RESEARCH Open Access On Friedrichs-type inequalities in domains rarely perforated along the boundary Yulia Koroleva 1,2 , Lars-Erik Persson 2,3 and Peter Wall 2* * Correspondence: peter.wall@ltu.se 2 Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden Full list of author information is available at the end of the article Abstract This article is devoted to the Friedrich s inequality, where the domain is periodically 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 H 1 . AMS 2010 Subject Classification: 39A10; 39A11; 39A70; 39B62; 41A44; 45A05. Keywords: Friedrichs-type inequ alities, homogenization, perforated along the boundary 1 Introduction This article deals with Friedrichs-type inequalities f or functions defined on domains which have a periodic perforation along the boundary. The size, shape and distribution of the perforation are described by a small parameter. It is assumed that the perfora- tion is “rare”, i.e., the size of the local perforation is much smaller than the period of perforation.Moreover,weconsiderthecasewherethefunctionssatisfyahomoge- neous Neumann condition on the part of the boundary corresponding to the domain without perfo ration and vanish on the perforation. The questions we are interested in are; how does the best constant in the Friedrichs-type inequality depend on the small parameter and what happens in the limit case where the parameter tends to zero? In particular, we will prove that the best constant converges to th e best constant in a dif- ferent type of Friedrichs inequality. The limit inequality is valid for all functions in the Sobolev space H 1 . Similar questions, with different types of microheterogeneities in a neighborhood of the boundary, were studied in [1-5]. In [1] (see also [2]), domains with a periodical perforation along the boundary were considered and the precise asymptotics of the best constant in a Friedrichs-type inequality was established. It was assumed that the size of perforation and the period were of the same order. Two different cases with non-periodical perforation were considere d in [4,5]. The convergence of the constant, as the size of perforation tends to zero, to the constant in the limit inequality was proved. In [3], a Friedrichs-type inequality was proved for functions vanishing on small periodically alternating pieces of t he boundary. The length of the pieces where the Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 © 2011 Koroleva et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativec ommons.or g/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, pro vided the original work is properly cited. functions vanish was assumed to be o f the same order as the length of the period. In particular, the precise asymptotics of the best constant, with respect to the small para- meter describing the heterogeneous boundary condition, was derived. 2 Preliminaries and statement of the problem Let Ω ⊂ ℝ 2 be a bounded domain such that the boundary, ∂Ω, is Lipschitz continuous. Suppose that coordinates (x 1 , x 2 )areusedinΩ. Introduce local coordinates (s, t)ina small neighborhood of ∂Ω in the following way: choose the origin O Î ∂Ω and a point P(s, t) in a neighborhood of ∂Ω.Then,t is the distance from P to the boundary and s is the counter clock-wise length of the boundary from O to the point P 1 (s, t), where P 1 is the point for which t = | PP 1 | (see Figure 1). Consider a semi-strip B ={ξ Î ℝ 2 :0<ξ 1 <1,ξ 2 >0}andaclosedsetT μ ⊂ B depending on a small parameter μ Î (0, 1] which characterizes the