Lower bounds for the spectrum of the Laplace and Stokes operators

We prove Berezin--Li--Yau-type lower bounds with additional term for the eigenvalues of the Stokes operator and improve the previously known estimates for the Laplace operator. Generalizations to higher-order operators are given.


Introduction
Sharp lower bounds for the sums of the first m eigenvalues of the Dirichlet Laplacian −∆ϕ k = µ k ϕ k , ϕ k | ∂Ω = 0 were obtained in [10]: m k=1 µ k ≥ n 2 + n (2π) n ω n |Ω| 2/n m 1+2/n . (1.1) Here |Ω| < ∞ denotes the volume of a domain Ω ⊂ R n and ω n denotes the volume of the unit ball in R n . It was shown in [9] that the estimate (1.1) is equivalent by means of the Legendre transform to an earlier result of Berezin [3].
In view of the classical H. Weyl asymptotic formula µ k ∼ (2π) n ω n |Ω| 2/n k 2/n as k → ∞, the coefficient of m 1+2/n in (1.1) is sharp, However, an improvement of the Li-Yau bound with additional term that is linear in m was obtained in [11]: In the theory of the attractors for the Navier-Stokes equations (see, for example, [2,4,15] and the references therein) lower bounds for the sums of the eigenvalues {λ k } ∞ k=1 of the Stokes operator are very important. In the case of a smooth domain the eigenvalue problem for the Stokes operator reads: Li-Yau-type lower bounds for the spectrum of the Stokes operator were obtained in [6]: m k=1 λ k ≥ n 2 + n (2π) n ω n (n − 1)|Ω| 2/n m 1+2/n . (1.5) The coefficient of m 1+2/n here is also sharp in view of the asymptotic formula ([1] (n = 3), [12] (n ≥ 2)): λ k ∼ (2π) n ω n (n − 1)|Ω| 2/n k 2/n as k → ∞. (1.6) The main results of this paper are twofold. First, we extend the approach of [11] to the case of the Stokes operator and, secondly, we obtain the exact solution of the corresponding minimization problem, thereby giving a much better value of the constant c n in (1.2) (in fact, the sharp value in the framework of the approach of [11]).
Here and in what follows I is defined in (1.3).
By orthonormality we always have R n F (ξ)dξ = m, and taking the first m eigenfunctions of the Laplace (or Stokes) operator for the ϕ k (or the u k , respectively) we get It was shown in [10] that the minimizer F * is radial and has the form shown in Fig. 1, where r * is defined by the condition R n F * (|ξ|)dξ = m: giving (1.1) upon substituting M = (2π) −n |Ω| for the Laplacian and giving (1.5) upon substituting M = (2π) −n (n − 1)|Ω| for the Stokes operator [6].
The additional regularity property of F (ξ): |∇F (ξ)| ≤ L gives a better lower bound [11]: (1.8) Clearly Σ M,L (m) ≥ Σ M (m) and Lemma 1 in [11] (in the notation our paper) reads: giving (1.2) with c n = 1/(24(n+2)) by substituting M = (2π) −n |Ω| and L = 2(2π) −n |Ω|I. In §3 we find the exact solution of the minimization problem (1.8): where t(m * ) is the unique positive root of the equation We also find the first three terms of the asymptotic expansion of the solution Σ M,L (m) in the following descending powers of m: m 1+2/n , m, m 1−2/n , m 1−4/n , . . . . Namely, 10) which shows that the second term is for all n linear with respect to m and positive with coefficient that is n(n+2)/2 times greater than that in (1.9), while the third term is always negative.
Dropping the third term and using the expressions for M and L we obtain the following asymptotic lower bounds. Accordingly, for large m the coefficient of m in the second term on the right-hand side in (1.11) is n(n + 2)/2 times greater than that in (1.2).
Then in §4 we turn to the analysis of the particular cases n = 2, 3, 4. The main result consists in the explicit formulas for Σ M,L (m). The case n = 2 is the simplest and we find (see Lemma 4.1) the explicit formula for the exact solution which coincides with the first three terms of its asymptotic expansion (The inequality Σ M,L (m) ≥ Σ 0 (m) probably holds for any n, not only for n = 2, 3, 4.) Then the negative contribution from the third term in (1.10) is compensated by a (1 − β)part of the positive second term (where 0 < β < 1 and β is sufficiently close to 1) and we obtain the following theorem.
Remark 1.1. Two term lower bounds for the 2D Laplacian with the second term of growth higher than linear in m were obtained in [7]. They depend on the shape of ∂Ω.

