Hadamard asymptotics for eigenvalues of the Dirichlet Laplacian
Introduction
The question of how eigenvalues change when the domain is slightly perturbed is a classical problem probably going at least as far back as Rayleigh [14], who studied eigenvalues and domain perturbation in connection with acoustics as early as in the nineteenth century. The approach given in this article owes to results by Hadamard [4], who in the early 20th century studied perturbations of domains with smooth boundary, where the perturbed domain is represented by where , is the signed distance to the boundary ( for ), h is a smooth function, ε is a small parameter, and is the reference domain. Hadamard's result for the first eigenvalue of the Dirichlet-Laplacian is given by where dS is the surface measure on and φ is an eigenfunction corresponding to such that . It is worth noting, that the problem of how eigenvalues change when the domain is perturbed, is closely related to shape optimization. We refer here to Henrot [5], and Sokołowski and Zolésio [15], and references found therein.
The aim of this article is to find minimal assumptions on the smoothness of the boundary when the Hadamard formula is still valid. A large quantity of studies of the Dirichlet problem already exists in the literature; see, for instance, Grinfeld [3], Henrot [5], Kozlov [10], [13], Kozlov and Nazarov [11], [12], and references found therein. In this article, we present an asymptotic formula of Hadamard type for perturbations in the case when the domains are of class or , respectively. The first class of domains is optimal for validity of the Hadamard formula. For the second class of domains, we give an optimal estimate of the remainder term. Let and be bounded domains in with boundaries and , respectively. We consider the spectral problems and In the case when the boundary is nonsmooth, we consider the corresponding weak formulation of the problem on the Hilbert spaces and with the usual inner product. Note though, that the techniques used are applicable to a wider class of partial differential operators. In particular to uniformly elliptic operators of second order.
It is known that if the two domains are close enough, both problems have the same number of eigenvalues in a small enough neighborhood of as the multiplicity of the eigenvalue. This means that for a fixed eigenvalue of (1.2) of multiplicity m, there are precisely m eigenvalues of (1.3) (counting multiplicity) near . This is a consequence of the continuous dependence of eigenvalues on the domain; see, e.g., Kato [7] (Sections IV.3 and V.3) or Henrot [5] and references therein. The explicit result in terms of quantities used in this article can be found in Kozlov [10] (Proposition 3). We will denote by the eigenspace corresponding to the eigenvalue and denote the dimension of by . For our results, we will characterize how close the two domains are in the sense of the Hausdorff distance between the sets and , i.e., We do not assume that one domain is a subdomain of the other. It should be noted however, that the abstract result presented below in Section 2 permits a more general type of proximity quantity for the two domains.
We consider three cases of regularity of the boundary , namely , , and Lipschitz boundaries. Let us first consider the Lipschitz case. Then there exists a positive constant M such that the boundary can be covered by a finite number of cylinders , , where there exist orthogonal coordinate systems in which where the center of is at the origin and Here, is the -dimensional ball of radius and with the center 0. We assume that and that are Lipschitz functions, i.e., This class of domains defined by a constant M and cylinders , , will be denoted in what follows by .
We assume that is close to in the sense that can be described by where are also Lipschitz continuous with Lipschitz constant M and is assumed to be small. One can show that there exist positive constants and , such that .
The case when is a - or -domain is defined analogously, with the following additional assumptions.
-assumption. We assume that such that and
-assumption. We assume that and that (1.5) holds. Furthermore,
Note that are only assumed to be Lipschitz continuous in both cases and satisfy (1.6) or (1.7), respectively.
Let us define the function σ on the surface in the case of the -assumption on the boundary. Let be the unit outward normal to the boundary at the point P. For we introduce the number . It is the smallest (in the absolute value) root of the equation where Q is the nearest point on to P lying on the line passing through P with the direction ν. Clearly σ is positive if Q is outside and negative if Q is inside. One can verify that this function is Lipschitz continuous on with Lipschitz constant depending on M and .
Theorem 1.1 Suppose that is a -domain with and that is as described above. Then for every . Here is an eigenvalue of the problem where . Moreover, in (1.9) run through all eigenvalues of (1.10) counting their multiplicities.
Theorem 1.2 Suppose that is a -domain, and is as described above. Then for every .
We note that if u satisfies (1.2) for a Lipschitz domain , then the normal derivative and so the integral in the right-hand side of (1.10) is defined. Despite this fact, we have demonstrated in Section 7 that the formula presented in (1.9) does not hold for Lipschitz domains and their small perturbations. Indeed, we show in Section 7 that the remainder is optimal for -perturbations. The result similar to Theorem 1.1 for the Neumann problem is proved in [9] and an analogue of Theorem 1.2 for the Neumann problem, albeit with a more complicated expression for the leading term , is derived in [16].
The paper is organized as follows. In Section 2 we present an abstract approach developed in [10], which concerns the asymptotics of eigenvalues of unbounded operators when their domains are changed. As a result of this approach, a theorem on asymptotics of eigenvalues is presented. It involves different terms and these terms are estimated in the remaining parts of the paper. The analysis is based on the theory of elliptic boundary value problems in Lipschitz, - and -domains that was developed in particular in the works of Dahlberg, Fabes, Jodeit, Kenig and Riviére. In the last section, we present an example which demonstrates the sharpness of Theorem 1.1, Theorem 1.2.
Section snippets
Preliminary results and definitions
Here we present an abstract result from [10], which will play an important role in the proofs of Theorem 1.1, Theorem 1.2.
We suppose that and are open Lipschitz domains and that D is an open ball such that and . Put , and finally . We extend functions by zero outside their respective domains. We let and denote the inner products on H given by respectively. Moreover, let and be the norms induced by
Estimates for the functions φ, and Tφ
In this section we give estimates and some representations for the terms appearing in (2.6) and (2.7). All of them are valid for Lipschitz domains.
We will use the following notation. Let Ω be a Lipschitz domain. The truncated cones at are given by, e.g., and the non-tangential maximal function is defined on the boundary ∂Ω by and We refer to Kenig [8] for further details.
We will use the short-hand notation for the tangential gradient.
From the definition of in (2.4), it
Two lemmas
In the case that the boundary of is of class (or ), it is more convenient to use another equivalent (for small d) description of the closeness of and : there exists a positive constant such that for every , in a cartesian coordinate system with the center at P and the tangent plane to the boundary at P given by , the domain is given by where If we shall use the notation for
Proof of Theorems 1.1 and 1.2
By (3.6) and (3.7) combined with Lemma 4.2, we obtain that Moreover, due to (3.8) and Lemma 4.2, Having in mind these relations and (3.23), we can write formula (2.5) as where is an eigenvalue of the problem where and run through
The end of the proof of Theorems 1.1 and 1.2
The function satisfies Clearly, and in the -case, and in the -case, we have and . Applying Lemma 5.1, we obtain in the -case and with the remainder replaced by in the -case. We note that in order to apply Lemma 5.1, it is useful to use the relation since formula (5.3) only contains squares of functions. Equality
Counterexamples
Let be the domain in given by The domain is given by where η is a positive, periodic, -function such that . We assume that δ and d are small parameters, , and that , where N is a large integer. We will consider three cases: is a Lipschitz perturbation, if we are dealing with a -perturbation and if , the perturbation is of the class .
Consider the eigenvalue problems
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