size and the shape of the perforation (see Figure 2). We study the case when T μ is shrinking in a uniform way as μ goes to zero. More- over, we assume that T μ is uniformly bounded with respect to μ, i.e., there exists r Î ℝ, r > 0 such that T μ ⊂ {ξ Î ℝ 2 :0<ξ 1 <1,0<ξ 2 <r} for all μ Î (0, 1]. Let T 1 μ be 1-periodic extension of T μ with respect to ξ 1 and T ε μ is the image of T 1 μ under the mapping s = εξ 1 , t = εξ 2 ,whereε is a small parameter, 0 <ε 1, 1 ε ∈ N . Define the domain ε = \T ε μ (see Figure 2). Further, we assume that μ = μ(ε) and that μ = μ ( ε ) → 0asε → 0 . (1) Hence, ε > 0 is a parameter which describes both the size of the perforation and the length of the period. Consider the following spectral problem: ⎧ ⎪ ⎨ ⎪ ⎩ −u ε = λ ε u ε i n ε , u ε =0 onT ε μ , ∂u ε ∂ ν =0 on∂ . (2) Here ν denotes the unit outward normal to Ω. The limit problem for (2) depends on how fast the size of the perforation goes to zero relative the length of the period. It was proved in [6] (see also [7]) that if the perforation is “rare”, i.e., the size of the local perforation goes to zero much faster than the period of perforation, then the limit pro- blem for (2) is the Robin boundary value problem P ( s; t ) t s P 1 (s;t) O : : W ¶ W : Figure 1 The local coordinate system. Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 2 of 12 −u 0 = λ 0 u 0 in , ∂u 0 ∂ ν + pu 0 =0on∂ , (3) where 0 <p < ∞. The precise meaning of that the perforation is rare is given in (14) later on. The faster the size of the local perforation goes to zero the smaller p will be. There is a critical speed which gives that p is equal to zero, see [6,7]. In such situations the limit problem is of N eumann type. The limit problem is of Dirichlet type (p = ∞) when the size of the local perforation does not go to zero “fast enough” relative the period of perforation. According to the general theory o f elliptic operators, there exist countable sets {λ k ε } and {λ k 0 } of eigenvalues of (2) and (3) which satisfy 0 <λ 1 ε ≤ λ 2 ε ≤···≤λ k ε ≤···,0<λ 1 0 ≤ λ 2 0 ≤···≤λ k 0 ≤··· . Using the same arguments as in [4], it follows that λ 1 ε > 0 .Thistogetherwiththe variational formulation of the smallest eigenvalue of (2) lead to the following Frie- drichs-type inequality for functions u Î H 1 (Ω) which vanish on T ε μ : ε u 2 dx ≤ K ε ε |∇u| 2 dx , (4) where K ε is the best constant and is given by K ε = 1 λ 1 ε . (5) In the case with p = ∞ (Dirichlet boundary conditions in the limit problem) the smallest eigenvalue λ 1 0 for the limit problem is related to the best constant in the Frie- drichs inequality f or functions in H 1 0 ( ) . Indeed, via the variational formulation of λ 1 0 we have that u 2 dx ≤ K 0 |∇u| 2 dx , (6) where the best constant is given by K 0 =1/λ 1 0 . » 2 » 1 § B T ¹ 1 e W 0 Figure 2 Geometry of the perforated domain. Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 3 of 12 A geometrical proof of that K ε ® K 0 was presented in [4] for the case p = ∞.The goal of this article is to answer the following questions, in the case 0 <p < ∞:(1)Is there a Friedrichs-type inequality related to the limit Robin boundary value problem? (2) If the answer on the first questions is yes, how is then K ε related to the best con- stant, K 0 , in this Friedrichs-type inequality. We will see that there is such a Friedrichs- type inequality and we p resent a result describing the asymptotic relation between K ε and K 0 . Moreover, as a result of our analysis we also obtain the convergence o f the eigenvalues, λ k ε → λ k 0 , for the case 0 <p < ∞. 