Strict Monotonicity and Unique Continuation for the Third-Order Spectrum of Biharmonic Operator

and Applied Analysis 3 eigensurfaces for problem 1.1 holds if some unique continuation property is satisfied by the corresponding eigenfunctions. 2. Preliminaries Let H be a finite dimensional separable Hilbert space. We denote by ·, · and ‖ · ‖ the inner product and the norm of the space H, respectively. Let T : H → H be a compact operator. Lemma 2.1. All nonzero eigenvalues of the operator T are obtained by the following characterizations: μn sup Fn∈Fn H min{ T u , u such that ‖u‖ 1; u ∈ Fn}, μ−n inf Fn∈Fn H Max{ T u , u such that ‖u‖ 1; u ∈ Fn}, 2.1 where Fn H denotes the class of n-dimensional subspaces Fn of H. Moreover, zero is the only accumulation point of the set of all eigenvalues of T . Here, the eigenvalues are repeated with its order of multiplicity, and the eigenfunctions are mutually orthogonal 10 . 3. Third-Order Spectrum of the Biharmonic Operator We define the third-order eigenvalue problem of the biharmonic operator as follows: Find ( β, α, u ) ∈ R × R ×H \ {0} such that Δ2u 2β · ∇ Δu ∣β∣2Δu αmu in Ω, u Δu 0 on ∂Ω. 3.1 If β, α, u is a solution of 3.1 then β, α is called third-order eigenvalue and u is said to be the associated eigenfunction. Lemma 3.1. Problem 3.1 is equivalent to the following problem: Find α, u ∈ R ×H \ {0} such that Δu αmeβ·xu in Ω, u Δu 0 on ∂Ω, 3.2 where Δ2,βu Δ eβ·xΔu . 4 Abstract and Applied Analysis Proof. For any β ∈ R , we have Δ ( eβ·xΔu ) ∇ ( ∇ ( eβ·xΔu )) ∇ ( βeβ·xΔu eβ·x∇ Δu ) eβ·x [ Δ2u 2 ( β · ∇ Δu ) ∣β∣2Δu ] . 3.3 Hence, problem 3.1 is equivalent to problem 3.2 Remark 3.2. Let u ∈ H; we denote by ∂u/∂ν the normal derivative defined by ∂u/∂ν ∇u|∂Ω · ν where ∇u|∂Ω ∈ L2 ∂Ω N and ∂u/∂ν ∈ L2 ∂Ω . Definition 3.3. A weak solution of 3.2 is a function u in H \ {0} witch satisfies, for β, α ∈ R N × R and for all φ ∈ H, ∫


Abstract and Applied Analysis
Based on the works of Anane et al. 1, 2 , we will determine the spectrum of 1.1 , which we call third-order spectrum for the biharmonic operator. This spectrum is defined to be the set of couples β, α ∈ R N × R such that the problem has a nontrivial solution u ∈ H. This spectrum, which is denoted by σ 3 Δ 2 , m , is an infinite sequence of eigensurfaces Γ ± 1 , Γ ± 2 , . . ., see Section 3. When β 0, the zero-order spectrum is defined to be the set of eigenvalues α ∈ R such that the problem has a nontrivial solution u ∈ H. In this case the spectrum is denoted by σ 0 Δ 2 , m . The eigenvalue problem 1.3 , which is studied by Courant and Hilbert 3 , admits an infinite sequence of real eigenvalues α n m n satisfying 1 α n m sup where F n H denotes the class of n-dimensional subspaces F n of H.
Definition 1.1. We say that solutions of problem 1.1 satisfy the unique continuation property U.C.P , if the unique solution u ∈ L 2 Loc Ω which vanishes on a set of positive measure in Ω is u ≡ 0.
In the literature there exist several works on unique continuation. We refer to the works of Jerison and Kenig 4 and Garofalo and Lin 5 , among others. The unique continuation property as defined above differs from the usual notions of unique continuation, see 6 for more details.
Here we use the notation < / ≡ to mean inequality almost everywhere together with strict inequality on a set of positive measure.
Since the pioneer works of Carleman 7 in 1939 on the unique continuation, this notion has been the interest of many researchers in partial differential equations, see for instance 4,5,8 . In 1992, de Figueiredo and Gossez 6 proved that strict monotonicity holds if and only if some unique continuation property is satisfied by the corresponding eigenfunction of a uniformly elliptic operator of the second order. In 1993, Gossez and Loulit 8 have proved the unique continuation property in the linear case of the laplacian operator. The unique continuation property of the biharmonic operator was proved recently by Cuccu and Porru 9 . Our purpose in the fourth section is to show that strict monotonicity of Abstract and Applied Analysis 3 eigensurfaces for problem 1.1 holds if some unique continuation property is satisfied by the corresponding eigenfunctions.

