Generalized Spectrum of Steklov-Robin Type Problem for Elliptic Systems ()
1. Introduction
We study the generalized Steklov-Robin eigenproblem. This spectrum includes the Steklov, Neumann and Robin spectra. We therefore generalize the results in [1] -[4] .
Consider the elliptic system
(1)
where
,
is a bounded domain with boundary
of class
,
Throughout this paper all matrices are symmetric. The matrix
![](//html.scirp.org/file/19-7402551x11.png)
verifies the following conditions:
(A1) The functions ![](//html.scirp.org/file/19-7402551x12.png)
(A2)
is positive semidefinite a.e. on Ω with
for
when
, and
when ![](//html.scirp.org/file/19-7402551x18.png)
The matrix
![](//html.scirp.org/file/19-7402551x19.png)
satisfies the following conditions:
(M1)
is positive semidefinite a.e. on Ω The functions
for
when
, and
when ![]()
is the outward (unit) normal derivative on
The matrix
![]()
verifies the following conditions:
(S1) The functions ![]()
(S2)
is positive semidefinite a.e. on
with
for
when
, and
when ![]()
and the matrix
![]()
(P1)
is positive semidefinite a.e. on
for
when
, and
when ![]()
We assume that
verify the following assumptions:
Assumption 1. 1)
is positive definite on a set of positive measure of Ω,
2)
is positive definite on a set of positive measure of ![]()
3)
is positive definite on a set of positive measure of Ω with ![]()
4)
is positive definite on a set of positive measure of ![]()
Remark 2. Assumption 1 is equivalent to
![]()
Remark 3. Since
satisfy (A2), (S2), (M1), (P1) respectively, then we can write them in the following form (i.e.; eigen-decomposition of a positive semi-definite matrix or diagonalization)
![]()
where
(
i.e.; are orthogonal matrices) are the normalized eigenvectors, I is the identity matrix,
is diagonal matrix and in the diagonal of
are the eigenvalues of J (i.e.;
) and ![]()
Remark 4. The weight matrices
and
may vanish on subsets of positive measure.
Definition 1. The generalized Steklov-Robin eigensystem is to find a pair
with
such that
(2)
Remark 5. Let
in (2) if there is such an eigenpair, then
and
![]()
Indeed, if
or
then
![]()
We have that
which implies that
and
this implies that
a.e. (with
) in Ω. This implies that
is not positive definite on a subset of Ω of
positive measure, and
then
a.e. with
on
This implies
that
is not positive definite on subset of
of positive measure. So we have that,
would be a constant vector function; which would contradict the assumptions (Assumption 1) imposed on
and ![]()
Remark 6. If
and
then
is an eigenvalue of the system (1) with eigenfunction
vector function on
.
It is therefore appropriate to consider the closed linear subspace (to be shown below) of
under Assumption 1 defined by
![]()
Now all the eigenfunctions associated with (2) must belong to the
-orthogonal complement
of this subspace in
We will show that indeed
is subspace of
Let
and
we wish to show that
and ![]()
![]()
Therefore
Now we show that ![]()
![]()
Since
it follows that
![]()
By setting
we get
![]()
Since
for a.e.
it readily follows that
![]()
that is, the vector
satisfies
![]()
or equivalently
![]()
Hence,
![]()
since
A similar arguments shows that
![]()
Therefore
so we have that
is a subspace of
Thus, one can split the Hilbert space
as a direct
-orthogonal sum in the following way
![]()
Remark 7. 1) If
in Ω, then the subspace
provided
on
.
2) If
in
and
, then the subspace
provided
on Ω.
・ We shall make use in what follows the real Lebesgue space
for
, and of the continuity and compactness of the trace operator
![]()
is well-defined, it is a Lebesgue integrable function with respect to Hausdorff
dimensional measure. Sometimes we will just use U in place of
when considering the trace of a function on
. Throughout, this work we denote the
-inner product by
![]()
and the associated norm by
![]()
(see [5] , [6] and the references therein for more details).
・ The trace mapping
is compact (see [7] ).
(3)
defines an inner product for
, with associated norm
(4)
Now, we state some auxiliary result, which will be need in the sequel for the proof of our main result. Using the Hölder inequality, the continuity of the trace operator, the Sobolev embedding theorem and lower semicontinuity of
, we deduce that
is equivalent to the standard
-norm. This observation enables us to prove the existence of an unbounded and discrete spectrum for the Steklov-Robin eigenproblem (1) and discuss some of its properties.
Definition 2. Define the functional
![]()
![]()
and
![]()
![]()
Lemma 1. Suppose (A2), (S2), (M1), (P1) are met. Then the functionals
and
are C1-functional (i.e.; continuously differentiable).