3 The main results The following Friedrichs-type inequality holds for functions in H 1 (Ω): Proposition 1 There exists a constant K 0 >0such that u 2 dx ≤ K 0 ⎛ ⎝ |∇u| 2 dx + p ∂ u 2 ds ⎞ ⎠ (7) for any u Î H 1 (Ω). Moreover, the best constant is K 0 =1/λ 1 0 ,where λ 1 0 is the smallest eigenvalue in the limit problem (3). Proof. The variational formulation of the smallest eigenvalue of the limit problem (3) is λ 1 0 = min u∈H 1 ()\{0} ⎧ ⎪ ⎨ ⎪ ⎩ |∇u| 2 dx + p ∂ u 2 ds u 2 dx ⎫ ⎪ ⎬ ⎪ ⎭ . For details, see paragraph 2.5 in [8]. From this, it is clear that the inequality (7) holds. It also follows that the best constant is 1/λ 1 0 . Let us define the following set of functions: W = u ∈ H 1 (): ∂u ∂ν + pu =0 on∂,0< p < ∞ . Note that solutions of the limit problem (3) belong to W. We remark that an inequality of the form (6) cannot be valid for functions in W. Indeed, Proposition 2 There is no C >0such that the inequality u 2 dx ≤ C |∇u| 2 d x (8) holds for all functions in W. Proof. We prove the statement by a counter example. Let K m = {x ∈ :dist ( x, ∂ ) ≥ 1/m, m ∈ N} . Define the function u m such that u m = m + p on K m and u m = m on ∂Ω.Itispossi- ble to construct a s mooth transition from K m to ∂Ω such that ∂u m /∂ν + pu m =0on ∂Ω and | ∇u m | 2 ≤ k m + p −m 1/m 2 = kp 2 m 2 Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 4 of 12 in Ω\K m for s ome constant k.Notethat∇u m =0onK m and t hat |Ω\K m |~1/m (~ means asymptotically equal to). We get that |∇u m | 2 dx u 2 m dx ≤ kp 2 m 2 |\K m | u 2 m dx ≤ kp 2 m 2 |\K m | m 2 || → 0 as m ® ∞. Thus, (8) cannot hold. We will now consider how to estimate the difference between the best constants, K 0 and K ε , in t he inequalities (7) and (4). First, observe that by Proposition 1 and (5) we have that | K ε − K 0 | = 1 λ 1 ε − 1 λ 1 0 . (9) To estimate |1/λ 1 ε − 1/λ 1 0 | we will use the i deas in the general method for estimating the difference between eigenvalues and eigenvectors of two operators defined on differ- ent spaces, which was introduced by Oleinik et al. [9], see also [10]. For the readers convenience, we review the main ideas in the method mentioned above. Indeed, let H ε and H 0 be separable Hilbert spaces with the inner products (u ε , v ε ) H ε ,(u, v) H 0 and norms | |u ε || H ε , ||u|| H 0 , respectively; assume that A ε Î L(H ε )and A 0 Î L(H 0 ) are linear continuous operators and Im A 0 ⊆ V ⊆ H 0 ,whereV is a linear subspace of H 0 . The following conditions are supposed to hold: C1 There exist linear contin uous operators R ε : H ε ® H 0 and a constant c >0such that (R ε f , R ε f ) H ε → c(f , f ) H 0 as ε → 0 for any f Î V. C2 The operators A ε : H ε ® H ε and A 0 : H 0 ® H 0 are positive, compact and self- adjoint. Moreover, sup ε || A ε || L ( H ε ) < + ∞ . C3 For all f ÎV it holds that | |A ε R ε f −R ε A 0 f || H ε → 0asε → 0 . C4 The sequence of operators A ε is uniformly compact in the following sense: if we take a sequence {f ε }, where f ε Î H ε , such that sup ε ||f ε || H ε < + ∞ , then there exist a sub- sequence {f ε k } and vector w 0 Î V such that | |A ε k f ε k − R ε k w 0 || H ε k → 0 as ε k ® 0. Let us also introduce the spectral problems for operators A ε , A 0 : A ε u k ε = μ k ε u k ε , μ 1 ε ≥ μ 2 ε ≥···, k =1,2, (u l ε , u m ε )=δ lm , (10) A 0 u k 0 = μ k 0 u k 0 , μ 1 0 ≥ μ 2 0 ≥···, k =1,2, (u l 0 , u m 0 )=δ lm , (11) where δ lm is the Kronecker symbol, the eigenvalues μ k ε , μ k 0 are repeated according to their multiplicities. The following lemma holds true (see [9, Chapter III]). Lemma 3 Suppose that the conditions C1-C4 are valid. Then, there is a sequence {β k ε } such that β k ε → 0 as ε → 0, 0 <β k ε <μ k 0 and the following estimate: | μ k ε − μ k 0 |≤ μ k 0 c − 1 2 μ k 0 − β k ε sup v∈N(μ k 0 ,A 0 ),||v|| H 0 =1 ||A ε R ε v −R ε A 0 v|| H ε (12) Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 5 of 12 holds, where N( μ k 0 , A 0 )={v ∈ H 0 : A 0 v = μ k 0 v } . Let us now give a more precise definition of that the perforation is rare. Indeed, introduce the space V μ(ε) as the closure of the set of functions in v Î C ∞ (ℝ 2 ∩ {ξ 2 > 0}) which are 1-periodic with respect to ξ 1 ,vanishingonT μ(ε ) and with finite B |∇v| 2 d ξ . The closure is with respect to the norm v = B |∇v| 2 dξ + v 2 dξ 1 .More- over, define the value η μ(ε) =inf v∈V μ(ε) \{0} B |∇v| 2 dξ v 2 dξ 1 , (13) where Σ:=∂B ∩ {ξ 2 = 0} (see Figure 2). Moreover, we define the number p as lim ε→0 η μ(ε) ε = p . (14) In fact, the number p corresponds to the ratio between measure of small set T μ and the length of p eriod, i.e., it describes how much of the Dirichlet condition pe r cell of periodicity we have. We will now prove the following estimate for |K ε - K 0 |: Theorem 4 Let K ε and K 0 be the constants in (4) and (7). If 0<p < ∞ is defined by (14), then there exists a constant C, independent of ε, such that | K ε − K 0 |≤C √ η μ(ε) + η μ(ε) ε − p + √ εη μ(ε) . (15) Proof. By (9) we will have an estimate of |K ε - K 0 | if we have an estimate of |1/λ 1 ε − 1/λ 1 0 | . In order to obtain such an estimate we will use the result in Lemma 3. Indeed, we introduce two auxiliary problems: ⎧ ⎪ ⎨ ⎪ ⎩ −u ε = f in ε , u ε =0 onT ε μ , ∂u ε ∂ν =0 on∂ (16) and the corresponding limit problem −u 0 = f in , ∂u 0 ∂ ν + pu 0 =0on∂ , (17) where f Î L 2 (Ω)andp satisfies (14). The fact that (17) is the limit problem for (16) for any f was established in [6]. More precisely, it was proved that if u ε Î H 1 (Ω ε )and u 0 Î H 1 (Ω) are weak solutions of (16) and (17), then u ε ⇀ u 0 weakly in H 1 (Ω)asε ® 0 which implies the convergence || u ε − u 0 || L 2 ( ) → 0asε → 0 . (18) Note that here and from now on, u ε is assumed to be defined in whole Ω and van- ishing on T ε μ . Let us now prove the following estimates for the solutions of (16) and (17): u ε H 1 ( ε ) ≤ k 1 f L 2 ( ε ) (19) Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 6 of 12 and | |u 0 || H 1 ( ) ≤ k 2 ||f || L 2 ( ) , (20) where k 1 and k 2 are independent of ε. First, we recall that using the technique devel- oped in [4] (see also [5]), one can prove the following Friedrichs-type inequality for functions w belonging to H 1 (Ω) and vanishing on T ε μ : w 2 dx ≤ K |∇w| 2 dx , (21) where K does not depend on ε. In particular, the inequality (21) implies that the solution of (16) satisfies the estimate ε u 2 ε dx ≤ K ε |∇u ε | 2 dx . (22) By choosing u ε as the test function in the weak formulation of (16), we have ε |∇u ε | 2 dx = ε fu ε dx . Using the Hölder inequality and (22), we obtain that | |∇u ε || L 2 ( ε ) ≤ √ K||f || L 2 ( ε ) . From this and (22) the estimate (19) follows, with k 1 = K(1 + K) . Let us now prove the estimate (20). Indeed, we start by showing that for any w Î H 1 (Ω)\{0} there exists a constant M which does not depend on w such that |∇w| 2 dx|p ∂ w 2 ds ≥ M||w|| 2 H 1 () . (23) Suppose that the contradiction holds: i.e., that for any m there exists w m Î H 1 (Ω)\{0} such that |∇w m | 2 dx + p ∂ w 2 m ds < 1 m ||w m || 2 H 1 () . Denote v m = w m /||w m || H 1 ( ) . Then, || v m || H 1 ( ) =1 (24) and |∇v m | 2 dx + p ∂ v 2 m ds < 1 m . (25) By the inequalities (7) and (25), we have that v 2 m dx ≤ K 0 ⎛ ⎝ |∇v m | 2 dx + p ∂ v 2 m ds ⎞ ⎠ < K 0 m . (26) Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 7 of 12 From (25) and (26), it follows that v m ® 0inH 1 ( Ω), which contradicts to (24). Thus, the estimate (23) is proved. Choosing u 0 as a test function in the weak formula- tion of (17) leads to the identity |∇u 0 | 2 dx + p ∂ u 2 0 ds = fu 0 dx . (27) By applying (23) to u 0 , using (27) and the Cauchy-Schwarz inequality, we get that | |u 0 || H 1 () ≤ 1 M ||f || L 2 () , (28) which is estimate (20) with k 2 =1/M. To estimate |1/λ 1 ε − 1/λ 1 0 | we will now use the method, which was described above, for estimating the difference between eigenvalues. Indeed, define the spaces H ε = L 2 (Ω ε ), H 0 = V = L 2 (Ω) and the restriction operator R ε : H 0 ® H ε . Define t he operators A ε and A 0 in the following way: A ε f = u ε and A 0 f = u 0 ,whereu ε and u 0 are the weak solutions of problems (16) and (17), respectively. Let us verify the conditions C1-C4. The condition C1 is valid with c = 1. Indeed, take f Î V. Then, (R ε f , R ε f ) ε = ε f 2 (x)dx → f 2 (x)dx =(f , f) 0 as ε ® 0 due to the fact that measure of T ε μ → 0 as ε ® 0. Let us verify the condition C2. First, we prove that the operator A ε is self-adjoint. Let f and g be functions in L 2 (Ω ε )anddefineu ε = A ε f and v ε = A ε g.Ifwechosev ε as test function in the weak formulation of (16) with f in the right-hand side and u ε as a test function in the case when the right-hand side is g, then we obtain that ε fv ε dx = ε ∇u ε ·∇v ε dx = ε gu ε dx . Hence, (A ε f , g) L 2 ( ε ) =(u ε , g) L 2 ( ε ) = ε ∇v ε ·∇u ε dx =(f, v ε ) L 2 ( ε ) =(f , A ε g) L 2 ( ε ) . Now, we prove the operator A 0 is self-adjoint. Define u 0 = A 0 f and v 0 = A 0 g, where f, g Î L 2 (Ω). According to the weak formulation of (17), we find that (A 0 f , g) L 2 () = u 0 gdx = ∇v 0 ·∇u 0 dx − ∂ u 0 ∂v 0 ∂ν ds = ∇v 0 ·∇u 0 dx + p ∂ u 0 v 0 ds = ∇u 0 ·∇v 0 dx − ∂ v 0 ∂u 0 ∂ν d s = fv 0 dx =(f, A 0 g) L 2 () . Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 8 of 12 That the operator A ε is positive follows from (A ε f , f ) L 2 () = u ε fdx = |∇u ε | 2 dx > 0 if u ε ≠ 0 (i.e., f ≠ 0). Similarly, we obtain that A 0 is positive. Indeed, the weak formu- lation of (17) gives (A 0 f , f ) L 2 () = u 0 fdx = |∇u 0 | 2 dx − ∂ u 0 ∂ u 0 ∂ν d s = |∇u 0 | 2 dx + p ∂ u 2 0 ds > 0 if u 0 ≠ 0. Next, we show that A ε and A 0 are compact operators. Let {f n } be a bounded sequence in L 2 (Ω ε ). Then, estimate (19) implies that there exists a constant c such that | |A ε f n || L 2 ( ε ) = ||u ε,n || L 2 ( ε ) ≤||u ε,n || H 1 ( ε ) ≤ k 1 ||f n || L 2 ( ε ) ≤ c . Hence, there exist a subsequence of {u ε,n }and ˜ u ε ∈ H 1 ( ε ) such that u ε,n k ˜ u ε weakly in H 1 (Ω ε ) and thus strongly in L 2 (Ω ε ) which exactly means that A ε is compact. Moreover, (19) implies that | |A ε f || L 2 ( ε ) ≤ k 1 ||f || L 2 ( ε ) for any f Î L 2 (Ω ε ). Hence, sup ε A ε L ( H ε ) ≤ k 1 . The compactness of A 0 can be proved analogously by applying esti- mate (20) instead of (19). Let us verify the condition C3 is fulfilled. Take f Î L 2 (Ω). It follows by (18) that | |A ε R ε f −R ε A 0 f || L 2 ( ε ) = ||A ε f −A 0 f || L 2 ( ε ) = ||u ε − u 0 || L 2 ( ) → 0 as ε ® 0. Let us verify that the condition C4 is satisfied. Consider a sequence { f ε }, where f ε Î L 2 (Ω ε ) such that sup ε ||f ε || L 2 ( ε ) < + ∞ . Then, | |A ε f ε || H −1 ( ε ) = ||u ε || H −1 ( ) ≤ k 1 ||f ε || L 2 ( ε ) < +∞ , due to (19). The Rellich imbedding theorem implies that the sequence {A ε f ε }iscom- pact in L 2 (Ω). Thus, there exists a subsequence {ε k } and w 0 Î L 2 (Ω) such that A ε k f ε k → w 0 as ε k → 0 . From this, we deduce that | |A ε k f ε k − R ε k w 0 || L 2 ( ε k ) → 0 as ε k ® 0. Hence, all the con- ditions C1-C4 are valid. Let l ε be an eigenvalue of the -Δ operator with the boundary conditions given in (16) and v ε the corresponding eigenvector. In this notation, we have that -Δv ε = l ε v ε and thus A ε (l ε v ε )=v ε . From this, it is evident that A ε v ε = (1/l ε )v ε . From this, it follows that μ k ε =1/λ k ε ( μ k ε is defined in (10)). In the same way, we can deduce that μ k 0 =1/λ k 0 . Using the estimate (12), we have 1 λ k ε − 1 λ k 0 ≤ 1 1 −λ k 0 β k ε sup v∈N(μ k 0 ,A 0 ),||v|| L 2 ( ) =1 ||A ε R ε v −R ε A 0 v|| L 2 ( ε ) . (29) Recall that N( μ k 0 , A 0 )={v ∈ H 0 : A 0 v = μ k 0 v } .Let v ∈ N(μ k 0 , A 0 ) .Ifwechoosef = v in the problem (17), then the solution, denoted by u v 0 , can be expressed as Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 9 of 12 u v 0 = A 0 v = μ k 0 v . Similarly, if we choose f = R ε v in the problem (16), then the solution, denoted by u v ε , is of the form u v ε = A ε R ε v . In this notation, (29) reads 1 λ k ε − 1 λ k 0 ≤ 1 1 −λ k 0 β k ε sup v∈N(μ k 0 ,A 0 ),||v|| L 2 ( ) =1 ||u v ε − R ε u v 0 || L 2 ( e ) . (30) In [6], it was proved that (17) is the limit problem corresponding to (16). By the results in [6], it follows that there exists a constant C 1 such that ||u ε − u 0 || L 2 () ≤ C 1 ||f || L ∞ () √ η μ + η μ ε − p + ||u 0 || L ∞ () √ εη μ (31) for any f Î L ∞ (Ω). In particular, for the present choice of f, f = v, we have | |u v ε − u v 0 || L 2 () ≤ C 1 ||v|| L ∞ () √ η μ + η μ ε − p + μ k 0 ||v|| L ∞ () √ εη μ . This together with the fact t hat eigenfunctions belong to C ∞ ( ¯ ) gives that there is a constant C 2 (which depends on k) such that sup v∈N(λ k 0 ,A 0 ),||v|| L 2 ( ) =1 ||u v ε − u v 0 || L 2 ( ε ) ≤ C 2 √ η μ + η μ ε − p + √ εη μ . (32) From this and (30), we obtain 1 λ k ε − 1 λ k 0 ≤ 1 1 −λ k 0 β k ε C 2 √ η μ + η μ ε − p + √ εη μ . By Lemma 3, we have that 1 −λ k 0 β k ε > 0 for sufficiently small values of ε (as β k ε → 0 ). Hence, there exists a constant C, independent of ε, such that 1 λ k ε − 1 λ k 0 ≤ C √ η μ + η μ ε − p + √ εη μ (33) and the proof is complete. As a consequence of the proof above we have the following result: Corollary 5 The eigenvalues λ k ε of (2) converge to the corresponding eigenvalue λ k 0 of (3). Proof. We note that by (33) | λ k 0 − λ k ε | = λ k ε λ k 0 1 λ k ε − 1 λ k 0 ≤ λ k ε λ k 0 C √ η μ + η μ ε − p + √ εη μ . It follows from (33) that {λ k ε } is bounded. Hence, | λ k 0 − λ k ε |≤λ k ε λ k 0 C √ η μ + η μ ε − p + √ εη μ → 0 as ε ® 0. Koroleva et al. Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 10 of 12 [...]... Chechkin, GA, Koroleva, YuO, Persson, LE: On the precise asymptotics of the constant in the Friedrich’s inequality for functions, vanishing on the part of the boundary with microinhomogeneous structure J Inequal Appl 2007, 13 (2007) Article ID 34138 4 Chechkin, GA, Koroleva, YuO, Meidell, A, Persson, LE: On the Friedrichs inequality in a domain perforated nonperiodically along the boundary Homogenization... Koroleva et al.: On Friedrichs-type inequalities in domains rarely perforated along the boundary Journal of Inequalities and Applications 2011 2011:129 