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International Journal of Differential Equations
Volume 2011 (2011), Article ID 619623, 22 pages
On Spectrum of the Laplacian in a Circle Perforated along the Boundary: Application to a Friedrichs-Type Inequality
1Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia
2Narvik University College, Postboks 385, 8505 Narvik, Norway
3Department of Engineering Science and Mathematics, Luleå University of Technology, 971 87 Luleå, Sweden
Received 24 May 2011; Accepted 30 August 2011
Academic Editor: Mayer Humi
Copyright © 2011 G. A. Chechkin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we construct and verify the asymptotic expansion for the spectrum of a boundary-value problem in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. As an application of the obtained results, the asymptotic behavior of the best constant in a Friedrichs-type inequality is investigated.
We study a two-dimensional eigenvalue problem for the Laplace operator in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. The asymptotic behavior of the spectrum of the considered boundary-value problem is investigated in this paper. We construct and verify the asymptotic expansion for the eigenvalues with respect to the small parameter describing the microinhomogeneous structure of the domain. A similar problem was considered in  for the case of perforation located along the plane part of the boundary. The case studied in this paper is much more complicated since the eigenvalues of multiplicity more than one can appear. The technique for asymptotic analysis of such kind of problem can be found, for example, in [2, 3].
The obtained results are used for asymptotic expansion of the best constant in a Friedrichs-type inequality for functions from the space , vanishing on the boundary of the perforation and satisfying homogeneous Neuman condition on the boundary of the circle. Analogous questions concerning the asymptotic behavior of the best constant in Friedrichs-type inequality in domains having microinhomogeneous structure in a neighborhood of the boundary were studied in [1, 4–11]. In the remaining part of this introduction, we will give a short description of some of the most important results in these papers to put the results obtained in this paper into a more general frame.
In paper , the authors proved a Friedrichs-type inequality for functions, having zero trace on the small periodically alternating pieces of the boundary of a two-dimensional domain. The total measure of the set, where the function vanishes, tends to zero. It turns out that for this case the constant in the Friedrichs-type inequality is bounded. Moreover, the precise asymptotics of the constant in the derived Friedrichs-type inequality is described as the small parameter characterizing the microinhomogeneous structure of the boundary, tends to zero.
Paper  is devoted to the asymptotic analysis of functions depending on the small parameter, which characterizes the microinhomogeneous structure of the domain where the functions are defined. The authors considered a boundary-value problem in a two-dimensional domain perforated nonperiodically along the boundary in the case when the diameter of circles and the distance between them have the same order. In particular, it was proved that the Dirichlet problem is the limit for the original problem. Moreover, some numerical simulations were used to illustrate the results. As an application, a Friedrichs-type inequality was derived for functions vanishing on the boundary of the cavities. It was proved that the constant in the obtained inequality is close to the constant in the inequality for functions from . The three-dimensional case of the same problem is considered in .
In paper , the author considered a three-dimensional domain, which is aperiodically perforated along the boundary in the case when the diameter of the holes and the distance between them have the same order. A Friedrichs-type inequality was derived for functions from the space vanishing on the boundaries of cavities. In particular, it was shown that the constant in the derived inequality tends to the constant of the classical inequality for functions from when the small parameter describing the size of perforation tends to zero.
Paper  (see also ) deals with the construction of the asymptotic expansion for the first eigenvalue of a boundary-value problem for the Laplacian in a perforated domain. This asymptotics gives an asymptotic expansion for the best constant in a corresponding Friedrichs-type inequality.
Paper , is devoted to the Friedrichs-type inequality, where the domain is periodically and rarely perforated along the boundary. It is assumed that the functions satisfy homogeneous Neumann boundary conditions on the outer boundary and that they vanish on the perforation. In particular, it is proved that the best constant in the inequality converges to the best constant in a Friedrichs-type inequality as the size of the perforation goes to zero much faster than the period of perforation. The limit Friedrichs-type inequality is valid for functions in the Sobolev space .
Some generalizations of Friedrichs-type inequalities are Hardy-type inequalities. There exist several books devoted to this topic, see [12–16]. The first attempts to generalize the classical results concerning Hardy-type inequalities in fixed domains to domains with microinhomogeneous structure one can find in [6, 10].
