EIGENVALUES HOMOGENIZATION FOR THE FRACTIONAL p-laplacian

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1 Electronic Journal of Differential Equations, Vol (2016), No. 312, ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu EIGENVALUES HOMOGENIZATION FOR THE FRACTIONAL -LAPLACIAN ARIEL MARTIN SALORT Abstract. In this work we study the homogenization for eigenvalues of the fractional -Lalace oerator in a bounded domain both with Dirichlet and Neumann conditions. e obtain the convergence of eigenvalues and the exlicit order of the convergence rates when eriodic weights are considered. 1. Introduction The urose of this article is to study the asymtotic behavior as ε 0 of the eigenvalues of the non-local roblem ( ) s u = λ,ε ρ ε (x) u 2 u u = 0 in R n \ in R n (1.1) where for ε > 0, the arameter λ,ε is the eigenvalue and 1 < <. The domain is assumed to be a bounded and oen set in R n, n 1. The weight functions ρ ε are ositive and bonded away from zero and infinity, i.e., for some constants ρ and ρ + it holds that 0 < ρ ρ ε (x) ρ + < x. (1.2) Here, for s (0, 1) we denote by ( ) s the fractional -Lalace oerator, which is defined as ( ) s u(x) u(y) 2 (u(x) u(y)) u(x) = 2. v. x y n+s dy. As ε 0 in (1.1), the following limit roblem is obtained R n ( ) s u = λ ρ 0 (x) u 2 u u = 0 in R n \ in (1.3) where ρ 0 (x) is the weak* limit in L () as ε 0 of the sequence {ρ ε } ε. For each fixed value of ε it is known that there exists a sequence of variational eigenvalues {λ ε k, } k 1 of (1.1) such that λ ε k, as k. Analogously, for the limit roblem (1.3), there exists a sequence of variational eigenvalues {λ 0 k, } k 1 such that λ 0 k, as k (see Section 2). e are interested in studying the behavior of the sequence {λ ε k, } k 1 as ε Mathematics Subject Classification. 35B27, 35P15, 35P30, 34A08. Key words and hrases. Eigenvalue homogenization; nonlinear eigenvalues; order of convergence; fractional -Lalacian. c 2016 Texas State University. Submitted Aril 27, Published December 10,

2 2 A. M. SALORT EJDE-2016/312 hen s = 1 and = 2, (1.1) becomes the eigenvalue roblem for the Lalacian oerator with Dirichlet boundary conditions. This roblem has been extensively studied and a comlete descrition of the asymtotic behavior of its sectrum was obtained in the 70 s. Boccardo and Marcellini [3], and Kesavan [12] roved that for each fixed k, lim ε 0 λε k,2 = λ 0 k,2. Later on, in [4] and [9] this result was extended to -Lalacian tye oerators. One of the uroses of our aer is to extend this results to non-local eigenvalue roblems. Our first result states the convergence of the k-th eigenvalue of roblem (1.1) to the k-th eigenvalue of the limit roblem (1.3) when a general family of weight functions is considered. Theorem 1.1. Let R n be an oen bounded domain and s (0, 1) and let ρ ε satisfying (1.2) such that ρ ε ρ0 weakly* in L () as ε 0. Let λ ε k, and λ0 k, be the k-th (variational) eigenvalues of (1.1) and (1.3), resectively. Then for each fixed k 1. lim ε 0 λε k, = λ 0 k, (1.4) A slight modification in the arguments of the roof in the revious result allow us to deal with the following non-local Neumann eigenvalue roblem considered recently in [7] L s, u + u 2 u = Λ,ε ρ ε (x) u 2 u in u s, (1.5) () where L s, denotes the regional fractional -Lalacian defined as u(x) u(y) 2 (u(x) u(y)) L s, := 2. v. x y n+s dy (1.6) for which, again, the min-max theory rovides a sequence of variational eigenvalues tending to +, denoted by {Λ ε k, } k 1. Analogously to the Dirichlet case, as ε 0, a limit roblem is obtained in terms of ρ 0, the weak* limit of ρ ε in L (), L s, u + u 2 u = Λ ρ 0 (x) u 2 u u s, () in (1.7) which has a sequence of variational eigenvalues denoted by {Λ 0 k, } k 1. Here s, () stands for a fractional order Sobolev sace, which is defined in Sections 2. The corresonding convergence result is stated as follows. Theorem 1.2. Let R n be an oen bounded domain and s (0, 1) and let ρ ε satisfying (1.2) such that ρ ε ρ0 weakly* in L () as ε 0. Let Λ ε k, and Λ0 k, be the k-th (variational) eigenvalues of (1.5) and (1.7), resectively. Then for each fixed k 1. lim ε 0 Λε k, = Λ 0 k, (1.8) Homogenization theory dates back to the late sixties with the works of Sagnolo and de Giorgi and it develoed very raidly during the last two decades. Homogenization theory tries to get a good aroximation of a macroscoic behavior of the

