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

Abstract

We will study the spectrum for the biharmonic operator involving the laplacian and the gradient of the laplacian with weight, which we call third-order spectrum. We will show that the strict monotonicity of the eigenvalues of the operator Δ²𝑢+2𝛽⋅∇(Δ𝑢)+|𝛽|²Δ𝑢, where 𝛽∈ℝ^𝑁, holds if some unique continuation property is satisfied by the corresponding eigenfunctions.

1. Introduction

We are concerned here with the eigenvalue problem: Find𝛽,𝛼,𝑢∈ℝ^𝑁×ℝ×𝐻suchthatΔ²𝑢+2𝛽⋅∇(Δ𝑢)+𝛽²Δ𝑢=𝛼𝑚𝑢inΩ,𝑢=Δ𝑢=0on𝜕Ω, (1.1) where Ω is a bounded domain in ℝ^𝑁  (𝑁≥1),  𝐻=𝐻²(Ω)∩𝐻¹₀(Ω), Δ² denotes the biharmonic operator defined by Δ²𝑢=Δ(Δ𝑢), and 𝑚∈𝑀={𝑚∈𝐿^∞(Ω)/meas{𝑥∈Ω/𝑚(𝑥)>0}≠0}.

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 Δ²𝑢+2𝛽⋅∇(Δ𝑢)+𝛽²Δ𝑢=𝛼𝑚𝑢inΩ,𝑢=Δ𝑢=0on𝜕Ω (1.2) has a nontrivial solution 𝑢∈𝐻. This spectrum, which is denoted by 𝜎₃(Δ²,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 Δ²𝑢=𝛼𝑚𝑢inΩ,𝑢=Δ𝑢=0on𝜕Ω (1.3) has a nontrivial solution 𝑢∈𝐻. In this case the spectrum is denoted by 𝜎₀(Δ²,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 1𝛼_𝑛(𝑚)=sup_{𝐹𝑛∈ℱ𝑛(𝐻)}min_{𝑢∈𝐹𝑛Ω}𝑚|𝑢|²𝑑𝑥∀𝑛≥1, (1.4) 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∈𝐿²_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.

Definition 1.2. We say that Γ_𝑘(𝛽,⋅) is strict monotone with respect to the weight if Γ_𝑘(𝛽,𝑚)>Γ_𝑘(𝛽,𝑚), for all 𝑚<_≢𝑚.

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 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: 𝜇_𝑛=sup_{𝐹𝑛∈ℱ𝑛(𝐻)}min(𝑇(𝑢),𝑢)suchthat‖𝑢‖=1;𝑢∈𝐹_𝑛,𝜇_−𝑛=inf_{𝐹𝑛∈ℱ𝑛(𝐻)}Max(𝑇(𝑢),𝑢)suchthat‖𝑢‖=1;𝑢∈𝐹_𝑛, (2.1) where ℱ_𝑛(𝐻) denotes the class of 𝑛-dimensional subspaces 𝐹_𝑛 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: Find𝛽,𝛼,𝑢∈ℝ^𝑁×ℝ×𝐻⧵{0}suchthatΔ²𝑢+2𝛽⋅∇(Δ𝑢)+𝛽²Δ𝑢=𝛼𝑚𝑢inΩ,𝑢=Δ𝑢=0on𝜕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}suchthatΔ^2,𝛽𝑢=𝛼𝑚𝑒^𝛽⋅𝑥𝑢inΩ,𝑢=Δ𝑢=0on𝜕Ω, (3.2) where Δ^2,𝛽𝑢=Δ(𝑒^𝛽⋅𝑥Δ𝑢).

Proof. For any 𝛽∈ℝ^𝑁, we have Δ𝑒^𝛽⋅𝑥Δ𝑢=∇∇𝑒^𝛽⋅𝑥Δ𝑢=∇𝛽𝑒^𝛽⋅𝑥Δ𝑢+𝑒^𝛽⋅𝑥∇(Δ𝑢)=𝑒^𝛽⋅𝑥Δ²𝑢+2𝛽⋅∇(Δ𝑢)+𝛽²Δ𝑢. (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, _Ω𝑒^𝛽⋅𝑥Δ𝑢Δ𝜑𝑑𝑥=𝛼_Ω𝑒^𝛽⋅𝑥𝑚𝑢𝜑𝑑𝑥. (3.4)

Definition 3.4. For (𝛽,𝛼)∈ℝ^𝑁×ℝ, we say that 𝑢∈𝐻 is a classical solution of problem (3.1) if 𝑢∈𝐶⁴(Ω).

