Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

Abstract

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

Keywords

Eigenvalue perturbation domain perturbation repeated eigenvalue Laplacian eigenvalues asymptotic expansion

1. Introduction

Many asymptotic formulae for the shift of eigenvalues of self-adjoint elliptic differential operators caused by small singular perturbations have been obtained in the case of multiplicity one (see for instance [15, Chapter 9], [3] and references therein, [10,19,20]). The generalization of such expressions to higher multiplicities often requires non-trivial calculations and is restricted to specific cases (see for example [5,14], [12, Theorem 2.5.8]). Moreover, such an effort is usually considered unnecessary due to genericity results of simple eigenvalues (see [16,17,23,24]). Nonetheless, higher multiplicities appear in many natural situations; for instance the Laplacian has non-simple spectrum whenever the domain presents some symmetries. In this article we derive a new tool to study the behaviour and properties of repeated eigenvalues for general types of perturbations. Namely, the main result is a variational characterization (Theorem 3.4) which allows the direct extension of asymptotic formulae which are valid for simple eigenvalues to non-simple ones. We then apply this result to study different domain perturbations of interest in applications.

The article is structured as follows. In Section 2, we recall some results regarding the spectral stability and generic simplicity for second-order self-adjoint elliptic operators. In Section 3, we derive the main variational characterization by a technique which involves a double perturbation at asymptotically different speeds. In Section 4, we specialize the variational characterization to asymptotic formulae which admit a separation of variables. Finally in Section 5, we consider some domain perturbations of particular interest: grounded inclusions, conductivity inclusions, and boundary deformations. We show how the general variational asymptotic formula applies in each of these cases and derive new interesting properties for the Dirichlet and Neumann Laplacian eigenvalues. We refer to Section 5.4 for a more specific summary of the results obtained.

2. Stability and simplicity of the spectrum of second-order self-adjoint elliptic operators

Let $L (Ω)$ be a second-order, self-adjoint, elliptic differential operator defined on Ω and let a be its associated coercive, continuous, and bounded bilinear form. We say that λ is an eigenvalue of $L (Ω)$ with associated eigenfunction $u \in V (Ω)$ if $\begin{matrix} a (u, v) = λ \int_{Ω} u v, \forall v \in V, \end{matrix}$ where $V (Ω) = H_{0}^{1} (Ω)$ in the case of homogeneous Dirichlet boundary conditions, or $V (Ω) = H^{1} (Ω) / R$ in the case of homogeneous Neumann boundary conditions. From standard results in spectral theory, we know that the eigenvalues of a self-adjoint elliptic operator defined on a Lipschitz domain have finite multiplicity and can be arranged in a non-decreasing sequence. We also assume that the associated eigenfunctions are orthonormal in $L^{2} (Ω)$ . With these conventions, we can state the following stability result.

Theorem 2.1.
For every $ε > 0$ , let L be a self-adjoint elliptic operator and let $ρ_{ε}$ be a smooth function with support contained in Ω such that its $C^{2}$ -norm is smaller than ε. For any $n \in N$ , let λ be the n-th eigenvalue of L and $λ_{ε}$ be the n-th eigenvalue of $L + ρ_{ε}$ , and let $u_{ε}$ be an associated eigenfunction. Then there exists a constant C such that for every ε small enough $\begin{matrix} | λ_{ε} - λ | ⩽ C ε, \end{matrix}$ and there exists u, an eigenfunction of λ, such that $\begin{matrix} | u_{ε} - u |_{H^{1} (Ω)} ⩽ C ε . \end{matrix}$

The proof is a consequence of standard results in perturbation theory (see for instance [21, Theorem 1 at p. 57]).

We recall that a family of domains $Ω_{ε}$ converges to Ω as $ε \to 0$ in Hausdorff distance, if $d_{H} (Ω_{ε}, Ω) \to 0$ , where $\begin{matrix} d_{H} (A, B) = max {sup_{a \in A} inf_{b \in B} | a - b |, sup_{b \in B} inf_{a \in A} | a - b |} . \end{matrix}$ We remark that although we will state all our results using Hausdorff convergence, they can be adapted to other types of convergence (for instance in Lebesgue measure or in $C^{k}$ -topology) whenever the domains under consideration satisfy enough regularity requirements.

We are interested not only in stability with respect to smooth perturbations of coefficients but also in singular perturbations of domains. In particular, we will need the following result.
Theorem 2.2.
Let Ω and $Ω_{ε}$ be two Lipschitz domains whose Hausdorff distance one from another is ε. Let $L (Ω)$ and $L (Ω_{ε})$ be second-order self-adjoint elliptic operators with homogeneous Dirichlet or Neumann boundary conditions imposed on $\partial Ω$ and $\partial Ω_{ε}$ . Let M be an upper bound to the Lipschitz constants of $\partial Ω$ and $\partial Ω_{ε}$ , and to the coercivity, continuity and boundedness constants of a. For any $n \in N$ , let λ be the n-th eigenvalue of $L (Ω)$ and $λ_{ε}$ be the n-th eigenvalue of $L (Ω_{ε})$ . Then there exists a constant C which depends only on M, n, and the dimension d such that $\begin{matrix} (1) & | λ_{ε} - λ | ⩽ C ε . \end{matrix}$