3 EJDE-2016/312 EIGENVALUES HOMOGENIZATION 3 heterogeneous material by letting the arameter ε 0. A case of relevant imortance is the study of eriodic homogenization roblems due to the many alications to hysics and engineering. The main references for the homogenization theory of (local) eriodic structures are the books by Bensoussan-Lions-Paanicolaou [1], Sanchez Palencia [17], Oleĭnik-Shamaev-Yosifian [16] among others. An interesting issue in the homogenization theory is to estimate the rates of convergence of the eigenvalues in (1.4) and (1.8); that is, to find bounds for the differences λ ε k, λ0 k, and Λε k, Λ0 k,. Since it is desirable to obtain the exlicit deendence on ε, we restrict our study to eriodic weights, i.e., we consider a family of weight functions ρ ε given in terms of a single-bounded Q eriodic function ρ in the form ρ ε (x) := ρ(x/ε), ε > 0, with Q being the unit cube of R n. The function ρ is assumed to satisfy the bounds (1.2). Under these assumtions it is well-known that ρ ε ρ in L () as ε 0, where ρ is the average of ρ on Q. In the local case, the rates of convergence for the eigenvalues of the -Lalace oerator were studied in several aers. The authors in [16] roved some estimates for the Dirichlet and the Neumann case when = 2 by using tools from functional analysis in Hilbert saces. Assuming that is a Lischitz domain they showed that there exists a constant C deending on k and such that λ ε k,2 λ 0 k,2 Cε 1 2. Later on, under the same assumtions on it was roved in [13] the following bounds for both Dirichlet and Neumann boundary conditions, λ ε k,2 λ 0 k,2 Cε log ε 1 2 +γ for any γ > 0, C deending on k and γ. hen the domain is more regular (C 1,1 is enough) in [11] exlicit deendence of the constant C on k was obtained. It was roved that λ ε k,2 λ 0 k,2 Cεk 3 1 n ε log ε 2 +γ for any γ > 0, C deending on γ. In both cases, when the domain is smooth, the logarithmic term can be removed. In [9] the results were extended to the local -Lalace oerator via non-linear techniques and the deendence on the constant was imroved. The authors in [9] roved that λ ε k, λ 0 k, Ck +1 n ε, Λ ε k, Λ 0 k, Ck 2 n ε (1.9) where C is a constant indeendent on k and ε which can be exlicitly comuted. To the best of our knowledge, no investigation was made on the homogenization and convergence rates for the weighted fractional -Lalacian eigenvalue roblem. In contrast with the -Lalacian oerator, the non-local nature of (1.1) makes more difficult to deal with the convergence rates. The main obstacle is how to manage the boundedness of fractional norms in order to obtain relations between the variational characterization of eigenvalues. In the two next results we obtain the rates of the convergence of the eigenvalues of roblems (1.1) and (1.5) when eriodicity assumtions are made on the weight family.

4 4 A. M. SALORT EJDE-2016/312 Theorem 1.3. Let R n be an oen and bounded domain and ρ L (R n ) be a Q eriodic function satisfying (1.2), Q being the unit cube of R n. Let λ ε k, and λ 0 k, be the k-th variational eigenvalues of (1.1) and (1.3), resectively. Then λ ε k, λ 0 k, Cε s (λ k, ) 1+ 1 for every k N and s (0, 1), λ k, being the k-th variational eigenvalue of the Dirichlet fractional -Lalacian of order s. The constant C deends only on, s, n, and the bounds of ρ. In the case = 2 the revious inequality becomes for every k N and s (0, 1). λ ε k,2 λ 0 k,2 Cε s k 3s/n Theorem 1.4. Let R n be an oen and bounded set with C 1 boundary and ρ L (R n ) be a Q eriodic function satisfying (1.2), Q being the unit cube of R n. Let Λ ε k, and Λ0 k, be the k-th variational eigenvalues of (1.5) and (1.7), resectively. Then Λ ε k, Λ 0 k, Cε s (Λ k, ) 2 for every k N and s ( 1, 1), Λ k, being the k-th variational eigenvalue of the regional Neumann fractional -Lalacian (1.7) with ρ 0 1. The constant C deends only on, s, n and the bounds of ρ. In the case = 2 the revious inequality becomes Λ ε k,2 Λ 0 k,2 Cε s k 4s/n for every k N and s ( 1, 1). Although the rates obtained in the two revious results are similar, in the Neumann case the range of allowed values for s is smaller, and more assumtions on the boundary of have to be made. Such restrictions arise from the use of trace arguments in the roof. Observe that the rates obtained in Theorems 1.3 and 1.4 are the natural generalization of the results for the local case stated in (1.9). This article is organized as follows: in Section 2 we introduce some definitions and roerties of the eigenvalues of non-local roblems meanwhile that in Section 3 we rove the results before stated. 2. Fractional Sobolev saces and fractional eigenvalues In this section we resent some well-known results about fractional Sobolev saces and the eigenvalues of non-local roblems. For more detailed information we refer to the reader, for instance to [8]. Let be a subset of R n, n 1. For any s (0, 1) and 1 we denote s, () the fractional Sobolev sace defined as follows s, () := { u L u(x) u(y) () : x y n +s L ( ) } endowed with the norm u s, () := u L () + [u] s, (), where [u] s, () is the so-called Gagliardo semi-norm of u defined as [u] s, () = u(x) u(y) x y n+s dx dy.

