Convergence rates of eigenvalue problems in perforated domains: the case of small volume

1.1 Motivations and main results

The homogenization theory of partial differential equations in perforated domains has been extensively studied in the past four decades due to its theoretical and numerical importance. In this paper, we are concerned with the Dirichlet eigenvalue problem of Laplace operator in periodically perforated domains: find ( ψ η , ε , λ η , ε ) ∈ H 0 1 ( Ω η , ε ) × R such that

where Ω _η,ɛ is a bounded and periodically perforated domain in R d , d ≥ 3. This problem was introduced by J. Rauch [1] in a crashed ice problem (also see [2]), wherein the first eigenvalue of (1.1) determines the rate of cooling for the evenly distributed crashed ice. Since −Δ is symmetric with compact resolvent in L ²(Ω _η,ɛ ), by the classical spectral theorem, the Dirichlet eigenvalues are positive, countable and can be listed in a nondecreasing order λ η , ε k : k = 1,2 , … (counted with multiplicity) such that λ η , ε k → ∞ as k → ∞. We denote by ψ η , ε k the normalized eigenfunction corresponding to λ η , ε k . We are interested in the asymptotic behaviors of λ η , ε k and ψ η , ε k as ɛ → 0.

The perforated domain Ω _η,ɛ is defined as follows. Let 0 < η, ɛ ≤ 1 be two small parameters. Let τ ⁱ (with i = 1, 2, …, m) be C ^1,1 connected domains with diameters comparable to 1. Define Y = [0, 1] ^d . Let τ η i ⊂ Y be a translated copy of ητ ⁱ satisfying dist ( τ η i , τ η j ) > c 0 for i ≠ j and dist ( τ η i , ∂ Y ) > c 0 . Let T η = ∪ i = 1 m τ η i ̄ ⊂ Y . In other words, T _η is the union of m holes contained in Y. Define

Note that the diameters of the holes in Ω _η,ɛ are roughly ηɛ and thus the holes only occupy a small portion (roughly O(η ^d )) of the volume of Ω.

We will impose a geometric assumption A: for each hole ε ( z + τ η i ) of T _η,ɛ , either the hole has at least a distance c ₀ ɛη away from ∂Ω, or ∂Ω intersects with the hole such that both Ω ∩ ε ( z + τ η i ) and Ω \ ε ( z + τ η i ) (in a neighborhood of ε ( z + τ η i ) ) are Lipschitz domains with a uniform Lipschitz character after rescaling. Let Γ _η,ɛ = ∂Ω ∩ ∂Ω _η,ɛ and Σ _η,ɛ = Ω ∩ ∂T _η,ɛ .

Under the above assumptions with η = 1, the asymptotic behavior of the Dirichlet eigenvalues was first studied by Vanninathan [3] and then the convergence rates were obtained by Oleĭnik, Shamaev and Yosifian [4]. We first describe the procedure of reduction for the case η ≪ 1 which is the same as η = 1 given in [3]. Let Y _η = Y\T _η , viewed as a flat torus with holes. Let ( ϕ η , λ ̄ η ) ∈ H 0 , p e r 1 ( Y η ) × R denote the first eigenvalue and eigenfunction of the cell problem,

with ‖ ϕ η ‖ L 2 ( Y η ) = 1 and ϕ _η = 0 in ∂T _η . The principal eigenfunction ϕ _η is nonnegative and satisfies ϕ η ( x ) ≈ min η − 1 dist ( x , T η ) , 1 , which means that ϕ _η degenerates proportional to a distance function near the holes. Let ϕ _η,ɛ (x) = ϕ _η (x/ɛ). As in the case η = 1, we have

where μ η , ε k denote the Dirichlet eigenvalues of the degenerate problem

with corresponding eigenfunctions ρ η , ε k ⊂ H ϕ η , ε , 0 1 ( Ω η , ε ) . The eigenvalues μ η , ε k can be given by the minimax principle,

Thus, to obtain the asymptotic behaviors of λ η , ε k , it suffices to study those of μ η , ε k . More details can be found in Section 2.