See [8] for the proof of Lemma 1.
Theorem 8.
is G-differentiable and convex. Then
is weakly lower-semi-continuous.
See [8] for the proof of Theorem 8.
2. Main Result
Theorem 9. Assume Assumption 1 as above, then we have the following.
1) The eigensystem (1) has a sequence of real eigenvalues
![]()
and each eigenvalue has a finite-dimensional eigenspace.
2) The eigenfunctions
corresponding to the eigenvalues
from an
-orthogonal and
- orthonormal family in
(a closed subspace of
).
3) The normalized eigenfunctions provide a complete
-orthonormal basis of
Moreover, each function
has a unique representation of the from
(5)
In addition,
![]()
Proof of Theorem 9. We will prove the existence of a sequence of real eigenvalues
and the eigenfunc-
tions
corresponding to the eigenvalues that from an orthogonal family in
.
We show that
attains its minimum on the constraint set
![]()
Let
by using the continuity of the trace operator, the Sobolev embedding theorem and
the lower-semi-continuity of ![]()
Let
be a minimizing sequence in W0 for
since
we have that
by the definition of
we have that for all
and for all sufficiently large l, then
by using the equivalent norm we have that, there is exist
such that
![]()
so we have that
![]()
Therefore, this sequence is bounded in
. Thus it has a weakly convergent subsequence ![]()
which convergent weakly to
in
. From Rellich-Kondrachov theorem this subsequence converges strongly to
in
so
in W0. Thus
as the functional is weakly l.s.c. (see Theorem 8).
There exists
such that
. Hence,
attains its minimum at
and
satisfies the following
(6)
for all
We see that
satisfies Equation (2) in a weak sense and
this im-
plies that
by the definition of W0. Now take
in Equation (6), we obtain that the eigenvalue
is the infimum
. This means that we could define
by the Rayleigh quotient
![]()
Clearly,
. Indeed assume that
then
on
hence
must be a constant and
with
that contradicts the assumptions imposed on
. Thus
.
Now we show the existence of higher eigenvalues.
Define
![]()
We know that the kernel of ![]()
![]()
Since W1 is the null-space of the continuous functional
on
W1 is a closed sub-
space of
, and it is therefore a Hilbert space itself under the same inner product
. Now
we define
![]()
Since
then we have that
. Now we define
![]()
we know that the kernel of ![]()
![]()
Since W2 is the null-space of the continuous functional
on
, W2 is a closed subspace of
, and it is therefore a Hilbert space itself under the same inner product
. Now
we define
![]()
Since
then we have that
Moreover, we can repeat the above arguments to show that ![]()
is achieved at some ![]()
We let
![]()
and
![]()
Since
then we have that
. Moreover, we can repeat the above arguments to show that ![]()
is achieved at some ![]()
Proceeding inductively, in general we can define
![]()
we know that the kernel of ![]()
![]()
Since Wj is the null-space of the continuous functional
on
, Wj is a closed subspace
of
, and it is therefore a Hilbert space itself under the same inner product
Now we define
![]()
In this way, we generate a sequence of eigenvalues
![]()
whose associated
are c-orthogonal and
-orthonormal in ![]()
Claim 1
as ![]()
Proof of claim 1. By way of contradiction, assume that the sequence is bounded above by a constant. Therefore, the corresponding sequence of eigenfunctions
is bounded in
By Rellich-Kondrachov theorem and the compactness of the trace operator, there is a Cauchy subsequence (which we again denote by
), such that
(7)
Since the
are
-orthonormal, we have that
, if ![]()
which contradicts Equation (7). Thus,
We have that each
occurs only finitely many times.
Claim 2
Each eigenvalue
has a finite-dimensional eigenspace.
See [8] for the proof of claim 2.
We will now show that the normalized eigenfunctions provide a complete orthonormal basis of
. Let
![]()
so that ![]()
Claim 3
The sequence
is a maximal
-orthonormal family of
. (We know that the set is
maximal
-orthonormal if and only if it is a complete orthonormal basis).
Proof of Claim 3. By way of contradiction, assume that the sequence
is not maximal, then there exists a
and
such that
and
, i.e.;
![]()
since
. Therefore
. We have that
. It follows from the definition of
that
![]()
Since we know from Claim 1 that
as
we have that
Therefore
a.e in Ω, which contradicts the definition of ξ. Thus the sequence
is a maximal
-orthonormal family of
so the sequence
provides a complete orthonormal basis of
that is, for any
,
with
and
![]()
![]()
Now let
![]()
Therefore,
![]()
and
![]()
Claim 4
We shall show that
![]()
Proof of Claim 4.
![]()
Thus
![]()