivatives with respect to h at h = iΩ , where the formulas are simpler.
2.3. Uniqueness Theorem. We often need the following uniqueness theorem for
the Bilaplacian.
Theorem 2.6. Let Ω ⊂ Rn be an open, connected, bounded, C 4 -regular region and
B a ball which meets ∂Ω in a C 4 hypersurface B ∩ ∂Ω. Assume u ∈ H 4 (Ω) and for
some constant K
|∆2 u| ≤ K |∆u| + |∇u| + |u| a.e. Ω with
(2.3)
∂u
∂2u
∂3u
u=
=
=
=
0
on
B
∩
∂Ω.
∂N
∂N 2
∂N 3
Then u ≡ 0 in Ω.
This theorem follows from [2, Theorem 8.9.1].
2.4. The Transversality Theorem. A basic tool for our results will be the
Transversality Theorem in the form below, due to D. Henry [1]. We first recall
some definitions.
A map T ∈ L(X, Y ) where X and Y are Banach spaces is a semi-Fredholm map
if 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
6
M. C. PEREIRA
index(T ) = ind(T ) = dim N (T ) − codim R(T ).
Let P be a property depending of a parameter x ∈ X, where X is a Baire
topological space. We say that P is generic (in x) if it holds for all x in a residual
set of X.
Let now X be a Baire space and I = [0, 1]. For any closed or σ-closed F ⊂ X
and any nonnegative integer m we say that the codimension of F is greater or
equal to m (codim F≥ m) if the subset {φ ∈ C(I m , X) | φ(I m ) ∩ F is non-empty }
is meager in C(I m , X). We say codim F = k if k is the largest integer satisfying
codim F ≥ m.
Theorem 2.7. Suppose given positive numbers k and m; Banach manifolds X, Y, Z
of class C k ; an open set A ⊂ X × Y ; a C k map f : A → Z and a point ξ ∈ Z.
Assume for each (x,y) ∈ f −1 (ξ) that:
(1) ∂f
∂x (x, y) : Tx X 7→ Tξ Z is semi-Fredholm with index < k.
(2) (α) Df (x, y) : Tx X × Ty Y 7→ Tξ Z is surjective
or n
o
(x,y))
(β) dim R(Df
≥ m + dim N ( ∂f
∂f
∂x (x, y)).
R(
(x,y))
∂x
(3) (x, y) 7→ y : f −1 (ξ) 7→ Y is σ-proper, that is f −1 (ξ) is a countable union of
sets 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 Lindelöf [every open cover has a countable
subcover] or, more specifically, if f −1 (ξ) is a separable metric space, or if X, Y are
separable metric spaces.
Let Ay = {x | (x, y) ∈ A} and
Ycrit = {y | ξ is a critical value of f (·, y) : Ay 7→ Z}.
−1
Then Ycrit is a meager
set in Y and, if (x,y) 7→ y : f (ξ) 7→ Y is proper, Ycrit is
∂f
also closed. If ind ∂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 iff codim Ycrit ≥ 1].
3. Genericity simplicity of the eigenvalues
We show in this section that the eigenvalues of the problem (1.1) are generically
simple in the set of open, connected, bounded C 4 regular regions of Rn , n ≥ 2.
More precisely, we show that the set
{h ∈ Diff 4 (Ω) | all the eigenvalues of (1.1) are simple in h(Ω)}
is a residual set in Diff 4 (Ω).
In order to apply transversality arguments, we first show that our generic property is equivalent of zero being a regular value for an appropriate mapping. More
precisely, we have
Proposition 3.1. Let Ω ⊂ Rn be an open, connected, bounded, C 4 -regular region,
V (·) a C α function defined in Ω and h ∈ Diff 4 (Ω). Then, all eigenvalues of (1.1)
GENERIC SIMPLICITY OF THE EIGENVALUES
7
are simple in h(Ω) if and only if zero is a regular value of the mapping Φh : H 4 ∩
3
H01 (Ω) × R 7→ L2 (Ω) × H 2 (∂Ω) defined by
Φh (u, λ) = h∗ (∆2 + V + λ)h∗ −1 u, h∗ ∆h∗ −1 u|∂Ω .