Preliminaries
Let H be a finite dimensional separable Hilbert space. We denote by ·, · and · the inner product and the norm of the space H, respectively. Let T : H → H be a compact operator. Lemma 2.1. All nonzero eigenvalues of the operator T are obtained by the following characterizations: Moreover, zero is the only accumulation point of the set of all eigenvalues of T . Here, the eigenvalues are repeated with its order of multiplicity, and the eigenfunctions are mutually orthogonal 10 .

Third-Order Spectrum of the Biharmonic Operator
We define the third-order eigenvalue problem of the biharmonic operator as follows: u Δu 0 on ∂Ω.

3.1
If β, α, u is a solution of 3.1 then β, α is called third-order eigenvalue and u is said to be the associated eigenfunction.

Abstract and Applied Analysis
Proof. For any β ∈ R N , we have Hence, problem 3.1 is equivalent to problem 3.2 where F n H denotes the class of n-dimensional subspaces F n of H and G Γ n m, · ⊂ R N × R is the graph of Γ n m, · .
Proof. Let β, α, u ∈ H \{0}, then β, α, u is a solution of 3.1 if and only if α, u is a solution of problem 3.2 . We prove that the map defines a scalar product on H H 2 Ω ∩ H 1 0 Ω equivalent to the usual scalar product Ω ΔuΔv dx.
The map l ·, · is a continuous symmetric bilinear form. Since Δ 2 satisfies the condition of the uniform ellipticity, then we have

3.10
where c min x∈Ω e β·x . Therefore, the bilinear form l ·, · is coercive. On the other hand, the operator is well defined, linear, symmetric, and compact on H. Then, problem 3.2 can be written as Note that α 0 is not an eigenvalue of 3.2 . It follows that α, β is an eigenvalue of 1.1 if and only if 1/α is eigenvalue of the operator T 2,β . By Lemma 2.1, the eigenvalues are given by the characterizations

3.13
In addition, we have 3.14 and u 2 u, u Δ 2,β u, u u 2 2,2,β , then relation 3.8 is satisfied. Since me β.x ∈ M, then we have Γ n m, β > 0 for all n ∈ N * . As zero is the only accumulation point of the sequence 1/α n n , it follows that Γ n m, β → ∞ when n → ∞. Therefore, the proof is completed.

Strict Monotonicity and Unique Continuation
In this section, we will show that strict monotonicity of eigensurfaces for problem 3.1 holds if some unique continuation property is satisfied by the corresponding eigenfunctions.
As a consequence of Theorems 4.1 and 4.2 we have the following result. where > 0 is chosen such that Γ i β, m < Γ i 1 β, m , which is possible by the continuous dependence of the eigenvalues with respect to the weight. We have Δ 2 u 2β · ∇ Δu β 2 Δu Γ i β, m mu Γ i β, m mu, 4.10 which shows that Γ i β, m is an eigenvalue for the weight m, that is, Γ i β, m Γ l β, m for some l ∈ N. Let us choose the largest l such that this equality holds. It follows from Γ i β, m < Γ i 1 β, m that l < i 1. Moreover, the monotone dependence, Γ i β, m ≤ Γ i β, m , implies l ≥ i. Then we conclude that l i. Hence, we have Γ i β, m Γ i β, m .