Generalized Spectrum of Steklov-Robin Type Problem for Elliptic Systems

We will study the generalized Steklov-Robin eigenproblem (with possibly matrix weights) in which the spectral parameter is both in the system and on the boundary. The weights may be singular on subsets of positive measure. We prove the existence of an increasing unbounded sequence of eigenvalues. The method of proof makes use of variational arguments.


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].
x M x P x Σ verify the following assumptions:

Assumption 1. ( )
A x is positive definite on a set of positive measure of Ω, or ( ) is positive definite on a set of positive measure of .∂Ω And ( ) M x is positive definite on a set of positive measure of Ω, or ( ) x M x P x Σ 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) if there is such an eigenpair, then 0 µ > and ( ) ( ) a.e.(with 0 ϕ ≠ ) in Ω.This implies that ( ) A x is not positive definite on a subset of Ω of positive measure, and ( ) , d 0, x ϕ ϕ Σ = a.e. with ( ) is not positive definite on subset of ∂Ω of positive measure.So we have that, ϕ would be a con- stant vector function; which would contradict the assumptions (Assumption 1) imposed on ( ) A x and ( ).
x Σ Remark 6.If ( ) 0 A x ≡ and ( ) 0 x Σ ≡ then 0 µ = is an eigenvalue of the system (1) with eigenfunction constant ϕ = vector function on Ω .It is therefore appropriate to consider the closed linear subspace (to be shown below) of ( )  Now all the eigenfunctions associated with (2) must belong to the ( ) : H Ω We will show that indeed and α ∈  we wish to show that ( ) ( ) M ∈ Ω that is, the vector H Ω Thus, one can split the Hilbert space ( ) H Ω as a direct ( ) .
, then the subspace • We shall make use in what follows the real Lebesgue space ( ) L ∂Ω for 1 q ≤ ≤ ∞ , and of the continuity and compactness of the trace operator ( ) ( ) ( ) is well-defined, it is a Lebesgue integrable function with respect to Hausdorff 1 N − dimensional measure.Sometimes we will just use U in place of U Γ when considering the trace of a function on ∂Ω .Through- out, this work we denote the ( ) ∫ and the associated norm by ( ) (see [5], [6] and the references therein for more details).• The trace mapping ( ) ( ) defines an inner product for ( ) 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 ( ) , .A Σ , we deduce that ( ) , .A Σ is equivalent to the standard ( ) H Ω -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.
is weakly lower-semi-continuous.See [8] for the proof of Theorem 8.

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 j ϕ corresponding to the eigenvalues j µ from an ( ) , A Σ -orthogonal and ( ) 3) The normalized eigenfunctions provide a complete ( ) has a unique representation of the from , , , .
In addition, .
Proof of Theorem 9. We will prove the existence of a sequence of real eigenvalues j µ and the eigenfunc- tions j ϕ corresponding to the eigenvalues that from an orthogonal family in ( ) We show that , A Σ Λ attains its minimum on the constraint set Λ by using the continuity of the trace operator, the Sobolev embedding theorem and the lower-semi-continuity of , .
= by the definition of α we have that for all 0 >  and for all sufficiently large l, then 2 , l A U α Σ ≤ + by using the equivalent norm we have that, there is exist , , , .
Therefore, this sequence is bounded in ( ) H Ω .Thus it has a weakly convergent subsequence { } H Ω .From Rellich-Kondrachov theorem this subsequence converges strongly to Û in ( ) = as the functional is weakly l.s.c. (see Theorem  8).
There exists 1 ϕ such that ( ) attains its minimum at 1 ϕ and 1 ϕ satisfies the following ( . , µ ϕ satisfies Equation (2) in a weak sense and 1 6), we obtain that the eigenvalue 1 This means that we could define 1 µ by the Rayleigh quotient ( ) ϕ must be a constant and ( ) , 0 A x ϕ ϕ = with 0 ϕ ≠ that contradicts the assumptions imposed on ( ) Now we show the existence of higher eigenvalues.Define ( ) We know that the kernel of 1  ( ) Since W 1 is the null-space of the continuous functional 1 , ., , and it is therefore a Hilbert space itself under the same inner product ., , and it is therefore a Hilbert space itself under the same inner product µ µ ≤ Moreover, we can repeat the above arguments to show that 3 µ is achieved at some .
: , 0 ≤ .Moreover, we can repeat the above arguments to show that 4 µ is achieved at some , and it is therefore a Hilbert space itself under the same inner product ( ) , .,. .

M P
Now we define In this way, we generate a sequence of eigenvalues By way of contradiction, assume that the sequence is bounded above by a constant.Therefore, the corresponding sequence of eigenfunctions j ϕ is bounded in ( ).
H Ω By Rellich-Kondrachov theo- rem and the compactness of the trace operator, there is a Cauchy subsequence (which we again denote by j ϕ ), such that  which contradicts Equation (7).Thus, .j µ → ∞ We have that each j µ occurs only finitely many times.Claim 2 Each eigenvalue j µ 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 ( ) ⋅∇ is the outward (unit) normal derivative on .∂Ω The matrix ( ) diagonal matrix and in the diagonal of the eigenvalues of J (i.e.;

Since W 2
is the null-space of the continuous functional 2 , orthonormal, we have that orthonormal if and only if it is a complete orthonormal basis).Proof of Claim 3. By way of contradiction, assume that the sequence { } 1