Proof. Observe that 0 is a regular value of Φh if and only if for all (u, λ) ∈ H 4 ∩
H01 (Ω) × R with Φh (u, λ) = 0
DΦh (u, λ)(u̇, λ̇) = h∗ (∆2 + V + λ)h∗ −1 u̇ + λ̇u, h∗ ∆h∗ −1 u|∂Ω
3
is onto since L0 : H 4 ∩ H01 (Ω) 7→ L2 (Ω) × H 2 (∂Ω) : u 7→ ( (∆2 + V + λ)u, ∆u|∂Ω )
is Fredholm with index zero and h∗ and h∗ −1 are isomorphisms. Let π : L2 (Ω) ×
3
H 2 (∂Ω) 7→ L2 (Ω) be the linear map given by π(u, ϕ) = u. Thus, Dφ(u, λ) is onto
if and only if
h
i
π R(∆2 + V + λ) ⊕ [u] = L2 (Ω),
that is, if and only if λ is simple eigenvalue of (3.1). It remains only to prove that
the problems (1.1) in h(Ω) is equivalent to
∗ 2
h (∆ + V + λ)h∗ −1 u(x) = 0 x ∈ Ω
(3.1)
u(x) = h∗ ∆h∗ −1 u(x) = 0
x ∈ ∂Ω
for all h ∈ Diff 4 (Ω). Let u ∈ H 4 ∩ H01 (Ω). Since h∗ and h∗ −1 are isomorphisms, we
have
h∗ (∆2 + V + λ)h∗ −1 u = 0 ⇐⇒ (∆2 + V + λ)h∗ −1 u = 0
and
h∗ ∆h∗ −1 u |∂Ω = 0 ⇐⇒ ∆h∗ −1 u |∂h(Ω) = 0.
It is clear that u = 0 in ∂Ω if and only if h∗ −1 u = 0 in ∂h(Ω). Thus u is a solution
of (3.1) if and only if v = h∗ −1 u is a solution of (1.1) in h(Ω).
It follows from the Proposition 3.1 that, in order to show generic simplicity of
the solutions of (1.1) is enough to show that 0 is a regular value of Φh , generically
in h ∈ Diff 4 (Ω). We show, using the Transversality Theorem, that 0 is a regular
value of
3
7→ L2 (Ω) × H 2 (∂Ω)
(u, λ, h) 7→
h∗ (∆2 + V + λ)h∗ −1 u, h∗ ∆h∗ −1 u|∂Ω (3.2)
F : BM × [−M, 0] × VM
where BM = {u ∈ H 4 ∩ H01 (Ω)\{0} | kuk ≤ M } and VM is an open dense set in
Diff 4 (Ω), for all M ∈ N. Taking the intersection of VM for M ∈ N, we obtain by
Baire Theorem the desired residual set.
Remark 3.2. For each M ∈ N, the set of h ∈ Diff 4 (Ω) such that all eigenvalues
λ ∈ (−M, 0) of (1.1) in h(Ω) is an open set in Diff 4 (Ω). In fact, there are only
finitely many such eigenvalues and each depends continuously on h (see [11, section
4]). To prove density, we may work with more regular (for example C ∞ ) regions.
If we try to apply the Transversality Theorem directly to the function F defined
in (3.2) we do not obtain a contradiction. What we do obtain is that the possible
critical points must satisfy very special properties. The idea is then to show that
these properties can only occur in a small (meager and closed) set and then restrict
8
M. C. PEREIRA
the problem to its complement. In our case the ‘exceptional situation’ is characterized by the existence of solutions u and v of (1.1) associated with the eigenvalue
∂ 2 u ∂ 2 ∆v
λ satisfying the additional property ∂N
2 ∂N 2 ≡ 0 on ∂Ω. We show in Lemma 3.4
that this situation is really ‘exceptional’, that is, it can only happen when h is
outside an open dense subset of Diff 4 (Ω) (for u, v and λ restricted to a bounded
set).
We will need the following ‘generic unique continuation result’.
Lemma 3.3. Let Ω ⊂ Rn n ≥ 2 be an open, connected, bounded, C 6 regular region
and M ∈ N. Then there exists an open dense subset OM ⊂ Diff 6 (Ω) such that for
all h ∈ OM if w ∈ H 5 (h(Ω)) with kwk ≤ M satisfies
(∆2 + V (x) + λ)w(x) = 0
x ∈ h(Ω)
∂2w
x ∈ ∂h(Ω)
w(x) = ∂N
2 (x) = ∆w(x) = 0
or
(∆2 + V (x) + λ)w(x) = 0
x ∈ h(Ω)
∂ 2 ∆w
w(x) = ∆w(x) = ∂N 2 (x) = 0 x ∈ ∂h(Ω)
for some λ ∈ [−M, 0] then w is identically null.