For the case η = 1, it was shown in [4] that | μ 1 , ε k − μ 1 k | ≲ ε , where μ 1 k are the eigenvalues of

and A ̄ 1 is a constant coefficient matrix satisfying the ellipticity condition. In view of (1.4), we have

Note that the homogenized eigenvalue problem (1.7) is posed in the domain Ω without holes, which is well-understood. The convergence rates of O(ɛ) for the eigenfunctions were also obtained in [4]. We point out that the convergence rate of order O(ɛ) obtained in [4] is optimal in the case of η = 1. However, due to several technical reasons, the implicit constants obtained via their approach is not quantitative (uncomputable) and depends essentially on the spectral gaps (appearing in the denominator).

For the small holes with η ≪ 1, a special case with d = 3 and η = ɛ ² was also studied in [4]. It was shown that as ɛ → 0, the eigenvalues of (1.5) converge to the Dirichlet eigenvalues of −Δ in Ω. More precisely, for any k ≥ 1,

where ξ 0 k is the kth Dirichlet eigenvalue of −Δ in Ω, or equivalently, ξ ̄ + ξ 0 k is the eigenvalue of − Δ + ξ ̄ , and ξ ̄ > 0 is a fixed number independent of ɛ.

In this paper, we revisit (1.1) with holes of arbitrary size, particularly including the case of small volume. We will show the optimal quantitative error estimate of eigenvalues under a refined asymptotic expansion. The following is the main result of this paper.

Compared with the results in [4], our result in Theorem 1.1 has two major improvements.

1. In the case η ≪ 1, our result shows that, compared to −Δ, the refined operator − ∇ ⋅ ( A ̄ η ∇ ) is a more accurate effective operator. In particular, we may compare the result of the special case d = 3 and η = ɛ ² with that of [4] mentioned before. In this case, (1.10) is reduced to

   (1.12) | λ ε , η k − ε − 2 λ ̄ η − μ η k | ≤ C k ε 2 ,

   where λ ̄ η ≈ η d − 2 = ε 2 . Definitely, the error in (1.12) is much sharper than (1.9). Moreover, it can be shown that | A ̄ η − I | ≤ C η d − 2 2 = C ε and therefore | μ η k − ξ 0 k | ≤ C k η d − 2 2 = C k ε , which explains why the improvement is reasonable by using the refined operator − ∇ ⋅ ( A ̄ η ∇ ) instead of −Δ. The refined operator takes advantage of both the homogenization process at large scales, leading to the factor ɛ, and the perturbation due to the perforation of small η at small scales, leading to the factor η d − 2 2 in (1.10). We should point out that, from the point of view of numerical computation, the calculation for A ̄ η for a fixed η is a one-time cost for solving a cell problem similar to η = 1 which makes it practical in applications.

2. The constant C _k in Theorem 1.1 and Theorem 1.2 below is quantitative in the sense that its dependence on k and the geometric characters can be determined. In particular, C _k is bounded by C ₀ k ^a for some computable a = a(d) > 0, where C ₀ is independent of k. However, the constants given by [4] are not fully quantitative and may rely on the spectral gaps.

Our second result is concerned with the convergence rates of eigenfunctions, which are typically more difficult due to the multiplicity of eigenvalues or the small spectral gaps between them; see [5], [6], [7] for related problems. As such, it seems natural to consider the asymptotic behaviors of an eigenspace corresponding to a collection of eigenvalues in a narrow spectral band.

To describe the result, recall that ρ η k are the normalized eigenfunctions of (1.15) corresponding to μ η k . For θ > 0, t > 0, define

In other words, S _η (θ; t) is the eigenspace of the homogenized eigenvalue problem (1.15) corresponding to the spectrum in the band [θ − t, θ + t].

