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, ∫