Abstract. In this paper we consider spectral properties of the operator pencils generated by boundary-value problems for second order elliptic equation
depending parameter in a plane angle. We receive explicit formula for the
eigenvalues and the eigenvectors of the operator pencil, the multiplicity of
the eigenvalues, and the differentiability of the eigenvalues and the eigenvectors with respect to the parameter.

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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2008, Vol. 53, N
◦
. 5, pp. 41-48
SPECTRAL PROPERTIES OF THE OPERATOR PENCILS GENERATED
BY SECOND ORDER ELLIPTIC BOUNDARY-VALUE PROBLEMS
DEPENDING ON PARAMETER IN AN ANGLE
Nguyen Manh Hung and Nguyen Thanh Anh
Hanoi National University of education
Abstract. In this paper we consider spectral properties of the operator pen-
cils generated by boundary-value problems for second order elliptic equation
depending parameter in a plane angle. We receive explicit formula for the
eigenvalues and the eigenvectors of the operator pencil, the multiplicity of
the eigenvalues, and the differentiability of the eigenvalues and the eigen-
vectors with respect to the parameter.
Keywords: Elliptic equation, boundary-value problem, nonsmooth domains,
pencil of operators.
1. Introduction
Boundary value problems and initial boundary value problems have been con-
sidered for various types of partial differential operators in domains with various
types of nonsmooth boundaries. For the boundary-value problems for elliptic equa-
tion in domains with conical points, the solvability of the problem, the smoothness
and the asymptotic of the solutions are determined by the spectral properties the op-
erator pencils generated by the problems such as the location and the multiplicities
of the eigenvalues (see [1,2,3]). Elliptic boundary-value problems depending on a pa-
rameter arise when ones consider initial-boundary-value problems for nonstationary
equations with coefficients depending on time (see, for example, [4,5]). In general,
it is not easy to locate the eigenvalues and calculate thei multiplicities. However,
for the second order equation in a plane angle, these can be done by establishing
explicit formulas for the eigenvalues and the eigenvectors. This is the main purpose
of the present paper.
Our paper is organized as follows. In Section 2, we introduce some needed
notations and definitions. The main results are presented in Section 3 and 4.
2. Preliminaries
Let K = {x = (x1, x2) ∈ R2 : r > 0, 0 < ω < ϕ} be a plane angle with
vertex at the origin. Here r, ω are the polar coordinates of the point x = (x1, x2)
and 0 < ϕ < 2pi, ϕ 6= pi. Let T be a positive real number or T = +∞. Set Γ0 = {x ∈
41
Nguyen Manh Hung and Nguyen Thanh Anh
R2 : r > 0, ω = 0}, Γ1 = {x ∈ R2 : r > 0, ω = ϕ}, ΓjT = Γ
j × [0, T ], j = 0, 1. For
each multi-index α = (α1, α2), set |α| = α1 + α2 and ∂
α
x = ∂
|α|/∂α1x1 ∂
α2
x2
.
Let L = L(t, ∂x) be a homogeneous elliptic operator of second order in K with
coefficients depending on a parameter t ∈ [0, T ]
L = L(t, ∂x) =
2∑
j,k=1
ajk(t)∂xj∂xk .
We consider the elliptic boundary-value problem depending on parameter:
L(t, ∂x)u = f in KT , (2.1)
Bj(t, ∂x)u = gj on ,Γ
j
T j = 0, 1 (2.2)
where Bj(t, ∂x) are homogeneous differential operators on the boundaries Γ
j
T
Bj(t, ∂x) =
∑
|α|=µj
bj,α(t)∂
α
x , µj = 0 or µj = 1.
We can write L(t, ∂x), Bj(t, ∂x) in the form
L(t, ∂x) = r
−2
L(ω, t, ∂ω, r∂r),
Bj(t, ∂x) = r
−µjBj(ω, t, ∂ω, r∂r).
The operator
U(λ, t) = (L(ω, t, ∂ω, λ),B0(ω, t, ∂ω, λ),B1(ω, t, ∂ω, λ)), λ ∈ C, t ∈ [0, T ]
is called the pencil of operators generated by the problem (2.1)-(2.2). For every fixed
λ ∈ C, t ∈ [0, T ] this operator continuously maps
X := H l((0, ϕ)) into Y := H l−2((0, ϕ))× C2 (l > 2),
where H l((0, ϕ)) is the usual Sobolev space of functions defined in (0, ϕ).
We mention now some well-known definitions in [2]. Let t0 ∈ [0, T ] be fixed.
Let λ0 be an eigenvalue of U(λ, t0) with geometric multiplicity I and ϕ0 ∈ X be
an eigenvector corresponding to λ0, that means, ϕ0 6= 0,U(λ0, t0)ϕ0 = 0 and I =
dimkerU(λ0, t0). If the elements ϕ1, . . . , ϕs of X satisfy the equations
σ∑
q=0
1
q!