Roughly speaking, the above theorem states that an eigenfunction of (1.1) corresponding to the eigenvalue λ ε k = ε − 2 λ ̄ + μ η , ε k can be well approximated by ϕ _η,ɛ u _η , where ϕ _η,ɛ (x) = ϕ _η (x/ɛ) and u _η is a finite linear combination of the eigenfunctions of the homogenized operator corresponding to the eigenvalues in a narrow band (with width 2t) around μ η k . The main novelty here is that one is free to choose the width of the band and the estimate is independent of the spectral gap near μ η k . In particular, if μ η k is simple and has large spectral gaps from μ η k − 1 and μ η k + 1 (e.g., the principal eigenvalue μ η 1 ), then ψ ε k converges to ϕ η , ε ρ η k in L ² (Ω _η,ɛ ) with an error of order O ( ε 1 2 η d − 2 2 ∧ ε ) . This rate may not be optimal in general which is caused by the suboptimal estimates of boundary layers (two different approaches will be used). The optimal boundary layer estimate in a perforated domain is an interesting and challenging problem that is beyond the scope of this paper. On the other hand, in case of small t, the negative power t ⁻¹ in (1.13) cannot be avoided generally and the dominating term t − 1 ε η d − 2 2 (say t < ε 1 2 ) should be optimal.

1.2 Strategy of the proof

First of all, we point out that the optimal upper bounds of the eigenvalues actually can be proved directly by the minimax principle. However, the optimal lower bounds of the eigenvalues are much harder and we develop a new approach to resolve the problem.

The proofs of our main results (as well as the proof in [4] for η = 1) can be roughly divided into two stages. In the first stage, we need to obtain an a priori quantitative convergence rate (not optimal) of eigenvalues in dependent of the spectral gaps. By this and the classical Weyl’s law for the homogenized eigenvalue problem, we are able to locate some common large spectral gaps between eigenvalues (see Lemma 5.5), which will play a key role in the second stage. We mention that in [4] only a qualitative convergence was proved as their first stage and hence their results are not fully quantitative. Unlike the other homogenization problems in domains without holes (see, e.g., [8], [9]), our proof for the degenerate problems in perforated domains involves an intermediate problems as a bridge.

In fact, in order to study the eigenvalue problem (1.5), we need to consider the boundary value problem,

and the corresponding homogenized problem is given by

Unfortunately, the convergence from ϕ η , ε 2 f to f is very weak if f ∈ L ^p (Ω) and therefore it is impossible to establish a quantitative convergence rate from u _η,ɛ to u _η under the assumption f ∈ L ^p (Ω) for some p ≤ ∞, which is a necessary regularity assumption in the application of minimax principle to (1.5). To handle this difficulty, we introduce an intermediate eigenvalue problem,

and the corresponding equation

Observe that (1.17) only has an oscillating factor ϕ η , ε 2 on the right-hand side. Thus, the solution u ̃ η , ε , as well as the eigenfunctions in (1.16), admits better regularity (indeed, u ̃ η , ε ∈ W 2 , p ( Ω ) , provided f ∈ L ^p  (Ω _η,ɛ )). Now, we consider two steps of convergence via the intermediate problem: the convergence rate of the solutions from (1.14) to (1.17) can be established under the condition f ∈ L ^p  (Ω _η,ɛ ) for some p > d (see Theorem 4.6); while the convergence rate from (1.17) to (1.15) can only be shown under the stronger condition f ∈ W ^1,p (Ω) for some p > d (see Theorem 4.7). The regularity condition on f in either step is compatible with the a priori regularity of the corresponding eigenfunctions and thus allows us to use the minimax/maximin principle to derive the suboptimal convergence rates of eigenvalues (see Proposition 5.1 and Proposition 5.3). We emphasize that in order to extract the small factor η d − 2 2 in the convergence rates, we have to estimate the weight ϕ _η and correctors χ _η optimally (see Section 2) and carry out a careful analysis throughout the entire proof (see Section 4).

