Spectral Convergence of the Laplace Operator with Robin Boundary Conditions on a Small Hole

In this paper, we study a bounded domain with a small hole removed. Our main result concerns the spectrum of the Laplace operator with the Robin conditions imposed at the hole boundary. Moreover, we prove that under some suitable assumptions on the parameter in the boundary condition, the spectrum of the Laplacian converges in the Hausdorff distance sense to the spectrum of the Laplacian defined on the unperturbed domain.


Introduction
It is a common expectation that small perturbations of the physical situation will lead to only a small change in the spectrum.In the case of domain perturbations, this is largely true for Dirichlet boundary conditions, while the Neumann or Robin case is more delicate.In the recent literature, such questions have already received quite a few answers, starting with the seminal work of Rauch and Taylor on the spectrum of the Laplace operator of domains with holes [13].
An excellent shortcut to the recent work on the asymptotic behavior of the eigenvalues of the Laplace operator on the domain with small spherical obstacles imposing the Neumann condition at their boundary and the Dirichlet condition at the rest part of the boundary can be found in [11].
Maz'ya, Nazarov and Plamenewskii, see [10, Ch.9, vol.I], have considered the Laplace operator on the domain with obstacles, imposing the Dirichlet condition on their boundary and have proved the validity of a complete asymptotic expansion for the eigenvalues.
For a survey on more recent research in this subject, we refer the reader to [4], [5], [6] where authors have considered the Dirichlet Laplacians on Euclidean domains or manifolds with holes and studied the problems of the resolvent convergence.
The problems with small Neumann obstacles of more general geometry can cause abrupt changes in the spectrum.For example, such an effect is observed when the hole has a "split ring" geometry, see [14].The split ring (even if very small) can produce additional eigenvalues that have nothing in common with the eigenvalues of the Neumann Laplacian on the unperturbed domain.The problems with the Neumann obstacles having more general geometry have been studied in [5] and later in [2].
The Robin case for general self-adjoint elliptic operators was considered in [8] but with the restriction that the boundary of the unperturbed domain is C 2 -smooth.In the mentioned work shrinking the hole and scaling properly the parameter in the boundary condition, the authors obtain an operator family that converges, in the norm-resolvent sense, to an operator with a point interaction in the domain without the hole.Results on resolvent convergence for operators with Robin conditions in domains with small holes in higher dimensions were also considered in [3], but in this paper the original domain must again be C 2smooth and the Robin condition was used with the coefficient independent of the hole size.
In this paper we will focus our attention on the spectral properties of a Laplacian defined on a two-dimensional bounded domain with no additional assumptions on the smoothness of its boundary with a single hole K ε (for a fixed parameter ε) having the Lipschitz boundary.On the boundary of the original domain we impose the Dirichlet or Neumann condition, and on the boundary of the hole we impose the Robin condition with the coefficient depending on the size of the hole.
Our main result is the proof of the spectral convergence, in the Hausdorff distance sense, of the spectrum of the Laplacian defined on the perturbed domain to the spectrum of the Laplacian defined on the original domain.
Plan of the paper.The paper consists of 6 sections, besides this introduction.
In Section 2 we present the main results, and consider a general theorem, namely Theorem 2.4.We will use Theorem 2.4 in the proof of Theorem 2.1 about the spectral convergence for the Laplacians on Ω and Ω \ K ε .
Section 3 contains the main tools of the spectral convergence of operators on varying Hilbert spaces In Section 4 and Section 5 we prove our results to which we already alluded.
In Section 6 we give some auxiliary material established in [2].

Main results
In this section, we present our main results.These results are proven in the following sections.