Proof. We not prove it here since the argument is very similar to the one of [11,
Lemma 5.3] and Lemma 3.4 below. In fact, it is enough to consider the following
applications
K : BM × [−M, 0] × Diff 5 (Ω)
7→
(u, λ, h)
7→
G : CM × [−M, 0] × Diff 6 (Ω)
7→
(v, λ, h)
7→
3
L2 (Ω) × H 2 (∂Ω) × L1 (∂Ω)
“
∂ 2 ∗−1 ”
u
h
h∗ (∆2 + V + λ)h∗−1 u, h∗ ∆h∗−1 u|∂Ω , h∗
∂N 2
and
5
H 1 (Ω) × H 2 (∂Ω) × L1 (∂Ω)
“
∂ 2 ∆ ∗−1 ”
h∗ (∆2 + V + λ)h∗−1 v, h∗ ∆h∗−1 v|∂Ω , h∗
h
v
∂N 2
where BM = {u ∈ H 4 ∩ H01 (Ω)\{0} | kuk ≤ M } and CM = {u ∈ H 5 ∩
H01 (Ω)\{0} | kuk ≤ M } and to apply the Transversality Theorem.
Lemma 3.4. Let Ω ⊂ Rn n ≥ 2 be an open, connected, bounded, C 6 regular region
and M ∈ N. Consider the map
H : BM × CM × [−M, 0] × OM
7→
3
5
L2 (Ω) × H 2 (∂Ω) × H 1 (Ω) × H 2 (∂Ω) × L1 (∂Ω)
given by
H(u, v, λ, h)
=
h∗ (∆2 + V + λ)h∗ −1 u, h∗ ∆h∗ −1 u|∂Ω , h∗ (∆2 + V + λ)h∗ −1 v,
2
∂ 2 ∗ −1
∗ ∂ ∆ ∗ −1
h∗ ∆h∗ −1 v|∂Ω , h∗
h
u
h
h
v
∂N 2
∂N 2
where BM = {u ∈ H 4 ∩H01 (Ω)\{0} | kuk ≤ M }, CM = {v ∈ H 5 ∩H01 (Ω)\{0} | kvk ≤
M } and OM is given by Lemma 3.3.
Then
QM = {h ∈ Diff 6 (Ω) | (0, 0, 0, 0, 0) ∈ H(BM × CM × [−M, 0], h)}
is a meager closed subset of Diff 6 (Ω).
GENERIC SIMPLICITY OF THE EIGENVALUES
9
Proof. We apply the Transversality Theorem. The differentiability of H is easy
to establish, and its partial derivatives can be computed using Theorem 2.4 (see
examples 2.2 and 2.5). As explained in section 2.2, we may suppose that h = iΩ .
Then
DH(u, v, λ, iΩ )(·) =
DH1 (u, λ, iΩ )(·), DH2 (u, iΩ )(·), DH3 (v, λ, iΩ )(·),
DH4 (v, iΩ )(·), DH5 (u, v, iΩ )(·)
where
DH1 (u, λ, iΩ )(u̇, λ̇, ḣ)
=
(∆2 + V + λ)(u̇ − ḣ · ∇u) + λ̇u,
DH2 (u, iΩ )(u̇, ḣ)
=
∆(u̇ − ḣ · ∇u)|∂Ω ,
DH3 (v, λ, iΩ )(v̇, λ̇, ḣ)
=
(∆2 + V + λ)(v̇ − ḣ · ∇v) + λ̇v,
DH4 (v, iΩ )(v̇, ḣ)
=
DH5 (u, v, iΩ )(u̇, v̇, ḣ)
∆(v̇ − ḣ · ∇v)|∂Ω ,
∂∆v ∂2u h ∂2∆
(v̇ − ḣ · ∇v) + ∇
· Ṅ
=
2
2
∂N ∂N
∂N
∂ 2 ∆v i
∂∆v
N · Ṅ · N + ḣ · ∇
+∇
∂N
∂N 2
∂u ∂ 2 ∆v h ∂ 2
+
(u̇ − ḣ · ∇u) + ∇
· Ṅ
∂N 2 ∂N 2
∂N
∂u
∂ 2 u i
+∇
N · Ṅ · N + ḣ · ∇
∂N
∂N 2
with Ṅ = −∇∂Ω (ḣ · N ). [Observe that Ω ⊂ Rn C 6 regular implies ḣ · ∇u ∈ H 4 (Ω)
and ḣ · ∇v ∈ H 5 (Ω).]