In the second stage, we prove the optimal convergence rates of eigenvalues (and the convergence rates of eigenfunctions as a byproduct). To this end, we first prove an improved convergence rate (still not optimal) directly from (1.14) to (1.15) without relying on the intermediate problem (1.17) under the stronger assumption f ∈ W ^1,p (Ω) for some p > d, i.e.,

where T η , ε : f → u η , ε and T η : f → u η are the resolvents given by (1.14) and (1.15), respectively. Then, we apply a duality argument to show the optimal convergence rate in the weak sense, i.e., for any g ∈ W ^1,p (Ω),

Combining the above two estimates with the large spectral gaps found in the first stage and the almost orthogonality of eigenfunctions (see Lemma 6.8), we are able to analyze precisely the dimension of the eigenspace of T η , ε corresponding to the eigenvalues less than μ η k + C k ε η d − 2 2 for each k ≥ 1, which implies the optimal lower bounds of the eigenvalues of T η , ε . This argument relies essentially on the orthogonal structure of linear eigenspaces.

2.1 The cell eigenvalue problem

This subsection is devoted to the estimates of the eigenpair ( ϕ η , λ ̄ η ) .

Recall that Y _η = Y \ T _η . Define

Here and after, a function defined in Y _η is said to be Y-periodic if it has the same trace on the opposite faces of Y. Thus the functions in H 0, per 1 ( Y η ) can be identified as Y-periodic functions in R d with no jumps across the cell boundaries.

Consider the eigenvalue problem

with ( u , λ ) ∈ H 0, per 1 ( Y η ) × R , where Y _η should be understood as a flat torus with holes (i.e., the equation continues to hold across ∂Y). It is well-known that the principal eigenvalue of the above cell problem, denoted by λ ̄ , is positive and simple, and the corresponding eigenfunction, denoted by ϕ _η , does not change sign in Y _η . Without loss of generality, we assume

For convenience, we extend ϕ _η by zero in the interior of T _η and denote it still by ϕ _η .

Since ϕ _η is Y-periodic, by rescaling, we see that ϕ _η,ɛ (x)≔ϕ _η (x/ɛ) satisfies

Proof. First of all, the eigenvalue equation (1.3) gives ‖ ∇ ϕ η ‖ L 2 ( Y ) 2 = λ ̄ η ‖ ϕ η ‖ L 2 ( Y ) 2 ≈ η d − 2 , by Lemma 2.1. This proves the first part of (i). The remaining estimates will be divided into several steps.

Step 1: Establish (ii) and the upper bounds in (iii). To do this, we use the observation that if −Δu = λu in B _2r  (x ₀) and |λ|r ² ≤ 1, then

and

The estimates above follow from the standard elliptic estimates for −Δu = F by an iteration argument. We note that the estimates above continue to hold if we replace B _r  (x ₀) by B _r  (x ₀) ∩ Y _η , where x ₀ ∈ ∂T _η , 0 < r ≤ cη, and −Δu = λu in Y _η , u = 0 on ∂T _η . As a result, we see that if x ∈ Y _η and dist (x, T _η ) ≤ η, then applying (2.11) to ϕ _η gives

where we have used the fact ‖ ∇ ϕ η ‖ L 2 ( Y ) 2 = λ ̄ η ≈ η d − 2 . Since ϕ _η,ɛ = 0 on ∂T _η , this implies that

if x ∈ Y _η and dist (x, T _η ) ≤ η. A similar argument shows that if x ∈ Y and r = dist (x, T _η ) ≥ η, then

Note that (2.12) and (2.14) together leads to (ii). Moreover, it follows by integration that ϕ _η (x) ≤ C for dist (x, T _η ) ≥ η. This together with (2.13) gives the upper bound in (iii).

Step 2: Prove the second part of (i). Let L _η = ⨏ _Y ϕ _η . Then

It follows that

where we have used the upper bound ϕ _η ≤ C in Step 1. Since ∫ Y ( ϕ η ) 2 = 1 , we see that

This implies that | L η − 1 | ≤ C η d − 2 2 , by the positivity of L _η . Thus,

where p = 2 d d − 2 (for d > 2).