Paper  deals with a three-dimensional weighted Hardy-type inequality in the case when the domain is bounded and has nontrivial microstructure. It is assumed that the small holes are distributed periodically along the boundary. The main result is the validity of a weighted Hardy-type inequality for the class of functions from the Sobolev space having zero trace on the small holes under the assumption that a weight function decreases to zero in a neighborhood of the microinhomogenity on the boundary.
In paper , the author derived a new two-dimensional weighted Hardy-type inequality in a rectangle for the class of functions from the Sobolev space vanishing on small alternating pieces of the boundary. The dependence of the best constant in the derived inequality on the small parameter describing the size of microinhomogenity was established.
This paper is organized as follows: in Section 2 we give all necessary definitions and state the spectral problem. Section 3 is devoted to the construction of the leading terms of asymptotic expansion, while the complete expansions for the simple and multiple eigenvalues are constructed in Sections 4 and 5, respectively. The verification of the constructed asymptotics is given in Section 6. Finally, in Section 7, the obtained results are applied to describe the asymptotic behavior for the best constant in a Friederichs-type inequality considered in a perforated domain.
Consider a unit circle centered at the origin. We introduce the polar system of coordinates in . Introduce a small parameter , and consider the open set which is the union of small sets periodically distributed along the boundary. Each of these small sets can be obtained from the neighboring one by rotation about the origin through the angle . Finally, we define and , see Figure 1. Let us describe the geometry of in details. Consider the semi-strip: Let be an arbitrary two-dimensional open domain with a smooth boundary that is symmetric the vertical axis and lies in a disk of a fixed radius centered at the point , see Figure 2. Let be the union of the -integer translations of along the axis . Then we define as the image of under the mapping .
Consider the following spectral problem:
Remark 2.1. In particular, it can be proved that the number of eigenvalues (bearing in mind the multiplicities) of the original problem converging to the eigenvalue of the limit (homogenized) problem is equal to the multiplicity of the mentioned eigenvalue of the limit problem (for the method of proof see, e.g., ).
Remark 2.2. The limit spectral problem (2.3) is studied very well. In particular, if the eigenvalue is simple, then the corresponding eigenfrequency of (2.3) is the zero-point of the Bessel-function , and the corresponding eigenfunction has the form . One can find the definition of Bessel-functions, for example, in [20, Section 4.7].
The goal of this paper is to construct and verify the asymptotic expansion for the eigenvalues of (2.2). The obtained asymptotics is used for studying the behavior of the best constant in a Friedrichs-type inequality for functions belonging to the Sobolev class (see the definition of in Section 7). One of the main results of this paper is the following asymptotics for converging to : where are some fixed constants which can be calculated according to (4.23) and (4.15) in the case of simple and according to (5.10) and (4.15) when is of multiplicity two. In particular, which implies that .
3. Construction of the Leading Terms of the Asymptotic Expansion
Suppose that is the simple eigenvalue for (2.3) and the corresponding eigenfunction is normalized in . Our aim is to construct the leading terms of the asymptotic expansions for converging to as well as converging to . We use the method of boundary-layer functions (see ) for this purpose. We are looking for eigenvalues and eigenfunctions in the following form: where , and
Substituting the first expansion from (3.1) and the sum from the second expansion in (2.2) and equating terms at the same power of , we get the equation for : The existence of the solution for (3.3) is given in the following proposition.
Proposition 3.1. For any , there exists the smooth solution of (3.3) satisfying the boundary condition
Proof. The existence of the smooth solution follows from the classical results on regular solutions of elliptic equations (see e.g., ). In order to get as the unique solution, one can add the condition of mutual orthogonality:
By multiplying (3.3) by , integrating (3.3) over , and twice integrating by parts the obtained equation, we find that Taking into account the fact that is the normalized (in ) solution of (2.3) and since satisfies (3.5), we can deduce that Then (3.7) leads to (3.4) and the proof is complete.