A proof of this result can be obtained as a consequence of the theory of transition operators and its applications (see [7] for a survey of this technique). In what follows, we outline another approach which relies on stability results for boundary value problems.
Outline of the proof.
We adapt the argument in [13, Section 4.4] to our case. Let E be an arbitrary Lipschitz domain containing $Ω_{ε} \cup Ω$ and consider $u_{ε}$ extended to the whole E. Let u be the orthogonal projection of $u_{ε}$ from $V (E)$ onto $V (Ω)$ . Let $\overline{u}$ be the unique solution in $V (Ω)$ of $\begin{matrix} a (\overline{u}, v) = λ \int u v, \forall v \in V (Ω) . \end{matrix}$ From [22, Inequality (3.2)], we have that $\begin{matrix} | \overline{u} - u |_{H^{1} (E)} ⩽ C | λ u |_{L^{2} (E)} | λ u |_{V {(E)}^{'}} d_{H} (Ω, Ω_{ε}), \end{matrix}$ where C is a constant which depends only on the Lipschitz constant of Ω, and the constants involved in the continuity, boundedness and coercivity assumptions on a. Since $| u |_{V {(E)}^{'}} ⩽ | u |_{L^{2} (E)}$ , and by Weyl’s law, there is a constant $\tilde{C}$ which depends only on the area of Ω and the dimension d such that $λ ⩽ \tilde{C} n^{d / 2}$ . Then, it follows that $\begin{matrix} | \overline{u} - u |_{H^{1} (E)} ⩽ C \tilde{C} n^{2 / d} | u |_{L^{2} (E)}^{2} d_{H} (Ω, Ω_{ε}) . \end{matrix}$ Hence, by (4.31) and Lemma 14 of [13], we obtain estimate (1). □

The following result concerning stability of eigenfunctions is a particular case of [9, Theorem 1.2 and subsequent discussion].
Theorem 2.3.
Let $Ω_{ε}$ be a family of Lipschitz domains converging to Ω in Hausdorff distance as $ε \to 0$ . Suppose that at least one of the following hypotheses holds:
$Ω_{ε} \subseteq Ω$ for every ε;

the Lipschitz constants of $\partial Ω_{ε}$ are uniformly bounded.
Let λ and $λ_{ε}$ be as in the hypothesis of Theorem 2.2 and let $u_{ε, 1}, \dots, u_{ε, m}$ be an orthonormal basis of the eigenspace of $λ_{ε}$ . Then there exists $u_{1}, \dots, u_{m}$ an orthonormal basis of λ such that as $ε \to 0$ it holds $\begin{matrix} u_{ε, j} \to u_{j} in H^{1} (Ω), \forall j \in {1, \dots, m} . \end{matrix}$

We move on to the issue of genericity of simple eigenvalues.
Theorem 2.4.
For every $ε > 0$ and every open set $U \subseteq Ω$ there exists a smooth function ρ with support contained in U such that its $C^{2}$ -norm is smaller than ε and $L + ρ$ has simple spectrum.

This result is an immediate consequence of [24, Theorem 7]. We remark that the results which follow could also be proved using only domain deformations, but the arguments would require more sofistication due to a more complicated dependence of function spaces on the variation of the underlying domain.
3. Variational asymptotic formula

Recall that the eigenvalues and eigenfunctions of L can be characterized respectively as minima and minimizers of a quadratic functional F. More explicitly, if we indicate as $λ_{1} ⩽ λ_{2} ⩽ \dots$ the eigenvalues of L and $u_{1}, u_{2}, \dots$ some associated orthonormal eigenfunctions, we have that for $F u = {⟨ L u, u ⟩}_{V}$ it holds $\begin{matrix} (2) & u_{n} \in \underset{u \in U_{n}}{argim} F u, λ_{n} = F u_{n}, \end{matrix}$ where $U_{n} = {u \in V : u ⊥ {u_{1}, \dots, u_{n - 1}}, | u |_{V} = 1}$ . However, there is an ambiguity in the choice of eigenfunctions which is particularly relevant to us: if $λ_{n} = \dots = λ_{n + m}$ , any choice of orthonormal eigenfunctions in the linear space spanned by ${u_{n}, \dots, u_{n + m}}$ is still a basis of the eigenspace. Our variational characterization will select the right basis for the problem considered up to a predetermined asymptotic error. More precisely, we make the following assumptions.

Assumption 3.1.
We suppose that there is a family of self-adjoint, elliptic, second-order differential operators $L_{ε}$ such that
$L_{ε}$ converges spectrally to L as $ε \to 0$ , i.e. the n-th eigenvalue of $L_{ε}$ and any of its associated eigenfunctions converge respectively to the n-th eigenvalue of L and to a function in its eigenspace;

there exists an open set on which the differential operators L and $L_{ε}$ coincide for every ε.

Assumption 3.2.
We suppose that we already know an asymptotic expansion for simple eigenvalues. In particular, we assume that if λ is a simple eigenvalue of L with associated eigenfunction u, and if $λ_{ε}$ is an eigenvalue of $L_{ε}$ which converges to λ as $ε \to 0$ , then $\begin{matrix} (3) & λ_{ε} - λ = f (ε, u) + r (ε), \end{matrix}$ where f is a known function and by $r (ε)$ we indicate a function which is of order $o (f (ε, u))$ as $ε \to 0$ . We also require f to be continuous.

Let us isolate a simple property of our operators which will be useful in the upcoming proof.
Remark 3.3.
For any $ρ \in C_{c}^{\infty} (R^{d})$ , the same asymptotics as in (3) holds for simple eigenvalues, if we consider the family of operators $L_{ε} + ρ$ instead of $L_{ε}$ . This is immediate because, if we indicate as $λ_{ε}^{ρ}$ the eigenvalues of $L_{ε} + ρ$ , we have $\begin{matrix} (L_{ε} + ρ) u_{ε} = L_{ε} u_{ε} + ρ u_{ε} = (λ_{ε} + ρ) u_{ε}, \end{matrix}$ therefore the eigenfunctions of $L_{ε} + ρ$ and $L_{ε}$ are the same, and $λ_{ε}^{ρ} = λ_{ε} + ρ$ , for any ε, so $\begin{matrix} λ_{ε}^{ρ} - λ_{0}^{ρ} = λ_{ε} - λ_{0} = f (ε, u) + r (ε) . \end{matrix}$

Under these assumptions, we can derive the following result. Theorem 3.4 (Variational characterization).