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright... http://www.journalofinequalitiesandapplications.com/content/2011/1/129 References 1 Chechkin, GA, Gadyl’shin, RR, Koroleva, YuO: On the eigenvalue of the Laplace operator in a domain perforated along the boundary Dokl Acad Nauk 81(3), 337–341 (2010) 2 Chechkin, GA, Gadyl’shin, RR, Koroleva, YuO: On the asymptotic behavior of a simple eigenvalue of a boundary-value problem in a domain perforated along the boundary Diff Equ... AG: Homogenization of a Fourier boundary-value problem for Poisson equation in a domain perforated along the boundary Russ Math Surv 45(4), 123 (1990) 8 Sobolev, SL: Some applications of functional analysis in mathematical physics Translations of Mathematical Monographs, 90 American Mathematical Society, Providence (1991) 9 Oleinik, OA, Shamaev, AS, Yosifian, GA: Mathematical Problems in Elasticity and... Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129 Page 11 of 12 Remark 6 The result established in Theorem 4 is valid for a wide class of domains with perforation along the boundary It is possible to estimate the difference |Kε - K0| more precisely in some partial cases For some particular cases, it is in fact possible to construct... out the proofs All authors conceived of the study, and participated in its design and coordination All authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Received: 10 March 2011 Accepted: 5 December 2011 Published: 5 December 2011 Koroleva et al Journal of Inequalities and Applications 2011, 2011:129 http://www.journalofinequalitiesandapplications.com/content/2011/1/129... = εa , a > 0, then (34) gives that hμ(ε) /ε ® ∞ This means that the perforation is vanishing too slow in order to have Robin boundary conditions in the limit problems (3) and (17) Remark 8 The result obtained in this article can be generalized to higher dimensional domain The crucial step is to prove the estimate similar to (31) This is a good future research problem Acknowledgements The authors thank... Asymptotics in parabolic problems Russ J Math Phys 16(1), 1–16 (2009) doi, 10.1134/S1061920809010014 5 Koroleva, YuO: On the Friedrich’s-type inequality in a three-dimensional domain aperiodically perforated along a part of the boundary Russ Math Surv 65(4), 199–200 (2010) 6 Belyaev, AG: On singular perturbations of boundary-value problems PhD Thesis Moscow State University, Moscow (1990) (in Russian)... Elasticity and Homogenization North-Holland, Amsterdam (1992) 10 Oleinik, OA, Shamaev, AS, Yosifian, GA: On a limit behavior of spectrum of the sequence of operators, which are defined in different Hilbert spaces Russ Math Surv 44(3), 157–158 (1989) 11 Il in, AM: Matching of asymptotic expansions of solutions of boundary value problems Translations of Mathematical Monographs, 102 American Mathematical Society,... suggestions, which have improved the final version of this article Author details 1 Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia 2Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE971 87 Luleå, Sweden 3Narvik University College, Postboks 385, 8505 Narvik, Norway Authors’ contributions . 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. ration and vanish on the perforation. The questions we are interested in are; how does the best constant in the Friedrichs-type inequality depend on the small parameter and what happens in the. 45A05. Keywords: Friedrichs-type inequ alities, homogenization, perforated along the boundary 1 Introduction This article deals with Friedrichs-type inequalities f or functions defined on domains which