However, the approximation does not satisfy the condition on . This forces us to introduce an additional term in second expansion of (3.1) to satisfy the appropriate boundary condition. We assume that the function has exponential decay as and is -periodical with respect to . Under this assumption, “almost” does not destroy (2.2) in the sense that the norm of additional contribution is small. The rigorous explanation is given in Section 6. Proceeding, we have that Taking into account (2.3) and (3.3), we see that has to satisfy the equation
Rewrite in polar coordinates and pass to the -variables in the argument of : Finally, replacing formulas and with Taylor series with respect to , substituting the obtained formula for in (3.9), and equating terms at , we deduce that
Taking into account (3.4), we derive the boundary conditions for on and on :
The following lemma gives the conditions to obtain as an exponentially decaying function as .
Lemma 3.2. Assume that is -periodic with respect to function with exponential decay as , and let be a -periodic solution of the boundary-value problem: with finite Dirichlet integral in . Then there exists the unique weak solution, which has asymptotics . To obtain as a function with exponential decay as , it is necessary and sufficient to have
Proof. The existence of the solution with asymptotics follows from the classical results on elliptic boundary-value problems in cylindric domains (see, e.g.,  and [24, Chapters 2, 5]). Let us verify (3.20). Define and . By multiplying the equation from (3.15) by , integrating it over , and using the property of , we get that Passing to the limit as , we obtain that This can be rewritten as Then has exponential decay as if and only if which is equivalent to (3.20). The proof is complete.
In order to obtain as function with exponential decay as , one must have where we denote . However, (3.24) implies that Integrate the identities :
4. Complete Expansion in the Case of the Simple Eigenvalue
Assume that is the simple eigenvalue of the limit problem. Now we construct the complete expansion in the following form: where is a smooth cutoff function, which equals to one when and zero when : Here , are -periodic in functions with exponential decay as . One can easily show that (4.2) solves the equation: if and only if .
We are looking for , which solves the equation: If (4.4) and (4.5) are satisfied, then from (4.1) is the solution of where . Our aim is to construct so that will be of small order as . This is the reason why we need to have as exponentially decaying functions.
Now we derive the formula for the Laplacian in -variables:
By substituting the Taylor series for the functions in (4.7), we get the final formula for :
By equating terms of the same power of , we obtain that
Consider now the boundary conditions from (2.2). According to the property of , in a small neighborhood of . Moreover, on , it yields that
We assume that the function has asymptotics: and since , we can derive the following formulas for : Rewriting as a Taylor series with respect to , we have
Consider now the boundary conditions on small holes. Analogously,
Substituting the Taylor series for with respect to in the last formula, using (4.14), and equating the terms with the same powers of in equation on , we get the following boundary condition for , on : where are polynomials of power with respect to with coefficients which depend on . The precise formula for can be derived for each fixed . For example, we have that The following Lemma is useful for our analysis. For the proof see for example,.
Lemma 4.1. Suppose that and satisfy the conditions of Lemma 3.2. (a) If is even with respect to , then is even; (b) if is odd with respect to and , then is odd with respect to and decays exponentially as .
Theorem 4.2. There exist numbers and -periodic in functions with finite Dirichlet integral in and exponential decay as , such that these functions are solutions of the following boundary-value problems: Moreover, the constants are defined by the formula: In particular,
Proof. Let be the solution of boundary-value problem (3.15). It can be easily verified that is a solution of (4.22), (4.20), (4.17) for , and defined by (4.21), (4.18), and (4.24). For any boundary-value problem (4.22), (4.20), (4.17) for has a -periodic solution with finite Dirichlet integral. By Lemma 3.2 and (3.27), has exponential decay as if and only if is given by (4.23) for . Let us verify formula (4.25) without applying the general (4.23). It is obvious that By using that fact one can write the boundary-value problem for as It can be verified that the function is -periodic with finite Dirichlet integral in , has exponential decay as , and satisfies problem (4.28) for defined by (4.25). We can use the induction process to finalize the proof.
Theorem 4.2 implies the validity of the following useful result.
Theorem 4.3. For any integer , the function is the solution of the boundary-value problem where , and as .