Let λ be an eigenvalue of L of multiplicity m, and let $λ_{ε, 1} ⩽ \dots ⩽ λ_{ε, m}$ be the eigenvalues of $L_{ε}$ which converge to λ as $ε \to 0$ . Then, for any $j \in {1, \dots, m}$ , it holds $\begin{matrix} λ_{ε, j} - λ = f (ε, v_{j}) + r (ε), \end{matrix}$ where $\begin{matrix} (4) & v_{j} \in \underset{v \in U_{λ, j}}{argim} f (ε, v), U_{λ, j} = {v \in V : L v = λ v, v ⊥ {v_{1}, \dots, v_{j - 1}}, | v |_{V} = 1} . \end{matrix}$

Proof.
For clarity, we split the proof in two steps. In the first step, we consider two perturbations of L at asymptotically different speeds ε and δ, which will allow us to obtain a first estimate on the eigenvalue perturbation. In the second step, we specialize this formula to the case where δ converges to zero much faster than ε ( $δ ≪ ε$ ), and we argue that this leads to the characterization in the thesis.

Step 1. By Assumption 3.1.b, we can choose an open ball $B \subseteq R^{d}$ on which $L \equiv L_{ε}$ . For every $δ > 0$ , by Theorem 2.4 there exists $ρ_{δ} \in C_{c}^{\infty} (B)$ such that $| ρ_{δ} |_{C^{2}} < δ$ and $L + ρ_{δ}$ has simple spectrum. Let $λ_{1}^{δ} < \dots < λ_{m}^{δ}$ be the eigenvalues of $L + ρ_{δ}$ which converge to λ as $δ \to 0$ , and let $v_{1}^{δ}, \dots, v_{m}^{δ}$ be the associated eigenfunctions. Let $λ_{ε, j}^{δ}$ be the eigenvalues of $L_{ε} + ρ_{δ}$ which converge to $λ_{j}^{δ}$ as $ε \to 0$ . By the standard elliptic estimates of Theorem 2.1 applied to the operator $L + ρ_{δ}$ , there is a constants $C_{1}$ such that $\begin{matrix} (5) & | λ_{j}^{δ} - λ | ⩽ C_{1} δ, \end{matrix}$ and $v_{j}^{δ} \to v_{j}$ in norm as $δ \to 0$ , where $v_{j}$ is a normalized function in the eigenspace of λ. Again by Theorem 2.1, this time applied to $L_{ε} + ρ_{δ}$ , there exists a constant $C_{2}$ such that $\begin{matrix} (6) & | λ_{ε, j}^{δ} - λ_{ε, j} | ⩽ C_{2} δ . \end{matrix}$ From Remark 3.3 we also have $\begin{matrix} (7) & λ_{ε, j}^{δ} - λ_{j}^{δ} = f (ε, v_{j}^{δ}) + r (ε) . \end{matrix}$ Therefore, putting together (5)-(6)-(7), we can immediately rewrite $\begin{matrix} (8) & \begin{matrix} | λ_{ε, j} - λ - f (ε, v_{j}^{δ}) | & ⩽ | λ_{ε, j} - λ_{ε, j}^{δ} | + | λ_{ε, j}^{δ} - λ_{j}^{δ} - f (ε, v_{j}^{δ}) | + | λ_{j}^{δ} - λ | \\ ⩽ C_{1} δ + r (ε) + C_{2} δ . \end{matrix} \end{matrix}$

Step 2. For every $ε > 0$ , let us now choose the parameter δ as a function of ε, small enough so that $δ = O (r (ε))$ . Then, by the continuity of f and (8), as $ε \to 0$ we have the asymptotic expansion $\begin{matrix} (9) & λ_{ε, j} - λ = f (ε, v_{j}) + r (ε) . \end{matrix}$

To retrieve the variational characterization of $v_{j}$ , which for now we know only being a normalized element in the eigenspace of λ, recall that we assumed the ordering $λ_{ε, 1} ⩽ \dots ⩽ λ_{ε, m}$ for every ε. Therefore, to have $λ_{ε, 1} ⩽ λ_{ε, j}$ for all j, as $v_{j}$ is the only quantity which depends on the index j in the right hand side of (9), it must hold $\begin{matrix} λ_{ε, 1} - λ = min_{v} (f (ε, v) + r (ε)), \end{matrix}$ where the minimum is taken among all eigenfunctions of λ of norm 1. Let $v_{1}$ be such a minimizer.