The hypotheses (1) and (3) of the Transversality Theorem can be verified as in
the proof of [11, Lemma 5.3]. We prove (2β) by showing that
n R(DH(u, v, λ, h)) o
=∞
dim
∂H
R ∂(u,v,λ)
(u, v, λ, h)
for all (u, v, λ, h) ∈ H −1 (0, 0, 0, 0, 0). Suppose this is not true for (u, v, λ, h) ∈
H −1 (0, 0, 0, 0, 0). ‘Changing the origin’, we may assume that h = iΩ . Then, there
3
5
exist θ1 , ..., θm ∈ L2 (Ω) × H 2 (∂Ω) × H 1 (Ω) × H 2 (∂Ω) × L1 (∂Ω) such that for all
6
n
4
1
ḣ ∈ C (Ω, R ) there exist u̇ ∈ H ∩ H0 (Ω), v̇ ∈ H 5 ∩ H01 (Ω), λ̇ ∈ [−M, 0] and
scalars c1 , ..., cm ∈ R such that
DH(u, v, λ, iΩ )(u̇, v̇, λ̇, ḣ) =
m
X
ci θi ,
θi = (θi1 , ..., θi5 )
i=1
that is,
(∆2 + V + λ)(u̇ − ḣ · ∇u) + λ̇u
∆(u̇ − ḣ · ∇u)|∂Ω
(∆2 + V + λ)(v̇ − ḣ · ∇v) + λ̇v
=
=
=
m
X
i=1
m
X
i=1
m
X
i=1
ci θi1
(3.3)
ci θi2
(3.4)
ci θi3
(3.5)
10
M. C. PEREIRA
∆(v̇ − ḣ · ∇v)|∂Ω
=
m
X
ci θi4
(3.6)
i=1
and
m
X
ci θi5
=
i=1
∂∆v ∂∆v
∂2u h ∂2∆
(
v̇
−
ḣ
·
∇v)
+
∇
·
Ṅ
+
∇
N
·
Ṅ
·N
∂N 2 ∂N 2
∂N
∂N
∂u ∂ 2 ∆v i ∂ 2 ∆v h ∂ 2
+
(
u̇
−
ḣ
·
∇u)
+
∇
· Ṅ
+ḣ · ∇
∂N 2
∂N 2 ∂N 2
∂N
∂u
∂ 2 u i
+∇
.
(3.7)
N · Ṅ · N + ḣ · ∇
∂N
∂N 2
3
4
1
2
2
Let {u1 , ..., ul
} be a basis for the kernel
of L0 : H ∩ H0 (Ω) 7→ L (Ω) × H (∂Ω)
2
given by L0 u = (∆ + V + λ)u, ∆u|∂Ω and consider the operators
3
AL0 : L2 (Ω) × H 2 (∂Ω) 7→ H 4 ∩ H01 (Ω)
defined by
w = AL0 (z, g)
2
where (∆ + V + λ)w − z belongs to N (L0 ), w⊥N (L0 ) in L2 (Ω) and ∆w = g on
∂Ω. [We can prove that these operators are well defined like in [9].]
Take the open subset J = {x ∈ ∂Ω | H(x) 6= 0} of ∂Ω, where H is the mean
curvature of ∂Ω. Since ∂Ω is a compact manifold in Rn , J can not be empty.
Moreover, by [1, Theorem 1.13], we have that
0 = ∆∂Ω (∆v) = ∆(∆v) − H
∂∆v ∂ 2 ∆v
∂ 2 ∆v
∂∆v ∂ 2 ∆v
−
= −λv − H
−
=−
2
2
∂N
∂N
∂N
∂N
∂N 2
on ∂Ω\J, that is,
∂ 2 ∆v
≡ 0 on ∂Ω\J.