Let Ω ⊂ R 2 be a bounded domain and K ⊂ Ω be a compact simply connected set with Lipschitz boundary.We denote Ω K := Ω \ K.By using Lemma 4.2, the quadratic forms where µ is the measure on ∂K related to the arc length and γ K is a real number, are closed and semi-bounded from below and hence define unique self-adjoint operators H ΩK (γ K ) and H ΩK (γ K ) which act as the Laplacian on their domains.We will study the question of the convergence of the spectrums of the operators H ΩK (γ K ), H ΩK (γ K ) when the hole K converges to a point.We start by a rather important results in the following theorem.
where M ε explodes to infinity under the condition that M ε = o 1 ε 3/2 .Then, for sufficiently small ε, there exists η(ε) > 0 with η(ε) → 0 as ε → 0, so that the following spectral convergence occurs where d is defined in (3.9) and σ • (•) denotes either the entire spectrum, the essential spectrum, or the discrete spectrum.Furthermore, the multiplicity of the discrete spectrum is preserved.
The previous result motivates the following consequences: Corollary 2.2.Suppose that H N Ω has purely discrete spectrum denoted by λ N k (Ω) (repeated according to multiplicity), and let λ D k (Ω) be the discrete spectrum of H D Ω .Then the infimum of the essential spectrums of H ΩK ε , H ΩK ε tend to infinity and there exists 3 Main tool of the spectral convergence of operators on varying Hilbert spaces For the convenience of the reader, this section begins by reviewing some basic facts that ensure spectral convergence for two operators having different domains.For more information we refer the reader to [12].To a Hilbert space H with inner product (•, •) and norm • together with a non-negative, unbounded operator A we associate the scale of Hilbert spaces where I is the identity operator.We think of (H ′ , A ′ ) as some perturbation of (H, A) and want to relax the assumption so that the spectral properties are not the same, but still close.Definition 1. (see [12]) Suppose we have linear operators Let δ > 0 and k ≥ 1.We say that (H, A) and (H ′ , A ′ ) are δ-close of order k iff the following conditions are fulfilled: for all f, u in the appropriate spaces.Here, a and a ′ denote the sesquilinear forms associated to A and A ′ .
We denote by d Haussdorff (A, B) the Hausdorff distance for subsets A, B ⊂ R: where d(a, B) := inf b∈B |a − b|.We set for closed subsets of [0, ∞).
For the next result, which comes from the work of O. Post [12], we have the following spectral convergence theorem in terms of the distance d.
We now turn to the proof of Theorem 2.4.Since the proof is almost the same for both the Dirichlet and Neumann cases, we will restrict ourselves to the Neumann case.The only difference is Lemma 6.3, but its validity for the Dirichlet case can be easily checked from [2].

Proof of Theorem 2.4
At this stage, we divide the proof into two steps.
We can apply the technique of [2].It is easy to see that and We define Ju = J 1 u = u| χΩ Kε for all u ∈ H and J ′ u = uχ ΩK ε for all u ∈ H ′ .Now let us construct the mapping J ′ 1 : Without loss of generality, assume that the ball B ε mentioned in Theorem 2.1 and Theorem 2.4 is centered at the origin.Let ǫ ∈ (ε, 2ε) be a number to be chosen later and let B ǫ ⊃ B ε be the ball again centered on the origin and radius ǫ, Ω ǫ := Ω\B ǫ .
We will first construct the mapping J ′ 1 first for smooth functions.For each v ∈ C ∞ (Ω Kε ) we define where ṽ(r, ϕ) = v(r cos ϕ, r sin ϕ).Now let us construct the mapping J ′ 1 u for any u ∈ H ′ 1 .Using the approximation method described in [7, Thm.2, 5.3.2],for the fixed sequence {η k } ∞ k=1 converging to zero we construct the sequence To deal with ∂Kε |u| 2 dµ we will use the trace inequality [7].We present it immediately after an auxiliary result on Lipschitz bounds [7].where K depends only on the norm of µ in C 1 (Ω) and δ ∈ (0, 1).