Notice that if $v_{ε, j} \to v_{j}$ for every j, then $v_{1}, \dots, v_{m}$ must be linearly independent, since by assumption the multiplicity of λ is m. Therefore, for $λ_{ε, 2} ⩽ λ_{ε, j}$ to hold for any $j ⩾ 2$ , $v_{2}$ must be the minimizer of the right hand side of (9) on a subspace orthogonal to $v_{1}$ , or equivalently, among all normalized eigenfunctions orthonormal to $v_{1}$ . By repeating the same reasoning for $λ_{ε, 3}, \dots, λ_{ε, m}$ , we then have that (4) must hold for every j. □
Remark 3.5.
We point out that in the previous proof we have used no information regarding the speed of convergence of the eigenfunctions or of the operators. In case we had this additional information, we could take a different route to approximate the eigenvalue perturbation, based on various general results from the literature. For instance, if we knew that the inverse operators $L_{ε}^{- 1}$ converge in norm to $L^{- 1}$ , and in particular that $| L^{- 1} - L_{ε}^{- 1} |_{L^{2}} = f (ε) + r (ε)$ , from [18, Theorem 3] we would have that $\begin{matrix} | 1 / λ - μ_{ε} | & ⩽ C_{1} (\sum_{i = 1}^{m} {⟨ (L^{- 1} - L_{ε}^{- 1}) (ϕ_{i}), ϕ_{i} ⟩}_{L^{2}} + {| L^{- 1} - L_{ε}^{- 1} |}_{L^{2}}) \\ ⩽ C_{2} {| L^{- 1} - L_{ε}^{- 1} |}_{L^{2}} = C_{2} (f (ε) + r (ε)), \end{matrix}$ where $μ_{ε} = \sum_{j = 1}^{m} 1 / λ_{ε, j}$ , $ϕ_{1} \dots ϕ_{m}$ is an $L^{2}$ -orthonormal basis of the eigenspace of $1 / λ$ , $C_{1}$ and $C_{2}$ some constants. Notice however that with this method we are unable to explicitly obtain the leading order coefficient in the asymptotic expansion, but only the order of convergence.

4. Asymptotic formulae involving a bilinear function

It is useful to consider more carefully the case where the expression in the asymptotic formula admits a separation of variables as $\begin{matrix} (10) & f (ε, u) = E (ε) b (u, u), \end{matrix}$ with $b : V \times V \to R$ being a symmetric bilinear form. This happens for many useful types of domain perturbations (see Section 5). The advantage of this case is twofold: the minimizer of (4) is unique and can be easily computed as follows. Choosing $u_{1}, \dots, u_{m}$ an arbitrary orthonormal basis of the eigenspace of λ, condition (4) can be rewritten as $\begin{matrix} (11) & v_{n} = E (ε) argim {a \cdot B a : a \in R^{m}, | a | = 1, a_{1} u_{1} + \dots + a_{m} u_{m} ⊥ {v_{1}, \dots, v_{n - 1}}}, \end{matrix}$ where B is a symmetric matrix with elements $\begin{matrix} (12) & B_{i}^{j} = b (u_{i}, u_{j}) . \end{matrix}$ Then, by diagonalizing B, we obtain the following result.

Corollary 4.1.
If f can be rewritten as in ( 10 ), then the minimum and minimizer of ( 4 ) are respectively the n-th eigenvalue of B and $a_{1, n} u_{1} + \dots + a_{m, n} u_{m}$ , where $(a_{1, n}, \dots, a_{m, n})$ is the normalized eigenfunction of B associated to its n-th eigenvalue.

Notice that if the bilinear form b has the further decomposition $\begin{matrix} (13) & b (u_{i}, u_{j}) = l (u_{i}) l (u_{j}) \forall i, j, \end{matrix}$ where $l : V \to R$ is linear, an easy computation shows that the first $m - 1$ eigenvalues of B are zero and the m-th one is $l {(u_{1})}^{2} + \dots + l {(u_{m})}^{2}$ . Therefore, we have the following result. Corollary 4.2.
If the bilinear form b can be rewritten as in ( 13 ), then $\begin{matrix} λ_{ε, n} - λ = \{\begin{matrix} E (ε) \sum_{k = 1}^{m} l {(u_{k})}^{2} + r (ε) & if n = m, \\ r (ε) & if n < m . \end{matrix} \end{matrix}$ Moreover, the eigenfunction in the eigenspace of λ to which the eigenfunction of $λ_{ε, m}$ converges is given by $\begin{matrix} \frac{\sum_{n = 1}^{m} l (u_{n}) u_{n}}{\sqrt{\sum_{n = 1}^{m} l {(u_{n})}^{2}}} . \end{matrix}$

5. Applications to eigenvalues of the Laplacian

In this section we consider some domain perturbations of particular interest in applications, and derive explicit asymptotic formulae for repeated eigenvalues of the Laplacian. Different numerical experiments are provided to validate these formulae.

Let us first fix some useful notation. We indicate as $d ⩾ 2$ the dimension of the ambient space $R^{d}$ . We define $Γ_{0}$ as the fundamental solution for the negative Laplacian in radial coordinates, in particular we have for any $r > 0$ , $\begin{array}{l} (14) & Γ_{0} (r) & = \{\begin{matrix} - \frac{1}{2 π} log r & if d = 2, \\ \frac{1}{d (d - 2) | S^{d - 1} | r^{d - 2}} & if d ⩾ 3, \end{matrix} \end{array}$ where $S^{d - 1}$ is the $(d - 1)$ -dimensional unit sphere.

5.1. Perturbation by a grounded inclusion

Let B be a Lipschitz domain in $R^{d}$ with connected boundary, with volume $| B | = | Ω |$ , and centered at the origin in the sense that $\int_{\partial B} y_{1} d σ (y_{1}, y_{2}) = \int_{\partial B} y_{2} d σ (y_{1}, y_{2}) = 0$ . Fix a point $z \in Ω$ and consider a scaling coefficient $ε > 0$ . Suppose then that the domain Ω is perturbed into $Ω_{ε} = Ω ∖ D$ , by inserting an inclusion $D = z + ε B$ and requiring homogeneous Dirichlet conditions to hold on $\partial D$ .