∂N 2
2
So, using Lemma 3.3, there exists a nonempty open set J˜ ⊂ J such that ∂∂N∆v
2 (x) 6= 0
∂2u
∂ 2 ∆v ∂ 2 u
˜
˜
for all x ∈ J. Since ∂N 2 ∂N 2 ≡ 0 on ∂Ω, we have that ∂N 2 ≡ 0 on J. Using [1,
Theorem 1.13] again, we obtain
∂u
∂2u
∂u
−
= −H
∂N
∂N
∂N
˜ that is, ∂u ≡ 0 on J˜ since H 6= 0 on J.
˜ Therefore, if u̇ ∈ H 4 ∩ H 1 (Ω) and
on J,
0
∂N
˜ we obtain
ḣ ≡ 0 on ∂Ω\J,
0 = ∆∂Ω u = ∆u − H
u̇ − ḣ · ∇u = −ḣ · N
∂u
=0
∂N
∂u
on ∂Ω. [Note u ∈ H01 (Ω) implies ∇u = ∂N
N .]
6
n
˜ we obtain from (3.3)
Then, choosing ḣ ∈ C (Ω, R ) such that ḣ ≡ 0 on ∂Ω\J,
and (3.4) that
u̇ − ḣ · ∇u =
l
X
i=1
ξi ui +
m
X
i,j=1
ci cj AL0 (θj1 , θi2 )
(3.8)
GENERIC SIMPLICITY OF THE EIGENVALUES
11
2
∂ u
since u̇ − ḣ · ∇u ∈ H 4 ∩ H01 (Ω). Substituting (3.8) in (3.7) and using that ∂N
2 ≡ 0
˜ we conclude that
on J,
∂u
∂ 2 u i
∂ 2 ∆v h ∂u ∇
·
Ṅ
+
∇
N
·
Ṅ
·
N
+
ḣ
·
∇
˜
∂N 2
∂N
∂N
∂N 2
J
∂ 2 ∆v h ∂ 2 u
∂2u
∂2u ∂ 3 u i
=
N · Ṅ +
N · Ṅ +
∇ N · Ṅ · N + ḣ · N
2
2
2
2
∂N ∂N
∂N
∂N
∂N 3 J˜
2
3
∂ ∆v ∂ u
=
ḣ · N ∂N 2 ∂N 3
J˜
belongs to a finite dimensional subspace of L1 (∂Ω) for all ḣ ∈ C 6 (Ω, Rn ) satisfying
˜ Since ∂ 2 ∆v
˜
ḣ ≡ 0 on ∂Ω\J.
∂N 2 (x) 6= 0 for all x ∈ J, this is only possible (dim Ω ≥ 2)
∂3u
˜ So, u satisfies
if ∂N 3 ≡ 0 on J.
(∆2 + V + λ)u = 0
in Ω
(3.9)
∂u
∂2u
∂3u
u = ∂N
= ∂N
=
=
0
on J˜
2
∂N 3
that is, u satisfies the hypotheses of Cauchy’s Uniqueness Theorem 2.6. Thus u ≡ 0
in Ω and we reach a contradiction, proving the result.
Theorem 3.5. Generically in the set of open, connected, bounded C 4 regular regions of Rn n ≥ 2 the eigenvalues of (1.1) are all simple.
Proof. Consider the following differentiable map
3
7→ L2 (Ω) × H 2 (∂Ω)
(u, λ, h) 7→
h∗ (∆2 + V + λ)h∗ −1 u, h∗ ∆h∗ −1 u|∂Ω
F : BM × [−M, 0] × VM
where BM = {u ∈ H 4 ∩ H01 (Ω)\{0} | kuk ≤ M } and VM = OM \QM ; OM is the
open dense set given by Lemma 3.3 and QM is the meager closed set given by
Lemma 3.4. Observe that VM is an open dense subset of Diff 4 (Ω).
Using the Transversality Theorem, we show that the set
{h ∈ VM | (u, λ) 7→ F (u, λ, h) has 0 as a regular value }
is open and dense in VM . Therefore, we show our result taking intersection with
M varying in N.
By Remark 3.2, we can suppose that our regions are C 6 regular. The verification
of hypotheses (1) and (3) of the Transversality Theorem are simple, so we just show
that (2α) holds. We reason by contradiction.