Combining the above lemma with δ = 1 2 and the inequality (4.11) we get Let us mention that in view of the inequalities (4.19) and (4.25), which will be proved later, and the construction of the function J ′ 1 , it follows that for any smooth function v the integrals Ω |∇J ′ 1 v| 2 dx dy and Ω |J ′ 1 v| 2 dx dy can be estimated from above by v 2 1 multiplied by some constant.Combining this with the Lemma 4.2 we get where C(ε) is some constant.Due to the positivity of the coefficient γ Kε and Lemma 4.2 the completeness of the space H 1 is equivalent to the completeness of the Sobolev space H 1 .Thus, using (4.12) and (4.13) we can define Step 2. The conditions (3.1)-(3.7)hold for the mappings J, J ′ , J 1 , J ′ 1 .
Given our construction, we have To complete the proof of (3.5), we use the following lemma, applied with η = ε and Γ = ∅: Lemma 4.3.Let Ω be an open bounded domain in R 2 .Suppose that B ǫ ⊂ Ω is a ball with center at some point x 0 ∈ Ω and radius ǫ > 0. Suppose that Γ ⊂ B ǫ be a bounded simply connected compact set with Lipschitz boundary.Then for any function u ∈ H 1 (B ǫ \ Γ) the following inequality holds where C 1 > 0 is a constant that depends on the distance between the boundary of B ǫ and the boundary of Ω.
Using the construction of J ′ and J ′ 1 and the completeness of Considering that J ′ u = 0 on K ε , J ′ u = u on Ω Kε and J ′ 1 u = u on Ω \ B ǫ one has In view of Lemma 4.2 applied with δ = 1 2 , one estimates from above the right-hand side of (4.17) by 2Kǫ Ω\Bǫ (|∇u| 2 +|u| 2 ) dx dy.Then use the following obvious bound which holds due to the positivity of γ Kε : and the fact that ǫ ≤ 2ε we have Combining the above inequality together with (4.19), the right-hand side of (4.16) can be estimated as follows We now give the proof of the estimate (3.7), i.e. under the assumptions given in Theorem 2.4, the inequality (3.7) holds with k = 2 and δ = O ε 1/6 + γ 1/2 Kε ε 1/4 for sufficiently small ε.Thus, in view of (2.3), δ converges to zero as ε → 0.
As before without loss of generality suppose that u ∈ C ∞ (Ω Kε ).Since J ′ 1 u = u on Ω Bǫ and J 1 u = u on Ω Kε we have (4.21)Since f ∈ H 2 loc (Ω), then using Lemma 6.1 (see Appendix) applied with domain Ω ′ such that Ω ′ ⊂ Ω and B ǫ ⊂ Ω ′ , and the fact that ǫ ≤ 2ε, the first term on the right hand side of (4.21) can be estimated as follows To proceed further with the proof of an upper bound of (4.21), we need to estimate the integral Bǫ |∇(J ′ 1 u)| 2 dx dy.Passing to polar coordinates we get As in the proof of (3.6), we find that Thus, using Lemma 4.2 applied with δ = 1 2 and inequality (4.18), we have Now we come to the second term of (4.23).Given the Lemma 6.2 (see Appendix) used for g = u, there exists a number τ ∈ (ε, 2ε) such that where ũ(r, ϕ) := u(r cos ϕ, r sin ϕ) and τ ∈ (ε, 2ε) is some number.If τ belongs to the interval (ε, 3ε/2], then we take ǫ as the supremum of all such numbers in (ε, 3ε/2].In the opposite case if τ ∈ (3ε/2, 2ε), then let ǫ be the infimum of such numbers.Since u is a smooth function, the above inequality is satisfied with τ = ǫ.
Combining this together with (4.18) we get Thus, by virtue of (4.23), (4.24), the above estimate and using the fact that ǫ ≥ ε, we have Finally, using the above bound and inequality (4.22), we estimate the righthand side of (4.21) as follows Let us now consider the second term in (4.20).By virtue of (4.18) and (4.22) we get Finally we move on to the third term in (4.20).We have Let us first find the appropriate estimate for the first integral on in the right hand side of (4.28).