Let λ be an eigenvalue of Ω with associated eigenfunction u. Then $λ_{ε}$ is an eigenvalue of $Ω_{ε}$ perturbed from λ with associated eigenfunction $u_{ε}$ if $λ_{ε} \to λ$ as $ε \to 0$ and $\begin{matrix} \{\begin{matrix} Δ u + λ u = 0 & in Ω, \\ u = 0 or \partial u / \partial ν = 0 & on \partial Ω, \end{matrix} \{\begin{matrix} Δ u_{ε} + λ_{ε} u_{ε} = 0 & in Ω ∖ D, \\ u_{ε} = 0 & on \partial D, \\ u_{ε} = 0 or \partial u_{ε} / \partial ν = 0 & on \partial Ω . \end{matrix} \end{matrix}$

In [3, Chapter 3], the leading-order term for the perturbation of a simple eigenvalue is obtained in the case of dimensions 2 and 3. However, these computations can be repeated exactly in the same way for $d ⩾ 4$ , and the resulting asymptotic formula can be restated as follows.

Lemma 5.1.
Given λ a simple eigenvalue of Ω with associated eigenfunction u, and $λ_{ε}$ the eigenvalue of $Ω_{ε}$ perturbed from λ, then $\begin{matrix} (15) & λ_{ε} - λ = \frac{u {(z)}^{2}}{Γ_{0} (ε)} + o (1 / Γ_{0} (ε)) . \end{matrix}$

We seek to apply the variational characterization of Theorem 3.4 to our case. Assumption 3.1 holds thanks to Theorems 2.2 and 2.3, and the fact the perturbation considered is restricted to a small part of the domain. Although the expression in (15) is not continuous under $L^{2}$ convergence of u, we can easily rewrite it so that Assumption 3.2 holds. In fact, supposing $u_{n} \to u$ in $L^{2}$ , we can exploit the regularity properties of solutions to elliptic equations to rewrite $\begin{matrix} (16) & u_{n} {(z)}^{2} + o (1 / Γ_{0} (ε)) = \int_{B_{r_{ε}}} u_{n}^{2} \to_{n \to \infty}^{} \int_{B_{r_{ε}}} u^{2} = u {(z)}^{2} + o (1 / Γ_{0} (ε)), \end{matrix}$ where $B_{r_{ε}}$ is a small enough ball centered at z. By Corollary 4.2, we immediately obtain the following result. Proposition 5.2.
Let λ be an eigenvalue of multiplicity m of the negative Laplacian on Ω and $u_{1}, \dots, u_{m}$ some associated eigenfunctions orthonormal in $L^{2} (Ω)$ . Then, the largest perturbed eigenvalue behaves like $\begin{matrix} (17) & λ_{ε, m} - λ = \frac{\sum_{j} u_{j} {(z)}^{2}}{Γ_{0} (ε)} + o (1 / Γ_{0} (ε)), \end{matrix}$ while all the other eigenvalues behave like $\begin{matrix} λ_{ε, n} - λ = o (1 / Γ_{0} (ε)), for n < m . \end{matrix}$
Remark 5.3.
We collect some interesting consequences of Proposition 5.2.
For ε small enough, the largest perturbed eigenvalue $λ_{ε, m}$ will always be simple as long as at least one of the eigenfunctions $u_{1}, \dots, u_{m}$ is not zero in z.

The eigenfunction associated to $λ_{ε, m}$ converges to $\begin{matrix} \frac{u_{1} (z) u_{1} + \dots + u_{m} (z) u_{m}}{\sqrt{u_{1} {(z)}^{2} + \dots + u_{m} {(z)}^{2}}} . \end{matrix}$

It can be shown that in two dimensions the higher-order terms in formula (15) can be further computed as $\begin{matrix} (18) & λ_{ε} - λ = \frac{u {(z)}^{2}}{log (ε) + R (z)} + O (ε^{2}), \end{matrix}$ where R is a function of z which does not depend on u (see [8]). Therefore, from Corollary 4.2, we have the better approximation $\begin{array}{l} λ_{ε, n} - λ = \{\begin{matrix} \frac{| U (z) |^{2}}{log (ε) + R (z)} + O (ε^{2}) & if n = m, \\ O (ε^{2}) & if n < m . \end{matrix} \end{array}$

Example 5.4.
Let Ω be the unit square ${(0, 1)}^{2}$ and consider the Dirichlet eigenvalue $λ = 50 π^{2}$ with associated orthonormal eigenfunctions $u_{1}$ , $u_{2}$ , $u_{3}$ defined as $\begin{matrix} \{\begin{matrix} u_{1} (x, y) = 2 sin (π x) sin (7 π y), \\ u_{2} (x, y) = 2 sin (7 π x) sin (π y), \\ u_{3} (x, y) = 2 sin (5 π x) sin (5 π y) . \end{matrix} \end{matrix}$ Since for any point z in Ω there is at least one eigenfunction which is non-zero at z, the insertion at z of a small grounded inclusion will cause an eigenvalue bifurcation of λ. In particular, one perturbed eigenvalue will shift from λ as $1 / log (ε)$ while the other two will shift like $O (ε^{2})$ . The result of a numerical simulation is presented in Fig. 1.
Fig. 1.
A ${log}_{2}$ - ${log}_{2}$ plot of the behaviour of an eigenvalue bifurcation from $50 π^{2}$ as the size coefficient ε of the inclusion decreases. The original domain is the unit square ${(0, 1)}^{2}$ and the inclusion is a disk of radius ε centered at $(1 / 4, 1 / 4)$ .

5.2. Perturbation by a conductivity inclusion

In this section, we consider a perturbation of Ω obtained by the insertion of a small inclusion with a conductivity coefficient different from the background.