Proposition 3.5. If 𝑢∈𝐻 is a weak solution of (3.2) and 𝑢∈𝐶⁴(Ω), then 𝑢 is a classical solution of (3.2).

Proof. Let 𝑢∈𝐶⁴(Ω) be a weak solution of (3.2), then we have _Ω𝑒^𝛽⋅𝑥Δ𝑢Δ𝜑𝑑𝑥=𝛼_Ω𝑒^𝛽⋅𝑥𝑚𝑢𝜑𝑑𝑥∀𝜑∈𝐻. (3.5)
Using the Green formula, we obtain _ΩΔ𝑒^𝛽⋅𝑥Δ𝑢𝜑𝑑𝑥=−_Ω∇𝑒^𝛽⋅𝑥Δ𝑢⋅∇𝜑𝑑𝑥+_𝜕Ω𝜑𝜕𝑒𝜕𝜈^𝛽⋅𝑥Δ𝑢𝑑𝑥,_Ω𝑒^𝛽⋅𝑥Δ𝑢Δ𝜑𝑑𝑥=−_Ω∇𝑒^𝛽⋅𝑥Δ𝑢⋅∇𝜑𝑑𝑥+_𝜕Ω𝑒^𝛽⋅𝑥Δ𝑢𝜕𝜑𝜕𝜈𝑑𝑥. (3.6)
Then we have _ΩΔ𝑒^𝛽⋅𝑥Δ𝑢𝜑𝑑𝑥=_Ω𝑒^𝛽⋅𝑥Δ𝑢Δ𝜑𝑑𝑥+_𝜕Ω𝜑𝜕𝑒𝜕𝜈^𝛽⋅𝑥Δ𝑢𝑑𝑥−_𝜕Ω𝑒^𝛽⋅𝑥Δ𝑢𝜕𝜑𝜕𝜈𝑑𝑥. (3.7)
Thus, the prove is complete.

Theorem 3.6. Let 𝒮_𝛽={𝑢∈𝐻/‖𝑢‖²_2,2,𝛽=∫_Ω𝑒^𝛽⋅𝑥|Δ𝑢|²𝑑𝑥=1}, then we have (a)𝜎₃(Δ²,𝑚)=⋃^+∞_𝑛=1𝐺(Γ_𝑛(𝑚,⋅)), where the function Γ_𝑛(𝑚,⋅): ℝ^𝑁→ℝ is defined by 1Γ_𝑛𝑚,𝛽=sup_{𝐹𝑛∈ℱ𝑛(𝐻)}min_{𝑢∈𝐹𝑛Ω}𝑒^𝛽⋅𝑥𝑚|𝑢|²𝑑𝑥,𝑢∈𝒮_𝛽∩𝐹_𝑛∀𝛽∈ℝ^𝑁, (3.8) 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,⋅).(b)∫_Ω𝑒^𝛽⋅𝑥𝑚|𝑢|²𝑑𝑥≤(Γ₁(𝑚,𝛽))⁻¹‖𝑢‖²_2,2,𝛽. (c)For all ∈ℝ^𝑁∶lim_𝑛→+∞Γ_𝑛(𝑚,𝛽)=+∞.