Estimates for orthonormal vector functions
Throughout Ω is an open subset of R n with finite n-dimensional Lebesgue measure |Ω|: We recall the functional definition of the Stokes operator [4,8,14]: V denotes the set of smooth divergence-free vector functions with compact supports and H and V are the the closures of V in L 2 (Ω) and H 1 (Ω), respectively. The Helmholtz-Leray orthogonal projection P maps L 2 (Ω) onto H, P : L 2 (Ω) → H. We have (see [14]) , where the last inclusion becomes equality for a bounded Ω with Lipschitz boundary. The Stokes operator A is defined by the relation and is an isomorphism between V and V ′ . For a sufficiently smooth u where {v k } ∞ k=1 ∈ V are the corresponding orthonormal eigenvectors. Taking the scalar product with v k we have by orthonormality and (2.2) that In case when Ω is a bounded domain with smooth boundary the eigenvalue problem (2.3) goes over to (1.4). We recall that a family Proof. Given an suborthonormal system {ϕ i } m i=1 (with supports in Ω), we build it up to a orthonormal system is an arbitrary orthonormal system with supports in R n \ Ω. The condition (ψ k , ψ l ) = δ kl is satisfied if the we chose for the matrix a = a ij the symmetric non-negative The system {ψ i } m i=1 classically satisfies Bessel's inequality, and since (ψ k , f ) = (ϕ k , f ), this gives (2.6).
Suborthonormal families typically arise as a result of the action of an orthogonal projection [5].
k=1 is orthonormal and P is an orthogonal projection, then both families η k = P ϕ k and ξ k = (I − P )ϕ k are suborthonormal.
We now obtain some estimates for the Fourier transforms for (sub)orthonormal families.
Proof. This follows from (2. The next lemma [6] is essential for the Li-Yau bounds for the Stokes operator and says that under the additional condition div u k = 0 the factor n in the previous estimate is replaced by n − 1. Proof. First we observe that ξ · u k (ξ) = (2π) −n/2 i u k · ∇ x e −iξx dx = 0 for all ξ ∈ R n ξ since the u k 's are orthogonal to gradients (see (2.1)). Let ξ 0 = 0 be of the form: The general case reduces to the case (2.10) by the corresponding rotation of R n about the origin represented by the orthogonal (n × n)−matrix ρ. Given a vector function u(x) = (u 1 (x), . . . , u n (x)) we consider the vector function A straight forward calculation gives that div u ρ (x) = div u(y), where ρ −1 x = y. In addition, (u ρ , v ρ ) = (u, v). Combining this we obtain that the family {(u k ) ρ } m k=1 is orthonormal and belongs to H(ρΩ).

Minimization problem
There is not much difference now between the Laplace and the Stokes operators, and the problem of lower bounds for the eigenvalues reduces to the problem of finding Σ M,L (m) defined in the minimization problem (1.8).
We consider the symmetric-decreasing rearrangement F * (ξ) of the F (ξ). It is well known (see, for example, [13]) that 0 ≤ F * (ξ) ≤ M, F * (ξ) dξ = F (ξ) dξ = m and, in addition, |∇F * (ξ)| ≤ ess sup|∇F (ξ)|. Also, This inequality follows from the Hardy-Littlewood inequality where G(ξ) = G * (ξ) = R 2 − |ξ| 2 and without loss of generality we assume that the ball B R contains the supports of F and F * . Thus, we obtain a one-dimensional problem equivalent to (1.8): where F (r) is decreasing and without loss of generality we assume that F is absolutely continuous.
We consider the function Φ s (r) shown in Fig. 2: Proof. If F is an admissible function and F (s) = Φ s (s) (= M), then F ≡ Φ s . Hence for any admissible function F such that F = Φ s (and, hence, F (r 0 ) < M = Φ s (r 0 ) at some point r 0 , 0 ≤ r 0 < s), the graph of F intersects the graph of Φ s to the right of r 0 at exactly one point with r-coordinate a, where a is in the region s < a < s + M/L. In other words, F (r) ≤ Φ s (r) for 0 ≤ r ≤ a and F (r) ≥ Φ s (r) for a ≤ r < ∞. Therefore where the functions under the integral sings are non-negative.