Let B, z, ε, D be defined as in Section 5.1. Suppose that $- Δ$ is perturbed into $- Δ_{ε}$ by inserting inside Ω a small inclusion D of conductivity k. This causes the eigenvalue λ to split into m (possibly distinct) eigenvalues $λ_{ε, 1} ⩽ \dots ⩽ λ_{ε, m}$ such that the following system holds: $\begin{matrix} (19) & \{\begin{matrix} (Δ + λ_{ε, n}) u_{ε, n} = 0 & in Ω ∖ \overline{D}, \\ (k Δ + λ_{ε, n}) u_{ε, n} = 0 & in D, \\ u_{ε, n} continuous & along \partial D, \\ \frac{\partial u_{ε, n}}{\partial ν} |_{+} = k \frac{\partial u_{ε, n}}{\partial ν} |_{-} & on \partial D, \\ u_{ε, n} = 0 or \partial u_{ε, n} / \partial ν = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $n = 1, \dots, m$ and $u_{ε, n}$ are eigenfunctions associated to $λ_{ε, n}$ .

It has been shown in [4] that if λ is a simple eigenvalue with associated eigenfunction u, and $λ_{ε}$ is a perturbation of λ, then $\begin{matrix} λ_{ε} - λ = ε^{d} {⟨ \nabla u (z), \nabla u (z) ⟩}_{M} + O (ε^{d + 1}), \end{matrix}$ where ${⟨ x, y ⟩}_{M} = x \cdot M (k, B) y$ for any $x, y \in R^{d}$ , and $M (k, B)$ is a $d \times d$ matrix known as polarization tensor, which can be defined by $\begin{matrix} {(M (k, B))}_{i, j} = \int_{\partial B} {(\frac{k + 1}{2 (k - 1)} I - {(K_{B}^{0})}^{*})}^{- 1} (ν_{j}) y_{i} d σ (y) . \end{matrix}$ Therefore, in this case Corollary 4.1 specifies to the following result.

Proposition 5.5.
Let λ be an eigenvalue of multiplicity m of $- Δ$ and let $u_{1}, \dots, u_{m}$ be some associated eigenfunctions orthonormal in $L^{2} (Ω)$ . Let $λ_{ε, 1} ⩽ \dots ⩽ λ_{ε, m}$ be the eigenvalues perturbed from λ which are solutions to ( 19 ). Then, for every $n \in {1, \dots, m}$ , the $O (ε^{d + 1})$ approximation of $λ_{ε, n} - λ$ is given by the n-th eigenvalue of the matrix with element $\begin{matrix} ε^{d} {⟨ \nabla u_{i} (z), \nabla u_{j} (z) ⟩}_{M} \end{matrix}$ in position $(i, j)$ .
Remark 5.6.
We consider some extremal cases of Proposition 5.5 that are of particular interest.
If ${⟨ \nabla u_{i} (z), \nabla u_{j} (z) ⟩}_{M} = 0$ for all $i \neq j$ , then it is enough to reorder $u_{1}, \dots, u_{m}$ according to the magnitude of ${⟨ \nabla u_{1} (z), \nabla u_{1} (z) ⟩}_{M}, \dots, {⟨ \nabla u_{m} (z), \nabla u_{m} (z) ⟩}_{M}$ , to obtain that for any n it holds $\begin{matrix} λ_{ε, n} - λ = ε^{d} {⟨ \nabla u_{n} (z), \nabla u_{n} (z) ⟩}_{M} + O (ε^{d + 1}) . \end{matrix}$ We also remark that if the multiplicity m is larger than the dimension d, only d vectors can be linearly independent, and thus we will have that $λ_{ε, n} - λ = O (ε^{d + 1})$ for $n ⩽ m - d$ .

If $\nabla u_{i} (z)$ , $\nabla u_{j} (z)$ are parallel with respect to ${⟨ \cdot, \cdot ⟩}_{M}$ , then, by Corollary 4.2, we will have $\begin{array}{l} λ_{ε, n} - λ = \{\begin{matrix} ε^{d} \sum_{n = 1}^{m} {⟨ \nabla u_{n} (z), \nabla u_{n} (z) ⟩}_{M} + O (ε^{d + 1}) & if n = m, \\ O (ε^{d + 1}) & if n < m . \end{matrix} \end{array}$

Example 5.7.
Let Ω be the unit square ${(0, 1)}^{2}$ and consider the Neumann eigenvalue $λ = 100 π^{2}$ with associated eigenfunctions $u_{1}$ , $u_{2}$ , $u_{3}$ , $u_{4}$ defined as $\begin{matrix} \{\begin{matrix} u_{1} (x, y) = \sqrt{2} cos (10 π y), \\ u_{2} (x, y) = \sqrt{2} cos (10 π x), \\ u_{3} (x, y) = 2 cos (6 π x) cos (8 π y), \\ u_{4} (x, y) = 2 cos (8 π x) cos (6 π y) . \end{matrix} \end{matrix}$ Let B be the disk of radius $1 / π^{2}$ centered at 0. In this case, it can be explicitly computed that $M (k, B) = 2 (k - 1) I / (π^{2} (k + 1))$ . We can easily determine, reasoning as in Remark 5.6, whether the first term in the asymptotic expansion of $λ_{ε, n} - λ$ is zero; such behaviour will depend on the choice of z. For example:
for $z = (1 / 2, 1 / 2)$ , it holds $\nabla u_{n} (z) = 0$ for any n, and therefore $λ_{ε, n} - λ = O (ε^{3})$ ;