∂q
dλq
U(λ0, t0)ϕσ−q = 0 for σ = 1, . . . , s, (2.3)
then the ordered collection ϕ0, ϕ1, . . . , ϕs is said to be a Jordan chain corresponding
to the eigenvalue λ0 of the length s+1. The elements ϕ1, . . . , ϕs are called generalized
42
Spectral properties of the operator pencils generated by second order elliptic...
eigenvectors. The rank of the eigenvector ϕ0 (rankϕ0) is the maximal length of
the Jordan chains corresponding to the eigenvector ϕ0. The sum of ranks of all
eigenvectors linearly independent of U(λ0, t0) is called the algebraic multiplicity of
the eigenvalue λ0. An eigenvalue is said to be simple if its algebraic multiplicity is
equal to one.
It is well-known that for each t ∈ [0, T ] the spectrum of the operator pencil
U(λ, t) is an enumerable set of eigenvalues with the finite algebraic multiplicities
(see [2, Theo. 5.2.1]).
To investigate boundary-value problems for elliptic equations in domains with
conical points we are concerned with understanding properties of eigenvalues of the
pencil U(λ, t) such as the location, the multiplicity and the smoothness with respect
to the parameter t.
3. The Dirichlet boundary problem
We can take advantage that the operator L is written in the form
L(t, ∂x) = (∂x2 − a1(t)∂x1)(∂x2 − a2(t)∂x1),
where aj(t) (j = 1, 2) are complex-value functions defined on [0, T ] and
Im a1(t) > 0, Im a2(t) < 0 for all t ∈ [0, T ].
In this section, we consider the Dirichlet boundary problem depending on a param-
eter:
L(t, ∂x)u = f in KT , (3.1)
u =gj on Γ
j
T ,j = 0, 1. (3.2)
In the polar coordinates (r, ω) the problem (3.1), (3.2) has the form
r−2L(t, ω, ∂ω, r∂r)u = f in KT ,
u|ω=0 = g0,
u|ω=α = g1,
where
L(t, ω, ∂ω, λ) =
2∏
j=1
((
sinω − aj(t) cosω
)
λ+
(
cosω + aj(t) sinω
)
∂ω
)
. (3.3)
Thus, a complex number λ is an eigenvalue of the operator pencil U(λ, t) generated
by the problem (3.1)-(3.2) if the following problem
L(t, ω, ∂ω, λ)u = 0 in (0, ϕ), (3.4)
u|ω=0 = 0, (3.5)
u|ω=α = 0 (3.6)
43
Nguyen Manh Hung and Nguyen Thanh Anh
has a nontrivial solution.
If λ = 0, w1(ω, t) = 1, w2(ω, t) = ω are two linearly independent solutions
of the equation (3.4). We can verify directly that for each t ∈ [0, T ] there are
no constants c1(t), c2(t) such that the function u = c1(t)u1 + c2(t)u2 satisfies the
boundary conditions (3.5), (3.6). Hence, λ = 0 is not an eigenvalue of U(λ, t).
For λ 6= 0, by calculating directly we can see that the functions
vj(ω, t) = e
λFj(ω,t), j = 1, 2, (3.7)
are solutions of the equation (3.4), where
Fj(ω, t) =
∫ ω
0
fj(θ, t)dθ, fj(ω, t) =
− sinω + aj(t) cosω
cosω + aj(t) sinω
.
The Wronskian of v1(ω, t) and v2(ω, t) at ω = 0 is∣∣∣∣ 1 1λa1(t) λa2(t)
∣∣∣∣ = λ(a2(t)− a1(t)) 6= 0 for all t ∈ [0, T ].
Thus, for λ 6= 0, v1(ω, t), v2(ω, t) are two linearly independent solutions of the equa-
tion (3.4), and therefore, the its general solution is
u = c1(t)v1 + c2(t)v2.
Inserting this function into the boundary conditions (3.5), (3.6), we get the system{
c1(t) + c2(t) = 0
c1(t)e
λF1(α,t) + c2(t)e
λF2(α,t) = 0.
(3.8)
The coefficients determinant of this system is equal to D(λ, t) = eλF2(α,t) − eλF1(α,t).
A complex number λ is an eigenvalue of U(λ, t) if the system has nontrivial solutions
c1(t), c2(t) or, in other words, λ is a solution of the following equation
D(λ, t) = eλF2(α,t) − eλF1(α,t) = 0. (3.9)
This equation is equivalent to
eλ(F2(α,t)−F1(α,t)) = 1. (3.10)
Since Im a1(t) > 0, Im a2(t) < 0, t ∈ [0, T ], by calculating directly, we have (see [1,
Theo. 14.6])
Im
(
F2(α, t)− F1(α, t)
)
< 0 for all t ∈ [0, T ].