Let Π(d) = (−d, d) 2 , d > 0, be the maximum square belonging to Ω and containing K ε .For almost all x 0 belonging to the projection of K ε on the axis X, let y(x 0 ) ∈ (−d, d) be the point such that Let us fix any y ∈ (−d, d).Without loss of generality suppose that y > y(x 0 ).Then Without loss of generality assume that the boundary of the unperturbed set which is , where y 1 is some C 1smooth function.Then the parameterization of the boundary of K ε coincides with (x, εy 1 (x/ε)), x ∈ (−ε, ε).
Integrating |f (x, y)| 2 over ∂K ε and using the inequality (4.29) we get where To proceed with a proof we need the following auxiliary result [9]: Lemma 4.4.Let Π ′ ⊂ R n be a convex set and let G and Q be arbitrary measurable sets in Π ′ with µ (G) = 0.Then, for all v ∈ H 1 (Π ′ ), the following inequality holds: With the above bounds, one can show that sufficiently small values of ε, the following is true where and f H 2 (Π(d)) means the Sobolev H 2 (Π(d)) norm of f .Next we need the following interior regularity theorem [1]: Theorem 4.5.(Interior Regularity Theorem.)Suppose that h ∈ H 1 (Ω) is a weak solution of −∆h = w.Then h ∈ H 2 loc (Ω) and for each Ω 0 ⊂ Ω there exists a constant c = c(Ω 0 ) independent of h and w such that: (4.33) In view of the above theorem the right-hand side of (4.32) can be estimated as follows with some constant c = c(d) does not depend on ε.By virtue of inequalities (4.30), (4.34) and Lemma 6.3 (see Appendix) we have The above combined the fact that estimates the right-hand side of the inequality (4.28) as follows ∂Kε By virtue of (4.26), (4.27) and the above inequality the right-hand side of inequality (4.20) satisfies r.h.s.(4.20 which ends the proof.
Let P ǫ denote the projection of M ǫ onto the axis X.We get Returning to the inequality (5.1) and combining the above bound together with the fact that Now let us go to the subset (B ǫ \ Γ) \ M ǫ .For any (x 0 , y 0 ) ∈ (B ǫ \ Γ) \ M ǫ let (x 0 , y 3 (x 0 )) with y 3 (x 0 ) < y 0 , be a point of intersection of line l x0 with the boundary of Ω.One can easily check that there is y 4 (x 0 ) ∈ (y 3 (x 0 ), y 0 ) such that where diam(Ω) is the diameter of Ω and dist((x 0 , y 0 ), ∂Ω) is the distance between (x 0 , y 0 ) and the boundary of Ω and C = 2max 1 dist((x 0 , y 0 ), ∂Ω) , diam(Ω) .

Appendix
In this section we mention several useful lemmas proved in [2].
Theorem 2.1.Let Ω be an open bounded domain in R 2 .Suppose that B ε ⊂ Ω is a ball with center at some point x 0 ∈ Ω and radius ε > 0. Suppose that K = K ε ⊂ B ε be a bounded simply connected compact set with Lipschitz boundary.Let H N Ω and H D Ω be the Neumann and Dirichlet Laplacians defined on the unperturbed domain Ω and H ΩK ε , H ΩK ε be the operators generated by (2.1) and (2.2) on Ω Kε with the coefficient γ Kε > 0 satisfying ) for small enough ε.Here, λ k (Ω Kε ) and β k (Ω Kε ) denote the discrete spectrum of H ΩK ε and H ΩK ε (below the essential spectrum) repeated according to multiplicity.Corollary 2.3.The Hausdorff distance between the spectra of H ΩK ε and H N Under the assumptions of Theorem 2.1 the operators H N Ω and H ΩK ε are δ(ε) close of order 2 with δ(ε) → 0 as ε → 0. The same is true for the operators H D Ω and H ΩK ε .