for $z = (1 / 3, 1 / 2)$ , $\nabla u_{1} (z) = \nabla u_{3} (z) = 0$ while $\nabla u_{2} (z)$ , $\nabla u_{4} (z)$ are all parallel and non-zero, thus from Item ii of Remark 5.6 we have $λ_{ε, n} - λ = O (ε^{3})$ for $n = 1, 2, 3$ while $λ_{ε, 4} - λ = Θ (ε^{2})$ ;

for $z = (1 / 7, 1 / 7)$ computations of the gradient of the eigenfunctions at z show that $λ_{ε, n} - λ = O (ε^{3})$ for $n = 1, 2$ and $λ_{ε, n} - λ = Θ (ε^{2})$ for $n = 3, 4$ ;

by Item i of Remark 5.6 there is no $z \in Ω$ such that $λ_{ε, n} - λ = Θ (ε^{2})$ for more than two different indices n.
Example 5.8.
Let Ω be the unit square ${(0, 1)}^{2}$ and consider the Neumann eigenvalue $λ = 4 π^{2}$ with associated eigenfunctions $u_{1}$ , $u_{2}$ defined as $\begin{matrix} \{\begin{matrix} u_{1} (x, y) = \sqrt{2} cos (2 π y), \\ u_{2} (x, y) = \sqrt{2} cos (2 π x) . \end{matrix} \end{matrix}$ Let B be the disk of radius $1 / π^{2}$ centered at 0. Recall that in this case we have $M (k, B) = 2 (k - 1) I / (π^{2} (k + 1))$ . Although the first term in the asymptotic formula for $λ_{ε, n} - λ$ can be easily computed in this case, here we focus our attention only on the asymptotic order. We can easily determine, reasoning as in Remark 5.6, whether the first term in the asymptotic expansion of $λ_{ε, n} - λ$ is zero; such behaviour will depend on the choice of z. For example:
for $z = (1 / 2, 1 / 2)$ , we have that both eigenfunctions $u_{1}$ , $u_{2}$ have zero gradient at 0, and thus both eigenvalues shift from λ as $O (ε^{3})$ ;

for $z = (1 / 4, 1 / 2)$ , one of the eigenfunctions has zero gradient while the other has a non-zero entry, thus one eigenvalue shift behaves like $ε^{2}$ , the other like $O (ε^{3})$ ;

for $z = (1 / 4, 1 / 4)$ , the gradients of the two eigenfunctions are orthogonal and non-zero, thus both eigenvalues shift from λ as $ε^{2}$ .
Numerical results are presented in Fig. 2.

Fig. 2.
A ${log}_{2}$ - ${log}_{2}$ plot of the behaviour of an eigenvalue bifurcation as the size coefficient ε of the conductivity inclusion decreases. The original domain is the unit square ${(0, 1)}^{2}$ , the inclusion a disk of conductivity $k = 2$ , centered respectively at $(1 / 2, 1 / 2)$ (left graph), at $(1 / 4, 1 / 2)$ (center graph), at $(1 / 4, 1 / 4)$ (right graph).

5.3. Perturbation by boundary deformation

In this section, we consider $Ω_{ε}$ obtained from Ω by a boundary deformation. For simplicity, we suppose that Ω is globally the epigraph of a Lipschitz function φ, that is $Ω = {x \in R^{d} : φ (x) ⩽ 0}$ . Given $w \in C^{2} (\partial Ω)$ , we also suppose that the boundary perturbation is such that $Ω_{ε} = {x \in R^{d} : φ (x) + ε w (x) ν (x) ⩽ 0}$ . Recall that if λ is a simple eigenvalue of Ω with associated eigenfunction u, and $λ_{ε} \to λ$ , then Hadamard’s formula reads $\begin{matrix} (20) & \frac{\partial λ_{ε}}{\partial ε} |_{ε = 0} = \int_{\partial Ω} (| \nabla u |^{2} - λ u^{2} - 2 {(\partial u / \partial ν)}^{2}) w, \end{matrix}$ for Dirichlet or Neumann conditions on $\partial Ω$ (for its proof see [6] or [11]). Therefore, if λ has multiplicity m, an application of Theorem 3.4 provides us with the variational formula $\begin{matrix} (21) & λ_{ε, n} - λ = ε \underset{v \in S_{n}}{argim} \int_{\partial Ω} (| \nabla v |^{2} - λ v^{2} - 2 {(\partial v / \partial ν)}^{2}) w + O (ε^{2}) . \end{matrix}$

Example 5.9.
Let $Ω = {(0, 1)}^{2}$ and consider the Dirichlet eigenvalue $λ = 10 π^{2}$ with associated orthonormal eigenfunctions $u_{1} (x, y) = 2 sin (π x) sin (3 π y)$ , $u_{2} (x, y) = 2 sin (3 π x) sin (π y)$ . Suppose that the boundary of Ω is perturbed on the upper side of the square Ω with $w (x, y) = sin (π x)$ , that is $\partial Ω \cap {y = 1}$ is deformed into ${(x, y) : 0 ⩽ x ⩽ 1, y = 1 + ε sin (π x)}$ . In this case, the integral in (20) is analytically computable as $\begin{array}{l} - \int_{\partial Ω} {(a_{1} \partial u_{1} / \partial ν + a_{2} \partial u_{2} / \partial ν)}^{2} w = \frac{16 π}{35} (- 105 a_{1}^{2} + 14 a_{1} a_{2} - 9 a_{2}^{2}) . \end{array}$ Thus, by (21), we can calculate explicitly $\begin{matrix} (22) & \begin{array}{l} u_{ε, 1} \to c_{1} u_{1} + c_{2} u_{2}, λ_{ε, 1} - λ = C_{1} ε + O (ε^{2}), \\ u_{ε, 2} \to c_{2} u_{1} - c_{1} u_{2}, λ_{ε, 2} - λ = C_{2} ε + O (ε^{2}), \end{array} \end{matrix}$ where $c_{1} ≃ 0.997$ , $c_{2} ≃ - 0.0724$ , $C_{1} ≃ - 152$ , $C_{2} ≃ - 12.2$ . Numerical results are presented in Figs 3 and 4.