Lemma 3.2. By a straight forward calculation
Combining the above results we see that the minimizing function is given by (3.3) and the second condition in (3.2) becomes σ n ∞ 0 r n−1 Φ s (r)dr = m, which in view of (3.5) gives the equation for t (and s): It will be shown (see (3.11)) that for m ≥ 1 the right-hand side in (3.6) is greater than 1.
Since the left-hand side is a polynomial of order n (with positive coefficients) monotonely increasing from 1 to ∞ on R + , the equation (3.6) has a unique solution t = t(m * ) ≥ 0. Using (3.5) this time with γ = n + 1 we find the solution of (1.8), that is, Σ M,L (m). In other words, we have just proved the following result.
where t(m * ) is the unique positive root of the equation (3.6).
Remark 3.1. The shape of the minimizer (3.3) was found in [7]. We use it here to find the exact solution (3.7) of the minimization problem (1.8).
We give explicit expressions for Σ M,L (m) (and thereby explicit lower bounds for sums of eigenvalues of the Laplace and Stokes operators) for the dimension n = 2, 3, 4 in §4.
Meanwhile we obtain the asymptotic expansion for Σ M,L (m) valid for all dimensions n.
First, it is convenient to write the right-hand side in (3.6) in the form (3.9) The first term here is obvious, the second and the third terms can be found in the standard way. Therefore substituting (3.9) into the second factor in (3. it remains to substitute into (1.10) M S and L S from (2.13). This gives that m k=1 ∇u k 2 ≥ r. h. s of (1.12) and inequality (1.12) follows by taking the first normalized eigenvectors of the Stokes problem for the u k 's. The proof of (1.11) is totally similar.
We conclude this section by checking that both for the Laplace and Stokes operators m * ≥ 1, that is, (n + 1)L n ω n M n+1 ≥ 1. (3.11) (Geometrically this means that Φ s always has a horizontal part.) This follows from the inequality which, in turn, is (3.1) with F being the characteristic function of Ω. In fact, (3.12) and the formulas for M and L give much more than (3.11): for the Laplace and Stokes operators, respectively, in the sense that the right-hand sides in (3.13) tend to infinity as n → ∞.

4.
Lower bounds for the Laplace and Stokes operators for n = 2, 3, 4 The case n = 2. The two-dimensional case is the simplest and the results are the most complete. Proof. In view of (3.7) we only need to calculate the last factor there. The positive root t(m * ) of the equation (3.6) n=2 , which is the quadratic equation (t + 1) and using (3.8) we obtain The rest is a direct substitution. We note that Σ M,L (m) = Σ 0 (m) n=2 , see (1.10). Proof. We consider (4.3). In view of (2.14) we have M = M L = (2π) −2 |Ω| and L = L L = 2(2π) −2 |Ω|I, therefore (4.1) gives for the Laplacian where the last inequality follows from (3.12): |Ω| 2 /I ≤ 2π. The proof (4.4) is similar: The proof of this theorem (which is Theorem 1.2 n=2 ) is complete.
The case n = 4.
The case n = 3.
Proof of Theorem 1.2 n=3 . The proof immediately follows from (4.6). The proof of Theorem 1.2 is complete.

Further examples. Dirichlet bi-Laplacian
Other elliptic equations and systems with constant coefficients and Dirichlet boundary conditions can be treated quite similarly. We restrict ourselves to the Dirichlet bi-Laplacian: We consider the L 2 -orthonormal family of eigenfunctions {ϕ k } m k=1 ∈ H 2 0 (Ω). Then the function F (ξ) = m k=1 | ϕ k (ξ)| 2 satisfies the same three conditions: where as before M = (2π) −n |Ω| and L = 2(2π) −n |ΩI. Since m k=1 ν k = R n |ξ| 4 F (ξ) dξ, we have to find the solution Σ 4 M,L (m) of the minimization problem