Therefore, the equation (3.10) has solutions
λk(t) =
ik2pi
F2(α, t)− F1(α, t)
, k = ±1,±2, . . . (3.11)
44
Spectral properties of the operator pencils generated by second order elliptic...
Thus, the eigenvalues of U(λ, t) are λk(t), k = ±1,±2, . . . For each eigenvalue λk(t),
choosing c1(t) = 1, c2(t) = −1 in (3.8), we have the corresponding eigenvector is
ψk(ω, t) = e
λk(t)F1(ω,t) − eλk(t)F2(ω,t), k = ±1,±2, . . . (3.12)
For each t0 ∈ [0, T ] fixed, by lemmas 3.1.1 and 3.1.2 of [3], the algebraic multiplicity
of an eigenvalue λ0 of U(λ, t0) coincides with the multiplicity of the zero λ = λ0 of
the function D(λ, t0). Moreover, we have
d
dλ
D(λ, t)|λ=λk(t) = e
λk(t)F2(α,t)f2(α, t)− e
λF1(α,t)f1(α, t)
= eλk(t)F1(α,t)
(
f2(α, t)− f1(α, t)
)
= eλk(t)F1(α,t)
a2(t)− a1(t)
(cosω + a1(t) sinω)(cosω + a2(t) sinω)
6= 0 (3.13)
for all t ∈ [0, T ]. This means the zeros λk(t) of D(λ, t) have multiplicity one, and
therefore, the eigenvalues λk(t)(k = ±1,±2, . . .) ofU(λ, t) are simple for all t ∈ [0, T ].
From the facts above, we get
Theorem 3.1. Suppose that the functions a1(t), a2(t) belong to the class C
h([0, T ]),
where h is a natural number. Then all eigenvalues of the pencil U(λ, t) are defined by
the formula (3.11); these eigenvalues are simple and belong to class the Ch([0, T ]).
The corresponding eigenvectors are given by the formula (3.12); they are infinitely
differentiable with respect to the variable ω and h-times continuously differentiable
with respect to the variable t.
4. The oblique derivative problem
In this section, we consider the oblique derivative problem depending on a
parameter:
L(t, ∂x)u = f in KT , (4.1)
∂νju+ bj(t)∂τju = gj on Γ
j
T , j = 0, 1, (4.2)
where τj are the directions of the rays Γ
j
, and νj are the exterior normals to the
sides Γj of the angle K, bj(t) are real-value functions defined in [0, T ]. If bj ≡ 0,
j = 0, 1, then (4.1)-(4.2) is the Newmann boundary problem.
In the polar coordinates (r, ω) the problem (4.1), (4.2) has the form
r−2L(t, ω, ∂ω, r∂r)u = f in KT ,
r−1(−∂ω + b0(t)∂r)u = g0 on Γ
0
T ,
r−1(∂ω + b1(t)∂r)u = g1 on Γ
1
T ,
45
Nguyen Manh Hung and Nguyen Thanh Anh
where L(t, ω, ∂ω, λ) is defined as in (3.3). Thus a complex number λ is an eigenvalue
of the operator pencil U(λ, t) generated by the problem (4.1)-(4.2) if the following
problem
L(t, ω, ∂ω, λ)u = 0 in (0, ϕ), (4.3)
(−∂ωu+ b0(t)λu)|ω=0 = 0, (4.4)
(∂ωu+ b1(t)λu)|ω=α = 0 (4.5)
has a nontrivial solution.
If λ 6= 0, as in Section 3, the general solution of the equation (4.3) has the
form
u = c1(t)v1 + c2(t)v2, (4.6)
where v1, v2 are defined as in (3.7). Inserting the function (4.6) into the boundary
conditions (4.4), (4.5), we get the system{
λ(a1(t) + b0(t))c1(t) + λ(a2(t) + b0(t))c2(t) = 0,
λeλF1(α,t)(f1(α, t) + b1(t))c1(t) + λe
λF2(α,t)(f2(α, t) + b1(t))c2(t) = 0.
(4.7)
The coefficients determinant D(λ, t) of this system is equal to
λ2
(
(a1(t) + b0(t))(f1(α, t) + b1(t))e
λF2(α,t) − (a2(t) + b0(t))(f2(α, t) + b1(t))e
λF1(α,t)
)
.