So, suppose there exists a point critical (u, λ, h) ∈ F −1 (0, 0). By ‘change origin’,
3
we may suppose that h = iΩ . Then, there exist (v, θ) ∈ L2 (Ω) × H − 2 (∂Ω) such
that
Z
h
i D
E
0=
v (∆2 + V + λ)(u̇ − ḣ · ∇u) + λ̇u + θ, ∆(u̇ − ḣ · ∇u)
(3.10)
∂Ω
Ω
4
for all (u̇, λ̇, ḣ) ∈ H ∩
H01 (Ω)
6
n
× R × C (Ω, R ) where
3
DF (u, λ, iΩ ) : H 4 ∩ H01 (Ω) × R × C 6 (Ω, Rn ) 7→ L2 (Ω) × H 2 (∂Ω)
is given by
DF (u, λ, iΩ )(u̇, λ̇, ḣ) = (∆2 + V + λ)(u̇ − ḣ · ∇u) + λ̇u, ∆(u̇ − ḣ · ∇u)|∂Ω
3
3
and h·, ·i∂Ω indicates the duality of H − 2 (∂Ω) × H 2 (∂Ω).
12
M. C. PEREIRA
Taking ḣ = λ̇ = 0 in (3.10), we have
Z
0=
v (∆2 + V + λ)u̇ + hθ, ∆u̇i∂Ω ∀u̇ ∈ H 4 ∩ H01 (Ω).
Ω
Since Ω ⊂ Rn is C 6 regular, it follows by regularity results for uniformly elliptic
equations that v ∈ H 5 (Ω) ∩ C 4,α (Ω) for 0 < α < 1 satisfying
(∆2 + V (x) + λ)v(x) = 0 x ∈ Ω
v(x) = ∆v(x) = 0
x ∈ ∂Ω
R
3
∂v
for all g ∈ H 2 (∂Ω). If ḣ = u̇ = 0 and λ̇ varies, we see
and hθ, gi∂Ω = ∂Ω g ∂N
R
v u = 0; and when u̇ = λ̇ = 0 and ḣ varies, we find
Ω
Z
D
E
0 = −
v (∆2 + V + λ)(ḣ · ∇u) − θ, ∆(ḣ · ∇u)
∂Ω
Z hΩ
i Z
∂v
2
2
=
(ḣ · ∇u) (∆ + V + λ)v − v (∆ + V + λ)(ḣ · ∇u) −
(ḣ · ∇u)
∂N
∂Ω
ZΩ h
∂
∂∆
∂∆v
− ∆v
(ḣ · ∇u) − v
(ḣ · ∇u)
=
(ḣ · ∇u)
∂N
∂N
∂N
∂Ω
Z
∂v i
∂v
+∆(ḣ · ∇u)
−
∆(ḣ · ∇u)
∂N
∂N
∂Ω
Z
∂u ∂∆v
∀ḣ ∈ C 6 (Ω, Rn ).
=
ḣ · N
∂N ∂N
∂Ω
Therefore, we obtain
∂u ∂∆v
≡ 0 on ∂Ω.
(3.11)
∂N ∂N
Using equation (3) given by [1, Theorem 1.13], we have
∂u
∂2u
∂u
∂2u
−
= −H
−
2
∂N
∂N
∂N
∂N 2
on ∂Ω. Multiplying this relation by ∂∆v
∂N , we obtain by (3.11) that
0 = ∆∂Ω u = ∆u − H
∂ 2 u ∂∆v
≡ 0 on ∂Ω.
∂N 2 ∂N
Using (3) of [1, Theorem 1.13] again, we have
0
that multiplied by
Since i∂Ω
result.
(3.12)
∂∆v ∂ 2 ∆v
−
∂N
∂N 2
2
∂∆v ∂ ∆v
= −(V + λ)v − H
−
∂N
∂N 2
2
∂∆v ∂ ∆v
= −H
−
∂N
∂N 2
=
∂2u
∂N 2
∆∂Ω (∆v) = ∆(∆v) − H
implies, by (3.12), that
∂ 2 u ∂ 2 ∆v
≡ 0 on ∂Ω.
(3.13)
∂N 2 ∂N 2
∈ VM , (3.13) can not occur. So, we reach a contradiction, proving the
Acknowledgments. The author wishes to thank the Professors Antônio L. Pereira,
Luiz A. F. de Oliveira and J. M. Arrieta for useful discussions and helpful remarks.
GENERIC SIMPLICITY OF THE EIGENVALUES
13
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Marcone C. Pereira
Escola de Artes, Ciências e Humanidades, Universidade de São Paulo - São Paulo Brazil
E-mail address: [email protected]