Fig. 3.
Behaviour of an eigenvalue bifurcation as the scaling parameter ε of the boundary deformation decreases. The original domain is the unit square ${(0, 1)}^{2}$ and the boundary deformation is given by $ε sin (π x) ν (x, 1)$ on the upper side.

Fig. 4.
Eigenfunctions associated to the eigenvalues perturbed from $10 π^{2}$ . The perturbation consists in a boundary deformation $ε sin (π x) ν (x, 1)$ of the upper side for $ε = 1 / 10$ . Numerical computations obtained with the finite element method of the eigenfunctions are plotted in the left column, their limiting functions as $ε \to 0$ in the two-dimensional eigenspace of $4 π^{2}$ given by (22) in the right column.

Notice that in general, minimizing the expression in (21) is a computationally expensive task. However, we can still obtain some cheaper, qualitative information if we approximate to a more treatable form the considered domain perturbation. We showcase such an heuristic in the following example, where a local perturbation is “singularized” to obtain an asymptotic formula easier to analyze.
Example 5.10.
Suppose a small dent is present on the surface $\partial Ω$ at the point z, shaped as a cone with circular base of radius δ and height ε. Let at first λ be a simple Dirichlet eigenvalue with associated eigenfunction u. If we approximate $\begin{matrix} \int_{\partial Ω} {(\partial u / \partial ν)}^{2} w ≃ δ^{2} π {(\partial u (z) / \partial ν)}^{2}, \end{matrix}$ then for eigenvalues low enough we can estimate $\begin{matrix} (23) & λ_{ε} - λ ≃ - ε δ^{2} π {(\partial u (z) / \partial ν)}^{2} . \end{matrix}$ The right hand side in (23) is bilinear in u therefore, if we adopt such an approximation for $λ_{ε} - λ$ , by Corollary 4.2 we have that for any non-simple eigenvalue the largest perturbed eigenvalue will shift like $O (ε)$ while all the smaller ones will shift like $O (ε^{2})$ .
5.4. Summary of results

We summarize hereafter the main results obtained for each of the perturbations considered. For this purpose, we recall that Ω indicates an arbitrary Lipschitz domain in $R^{d}$ , λ an eigenvalue of the negative (Dirichlet or Neumann) Laplacian on Ω, $u_{1}, \dots, u_{m}$ an arbitrary orthonormal basis in $L^{2} (Ω)$ of the eigenspace of λ, and $λ_{ε, 1} ⩽ \dots ⩽ λ_{ε, m}$ the eigenvalues perturbed from λ.

When a hole D of volume $ε^{d}$ and centered at z is cut out from Ω and homogeneous Dirichlet boundary conditions are imposed on $\partial D$ , we have $\begin{array}{l} λ_{ε, m} - λ = C ε^{d - 1} \sum_{i = 1}^{m} u_{i} {(z)}^{2} + O (ε^{d}), \\ λ_{ε, n} - λ = O (ε^{d}) for n < m, \end{array}$ in the case where $d ⩾ 3$ (see (18) for the case $d = 2$ ), where C is a constant which depends only on the dimension d. Therefore, we have that the largest eigenvalue splits at a higher asymptotic order than all the others eigenvalues, as long as one among the quantities $u_{1} (z), \dots, u_{m} (z)$ is non-zero.

In the case of a conductivity inclusion, we do not have such an explicit formula, but we can still easily recover a first-order approximation by computing the eigenvalues of a finite matrix. More precisely, if we suppose to change the conductivity coefficient from 1 to k only in D, a small disk of radius ε centered at a point z, then for any $n \in {1, \dots, m}$ , $\begin{matrix} λ_{ε, n} - λ = 2 \frac{k - 1}{k + 1} ε^{d} μ_{n} + O (ε^{d + 1}), \end{matrix}$ where $μ_{n}$ is the n-th eigenvalue of the matrix with element $\nabla u_{i} (z) \cdot \nabla u_{j} (z)$ in position $(i, j)$ .

In the case of a normal boundary deformation of Ω with shape $w \in C^{2} (\partial Ω)$ , that is the perturbed domain boundary is given locally by $\partial Ω + ε w ν$ , to find $λ_{ε, n} - λ$ for any $n \in {1, \dots, m}$ , one has to find the minimizer $v_{n}$ of $\begin{matrix} J (v) = \int_{\partial Ω} (| \nabla v |^{2} - λ v^{2} - 2 {(\partial v / \partial ν)}^{2}) w, \end{matrix}$ among all v’s of unit $L^{2}$ -norm in the eigenspace of λ and perpendicular to $v_{1}, \dots, v_{n - 1}$ . Then $\begin{matrix} λ_{ε, n} - λ = ε J (v_{n}) . \end{matrix}$

As a final remark, let us point out that similar formulae can be derived for many other types of domain perturbations or other differential operators. For example, with the same approach of Section 5.3, it is immediate to generalize the asymptotic expansion of eigenvalues in the case of shape deformation of conductivity inclusions (see [1, Theorem 2.1]); or, with the same approach of Section 5.2, to generalize the asymptotic formulae for eigenvalues of the Lamé operator in the context of linear elasticity (see [2, Theorem 2.1]).

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