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 𝑙∶𝐻²(Ω)∩𝐻¹₀(Ω)()()=Δ→ℝ,𝑢,𝑣↦𝑙𝑢,𝑣^2,𝛽𝑢,𝑣 (3.9) defines a scalar product on 𝐻=𝐻²(Ω)∩𝐻¹₀(Ω) equivalent to the usual scalar product ∫_ΩΔ𝑢Δ𝑣𝑑𝑥.
The map 𝑙(⋅,⋅) is a continuous symmetric bilinear form. Since Δ² satisfies the condition of the uniform ellipticity, then we have 𝑙(𝑢,𝑢)=Δ^2,𝛽=𝑢,𝑢_Ω𝑒^𝛽⋅𝑥|Δ𝑢|²𝑑𝑥≥𝑐^′‖𝑢‖²_2,2, (3.10) where 𝑐^′=min_𝑥∈Ω𝑒^𝛽⋅𝑥. Therefore, the bilinear form 𝑙(⋅,⋅) is coercive. On the other hand, the operator 𝑇^2,𝛽∶𝐻²(Ω)∩𝐻¹₀(Ω)→𝐻²(Ω)∩𝐻¹₀(Ω),𝑢↦𝑇^2,𝛽(𝑢)=Δ^2,𝛽−1𝑚𝑒^𝛽⋅𝑥𝑢 (3.11) is well defined, linear, symmetric, and compact on 𝐻. Then, problem (3.2) can be written as 𝑇^2,𝛽(𝑢)=1𝛼𝑢,𝑢∈𝐻²(Ω)∩𝐻¹₀(Ω). (3.12)
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 1𝛼_𝑛=sup_{𝐹𝑛∈ℱ𝑛(𝐻)}min𝑙𝑇^2,𝛽(𝑢),𝑢suchthat‖𝑢‖=1;𝑢∈𝐹_𝑛,1𝛼_−𝑛=inf_{𝐹𝑛∈ℱ𝑛(𝐻)}max𝑙𝑇^2,𝛽(𝑢),𝑢suchthat‖𝑢‖=1;𝑢∈𝐹_𝑛. (3.13)
In addition, we have 𝑙𝑇^2,𝛽𝑢,𝑢=Δ^2,𝛽Δ^2,𝛽−1𝑚𝑒^𝛽⋅𝑥𝑢=,𝑢_Ω𝑚𝑒^𝛽⋅𝑥|𝑢|²𝑑𝑥, (3.14) and ‖𝑢‖²=(𝑢,𝑢)=⟨Δ^2,𝛽𝑢,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.

4. 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.

Theorem 4.1. Let 𝑚 and 𝑚 be two weights with 𝑚<_≢𝑚 and 𝑘∈ℕ. If the eigenfunctions 𝜑_𝑘 associated to Γ_𝑘(𝛽,𝑚) satisfy the (U.C.P) then Γ_𝑘(𝛽,𝑚)>Γ_𝑘(𝛽,𝑚).

Theorem 4.2. Let 𝑚 be a weight and 𝑘∈ℕ. If the eigenfunctions 𝜑_𝑘 associated to Γ_𝑘(𝛽,𝑚) do not satisfy the (U.C.P) then there exists a weight 𝑚 with 𝑚<_≢𝑚, such that, for some 𝑖∈ℕ with Γ_𝑖(𝛽,𝑚)=Γ_𝑘(𝛽,𝑚), one has Γ_𝑖(𝛽,𝑚)=Γ_𝑖(𝛽,𝑚).

As a consequence of Theorems 4.1 and 4.2 we have the following result.

Corollary 4.3. Let 𝑚∈𝐿^∞(Ω) and 𝑘∈ℕ. If Γ_𝑘(𝛽,1)<_≢𝑚<_≢Γ_𝑘+1(𝛽,1), then the only solution of the problem 𝑃_𝑚Δ²𝑢+2𝛽⋅∇(Δ𝑢)+𝛽²Δ𝑢=𝑚𝑢inΩ,𝑢=Δ𝑢=0onΩ (4.1) is 𝑢≡0.