Since Im a1(t) > 0, Im a2(t) < 0 for all t ∈ [0, T ] and b0(t) is a real function, then
aj(t) + b0(t) 6= 0 for all t ∈ [0, T ] (j = 1, 2). (4.8)
Moreover, we have
fj(ω, t) =
(− sinω + aj(t) cosω)(cosω + aj(t) sinω)
| cosω + aj(t) sinω|2
=
(|aj(t)|
2 − 1) sinω cosω +Re aj(t)(cos
2 ω − sin2 ω) + i Im aj(t)
| cosω + aj(t) sinω|2
(j=1,2). It follows from this equality that fj(ω, t) (j=1,2) are not real for all ω ∈ [0, α]
and all t ∈ [0, T ]. Therefore,
fj(α, t) + b1(t) 6= 0 for all t ∈ [0, T ] (j = 1, 2). (4.9)
Thus, D(λ, t) = 0 if and only if
eλ(F2(α,t)−F1(α,t)) =
(a2(t) + b0(t))(f2(α, t) + b1(t))
(a1(t) + b0(t))(f1(α, t) + b1(t))
. (4.10)
46
Spectral properties of the operator pencils generated by second order elliptic...
Denote by A(t) the right-hand side of (4.10) and set
Z(t) = ln |A(t)|+ iArgA(t),
where Arg z is the argument of the complex number z belonging to [0, 2pi). Then
the equation (4.10) is equivalent to
λ(F2(α, t)− F1(α, t)) = Z(t) + ik2pi, (4.11)
and therefore, the zeros of D(λ, t) except λ = 0 are
λk(t) =
Z(t) + ik2pi
F2(α, t)− F1(α, t)
, k = ±1,±2, . . . (4.12)
By calculating directly and using (3.13), (4.8) and (4.9), we get
d
dλ
D(λ, t)|λ=λk(t)
= λ2k(t)e
λk(t)F1(α,t)
(
a2(t) + b0(t)
)(
f2(α, t) + b1(t)
)(
f2(α, t)− f1(α, t)
)
6= 0
for all t ∈ [0, T ]. Hence, as in Section 3, the zeros λk(t) (k = ±1,±2, . . .) of D(λ, t)
have the multiplicity one, and therefore, they are simple eigenvalues of the pencil
U(λ, t) for all t ∈ [0, T ].
For each eigenvalue λk, choosing c1(t) = a2(t) + b0(t), c2(t) = −(a1(t) + b0(t))
in (4.7) we have the corresponding eigenvector is
ψk(ω, t) = (a2(t) + b0(t))e
λk(t)F1(ω,t) − (a1(t) + b0(t))e
λk(t)F2(ω,t). (4.13)
If λ = 0, as in Section 3, the equation (4.3) has the general solution in the
form
u(ω, t) = c1(t) + c2(t)ω.
Inserting this function into the boundary conditions (4.4), (4.5), we get c2(t) ≡ 0
and c1(t) is arbitrary. Thus, λ(t) = 0 is an eigenvalue of the pencil U(λ, t) with
the geometric multiplicity one, and u0 = 1 is an eigenvector corresponding to this
eigenvalue. To find a necessary and sufficient condition for the simplicity of the
eigenvalue λ(t) = 0, we consider the equation
U(0, t)u1 = −U
′(0, t)u0 (4.14)
where U′(0, t) = (L′(t, ω, ∂ω, 0), b0(t), b1(t)),
L
′(t, ω, ∂ω, 0) =
d
dλ
L(t, ω, ∂ω, λ)|λ=0
=
2∑
j,h=1
j 6=h
(sinω − aj(t) cosω)(cosω + ah(t) sinω)∂ω.
47
Nguyen Manh Hung and Nguyen Thanh Anh
Rewrite the equation (4.14) in the equivalent form
∂2ωu1 = 0 in (0, α), (4.15)
−∂ωu1|ω=0 = −b0(t) on Γ
0
T , (4.16)
∂ωu1|ω=α = −b1(t) on Γ
1
T . (4.17)
The equation (4.15) has the general solution u1(ω, t) = c1(t) + c2(t)ω. Inserting this
function into the boundary conditions (4.16), (4.17), we have
−c2(t) = −b0(t), c2(t) = −b1(t).
Thus, if b2(t)+b1(t) 6= 0, then the problem (4.15)-(4.17) has no solutions. Otherwise,
u1 = b0(t)ω is an its solution.
Summarizing the results above, we obtain
Theorem 4.1. Suppose that the functions a1(t), a2(t), b0(t), b1(t) belong to the class
Ch([0, T ]), where h is a natural number. Then
1) All nonzero eigenvalues of the pencil U(λ, t) are given by the formula (4.12);
these eigenvalues are simple and belong to class the Ch([0, T ]). The corresponding
eigenvectors are given by the formula (4.13), which are infinitely differentiable with
respect to the variable ω and h-times continuously differentiable with respect to the
variable t.
2) λ(t) ≡ 0 is an eigenvalue of the pencil U(λ, t) with a corresponding eigen-
vector u0 = 1. This eigenvalue is simple on [0, T ] if and only if b0(t) + b1(t) 6= 0 for
all t ∈ [0, T ].
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