Proof. According to the definition of , we have that
Passing from variables to polar coordinates , we get that
By using the fact that and due to the result of Theorem 4.2, we have that, for any , where is fixed. Hence, . Similarly, taking into account (4.34) and (4.35), we can deduce that Consequently, Furthermore,
According to the definition of , the support of and is the set . Summarizing, we have that and we can derive that Therefore, Consider now on : where are the coefficients of the Taylor series of the function . Hence, and which yields that as . Analogously, one can verify that as . The proof is complete.
5. Complete Expansion in the Case of Multiple Eigenvalue
In this section we consider the case when is of multiplicity two. The asymptotics of the eigenvalue were constructed in the form (2.4) and
In this case, where are polynomials of power with respect to with coefficients which depend on . Moreover, where
Theorem 5.1. There exist numbers and -periodic in even functions and odd functions with finite Dirichlet integral in , which have exponential decay as , such that these functions are solutions of the boundary-value problems (5.8), (5.4), (5.5), and (5.9), (5.7), respectively. Moreover, the constants are defined by the formula:
Proof. The problems (5.8), (5.5), (5.4) for functions coincide with problems (4.22), (4.17), and (4.20) (if one change by and by the respective ). Therefore the construction of and is just the same as the construction from the proof of Theorem 4.2. Due to (5.9), (5.7), the problem for is as follows: The function is even (due to (4.26)) and, hence, the right-hand side is odd in (5.11) and is even in (5.8). By Lemma 3.2 and Theorem 4.2, we conclude that there exists the even solution of (5.11) with exponential decay. Then we can use the iteration process to complete the proof.
Theorem 5.1 implies the validity of the following result.
Theorem 5.2. For any integer , the function is the solution of the boundary-value problem: where , and as .
Proof. The proof is analogous to the proof of Theorem 4.3. Hence, we omit the details.
6. Verification of the Asymptotics
Consider the boundary-value problem: where and is some fixed number.
Similarly to the techniques used in [3, 18], one can show that the boundary-value problem (6.1) has the solution and the following representation holds: for close to the simple eigenvalue of the problem (2.3) and for close to multiple eigenvalue of the problem (2.3). Here is normalized in eigenfunctions to (2.2) and is orthonormalized in eigenfunctions to (2.2). Moreover, where the constant is independent on and . It follows from (6.2) and (6.4) that Consider now the case of simple . Define the function: where and are the solutions of (4.31) and (3.15), respectively, and is given by (3.18). Then, by Theorem 4.3, is the solution of (6.1) if Taking into account (6.5), (6.7), and the fact that , we can conclude that for each fixed , Therefore the asymptotics constructed in Section 4 coincide with the expansion of . For the case of multiple , one can use the same technique. The difference is follows: one should use (6.3) instead of (6.2) and Theorem 5.2 instead of Theorem 4.3. The asymptotics of are completely verified.
7. Application to a Friedrichs-Type Inequality
Consider the sets , which were defined in Section 2.
Definition 7.1. The Sobolev class is the class of functions from having zero trace on .
Theorem 7.2. Let . Then a Friedrichs-type inequality holds, where the best constant has the asymptotics as . Here is the smallest root of the Bessel function and the constant is given by (3.18).
Proof. The geometric approach developed in [5, 9] allows us to state that there is a constant such that The idea and method of proof are exactly similar to the ones which were used in the mentioned papers. We are interested in the behavior of the best possible constant as . Clearly, the best constant , where is the smallest eigenvalue of the boundary-value problem (2.2) (due to the variational formulation of the smallest eigenvalue). Therefore, we can apply (2.4) and (3.28) to derive the asymptotic expansion for : Since we are interested in the smallest eigenvalue , we have to choose the smallest positive root of as , precisely, . Then, we get, after some simple calculations and using (4.15) and (3.28), The proof is complete.
This paper was completed during the stay of the second author as PostDoc at Luleå University of Technology in 2010-2011. The second author wants to express many thanks to Luleå University of Technology (Sweden) for financial support and wonderful conditions to work. The work of the first and the second authors was supported in part by RFBR (Project no. 09-01-00353) and the work of the forth author was supported by a grant from the Swedish Research Council (Project no. 621-2008-5186). The authors also thank referee for several suggestions, which have improved the final version of this paper.
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