Proof of Theorem 4.1. Let 𝑘∈ℕ; we define the space 𝐹_𝑘=𝜑₁,𝜑₂,…,𝜑_𝑘, (4.2) spanned by the eigenfunctions 𝜑_𝑖 associated to Γ_𝑖(𝛽,𝑚) with _Ω𝑒^𝛽⋅𝑥𝜑_𝑖²𝑑𝑥=1,for𝑖=1,…,𝑘. (4.3)
We have 1Γ_𝑘𝛽,𝑚=min_{𝑢∈𝐹𝑘Ω}𝑚𝑒^𝛽⋅𝑥|𝑢|²𝑑𝑥suchthat_Ω𝑒^𝛽⋅𝑥|Δ𝑢|²𝑑𝑥=1=_Ω𝑚𝑒^𝛽⋅𝑥𝜑²_𝑘𝑑𝑥. (4.4)
Let 𝑢∈𝐹_𝑘, with ∫_Ω𝑒^𝛽⋅𝑥|Δ𝑢|²𝑑𝑥=1. We have either 𝑢 achieves the infimum in (4.4) or not. In the case 𝑢 is an eigenfunction associated to Γ_𝑘(𝛽,𝑚), then by the (U.C.P) and since 𝑢²(𝑥)>0 a.e. 𝑥∈Ω, we have 1Γ_𝑘=𝛽,𝑚_Ω𝑚𝑒^𝛽⋅𝑥𝑢²𝑑𝑥<_Ω𝑚𝑒^𝛽⋅𝑥𝑢²1𝑑𝑥=Γ_𝑘𝑚.𝛽, (4.5)
Thus, Γ_𝑘(𝛽,𝑚)>Γ_𝑘(𝛽,𝑚). In the other case, we have 1Γ_𝑘<𝛽,𝑚_Ω𝑚𝑒^𝛽⋅𝑥|𝑢|²𝑑𝑥<_Ω𝑚𝑒^𝛽⋅𝑥|𝑢|²1𝑑𝑥=Γ_𝑘𝑚.𝛽, (4.6)
Thus, in both cases we have 1Γ_𝑘<𝛽,𝑚_Ω𝑚𝑒^𝛽⋅𝑥𝑢²𝑑𝑥. (4.7)
It follows that 1Γ_𝑘𝛽,𝑚<inf_{𝑢∈𝐹𝑘Ω}𝑚𝑒^𝛽⋅𝑥𝑢²𝑑𝑥;_Ω𝑒^𝛽⋅𝑥|Δ𝑢|²𝑑𝑥=1. (4.8)
This yields the desired inequality (1/Γ_𝑘(𝛽,𝑚))<(1/Γ_𝑘(𝛽,𝑚)). Hence, we have Γ_𝑘(𝛽,𝑚)>Γ_𝑘(𝛽,𝑚).

Proof of Theorem 4.2. Denote by 𝑢 an eigenfunction associated to Γ_𝑘(𝛽,𝑚) which vanishes on a set of positive measure. Take 𝑖 such that Γ_𝑘(𝛽,𝑚)=Γ_𝑖(𝛽,𝑚)<Γ_𝑖+1(𝛽,𝑚) and define 𝑚(𝑥)=𝑚(𝑥)(𝑥)𝑚(𝑥)(𝑥)if𝑢≠0,+𝜖if𝑢=0, (4.9) where 𝜖>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 Δ²𝑢+2𝛽⋅∇(Δ𝑢)+𝛽²Δ𝑢=Γ_𝑖𝛽,𝑚𝑚𝑢=Γ_𝑖𝛽,𝑚𝑚𝑢, (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).

Proof of Corollary 4.3. Suppose that (𝑃_𝑚) has nontrivial solution, that is, 1∈𝜎₃(Δ²,𝑚). From the inequality Γ_𝑘(𝛽,1)<_≢𝑚<_≢Γ_𝑘+1(𝛽,1) and the strict monotonicity, we deduce Γ_𝑘𝛽,Γ_𝑘𝛽,1>Γ_𝑘,Γ𝛽,𝑚_𝑘+1𝛽,𝑚>Γ_𝑘+1𝛽,Γ_𝑘+1𝛽,1. (4.11)
Since Γ_𝑘+1𝛽,Γ_𝑘+1𝛽,1=Γ_𝑘𝛽,Γ_𝑘𝛽,1=1 (4.12)
we deduce that Γ_𝑘𝛽,𝑚<1<Γ_𝑘+1𝛽,𝑚 (4.13) which is a contradiction. Hence, the proof is complete.

Acknowledgment

The authors would like to thank the anonymous referees for their constructive suggestions.