Step 3: Prove the lower bounds in (iii). Since T _η is a union of finite holes τ η i , it suffices to consider the behavior of ϕ _η near an arbitrary hole. Let x ₁ ∈ Y (view Y as a flat torus). We first claim that there exists C _∗ > 0 such that

Actually, if the above inequality is not true, then

This contradicts to (2.6) if we choose C _∗ such that 1 2 | B 1 | C ∗ d − 2 2 > C 1 . Hence the claim is proved.

We next employ a standard lifting technique to reduce the nonnegative eigenfunction ϕ _η to a nonnegative harmonic function so that the Harnack inequality applies. Define h ( x , s ) = exp λ ̄ η 1 2 s ϕ η ( x ) . Then Δ + ∂ s 2 h = 0 in Y _η × (−1, 1) and h = 0 on ∂T _η × (−1, 1). Now, if B 2 C * η ( x ) ∩ T η = ∅ , by the Harnack inequality for the nonnegative harmonic function h in Y _η × (−1, 1) and the previous result,

where we have used the fact h (x, s) ≈ ϕ _η (x) for any (x, s) ∈ Y _η × (−1, 1) and (2.15). This shows that ϕ _η (x) ≥ c ₁ for any x with dist (x, T _η ) ≥ 2C _∗ η.

Now if x ∈ Y _η with dist (x, T _η ) < 2C _∗ η, then by a standard barrier function argument and the lower bound for dist (x, T _η ) ≥ 2C _∗ η, we have

This can be shown by considering each hole τ η i and the boundary behavior of h near the C ^1,1 boundary ∂ τ η i × ( − 1,1 ) . The lower bound in (2.8) then follows.

Step 4: Prove (iv). We view ϕ _η − 1 as a solution of

Then for r = dist (x, T _η ) > η, we apply the interior estimate to get

where we have used (2.6) and (2.8) in the last inequality. Finally for dist (x, T _η ) ≤ η, (2.8) gives the desired bounded. □

2.2 Weighted sobolev spaces

Let

Let H ϕ η , p e r 1 ( Y η ) be the periodic ϕ _η -weighted Sobolev space defined by

Similarly, define the ϕ _η,ɛ -weighted Sobolev space in Ω _η,ɛ by

For simplicity, from now on, we will denote H ϕ η , ε , 0 1 ( Ω η , ε ) by V _η,ɛ .

The corresponding norms of the above spaces are given by

and

Let us first list some properties of the weighted Sobolev spaces V _η,ɛ and H ϕ , p e r 1 ( Y η ) . These properties guarantee the solvability of the degenerate equation associated with the operator L η , ε and the validity of the classical spectral theorem for compact self-adjoint operators in separable Hilbert spaces.

The proof of the qualitative properties (i), (iii) and (iv) in Proposition 2.3 can be found in [3, Proposition 5.1] and the references therein. The quantitative estimates independent of ɛ and η are special cases of Theorem 3.2 and Theorem 3.3 proved in Section 3.

The proof of Proposition 2.4 can be found in [3, Proposition 5.2] and the references therein. The independence of η of the constant C in (ii) can be seen by the Hardy’s inequality applied in a cell as in (3.8).

2.3 Two-scale expansion

One of the main tools to study the asymptotic behavior of the eigenvalues is the minimax principle (or the maximin principle). The minimax principle for the eigenvalue problem (1.1) states that

The following lemma, first discovered in [3], provides the two-scale relationship between the eigenvalues of (1.1) and the principal eigenvalue of (2.1).

See [3, Lemma 6.1] for a proof of the above lemma.

Dividing ∫ Ω η , ε ϕ η , ε 2 v 2 on both sides of (2.22), we get

By Proposition 2.3 (iii) and the minimax principle (2.21), we see