5 EJDE-2016/312 EIGENVALUES HOMOGENIZATION 5 Hereafter we denote X s, 0 () := {u : u = 0 in R n \ }. It is well-known that the sace s, is continuously embedded in s, when s s, (see for instance [8]). Lemma 2.1. Let [1, ) and 0 < s < s < 1. Let be an oen set in R n and u a measurable function. Then u s, () C u s, () for some ositive constant C = C(n, s, ). Here we denote by (v) U the average of the function v on the set U. This is useful tool to be used is the following fractional Poincaré inequality on cubes of side ε. Lemma 2.2. Let Q be the unit cube in R n, n 1. Then, for every u s, (Q ε ), 1 < < we have u (u) Qε L (Q ε) cε s [u] s, (Q ε), where Q ε = εq and c is a constant deending only on n. Proof. Given u s, (Q ε ), by using Jensen s inequality it follows that u (u) Qε dx = (u(x) u(y)) dy dx Q ε Q ε Q ε u(x) u(y) dy dx Q ε Q ε cε Qε s u(x) u(y) x y n+s dy dx, from where the result follows. It is readily seen that the following Poincaré-tye inequality holds: Q ε u L (Q ε) cε s [u] s, (Q ε) for all u X s, 0 (Q ε ), from where it follows that [ ] s, () is an equivalent norm in the sace X s, 0. Another result that we will use is the trace s inequality for fractional saces roved in [18], which it is necessary for our auxiliary comutations. Proosition 2.3. Let be a bounded C 1 domain and 1 < s < 1. Then u s 1, ( ) C u s, (), where C is a constant deending on s, and Fractional eigenvalues. Let R n be an oen bounded domain. Given a weight function ρ bounded away from zero and infinity, we consider the Dirichlet eigenvalue roblem ( ) s u = λ D ρ u 2 u in, u = 0 in R n \. (2.1) Because of the non-local nature of the roblem it is needed to consider the boundary condition not only on but in R n \. This roblem has a variational structure. e say that u X s, 0 () is a weak solution of (2.1) if R n R n u(x) u(y) 2 (u(x) u(y))(v(x) v(y)) x y n+s = λ D ρ(x) u 2 uv

6 6 A. M. SALORT EJDE-2016/312 for every v X s, 0 (). The following non-local Neumann eigenvalue roblem for the regional -Lalacian defined in (1.6) was considered recently in [7] L s, u + u 2 u = λ N ρ(x) u 2 u in, u s, (). (2.2) In this case, we say that a function u s, () is a weak solution of (2.2) if u(x) u(y) 2 (u(x) u(y))(v(x) v(y)) x y n+s + u 2 uv = λ N ρ u 2 uv for every v s, (). As in [10], a non-decreasing sequence of eigenvalues for (2.1) and (2.2) can be defined by means of cohomological index. e will denote by {λ Dk, } k 1 and {Λ N k, } k 1 such sequences, resectively. They can be written by using the following inf-su characterization: [u] λ Dk, = inf su, λ C D k u C ρ 1 u N k, = inf su L C () D k u C u s, () ρ 1 u L (), (2.3) where D k = { X s, 0 () : i() k} and D k = { s, () : i() k}. Here i denotes the cohomological index, see for instance [15] for the definition and further roerties. These formulas differ from the classical ones by the use of the index instead of the genus, but they coincide in the lineal case. In fact, when = 2 the sequence defined in (2.3) coincides with the sequence of variational eigenvalues which uses dimension instead of index (see for instance [19]) [u] 2 λ Dk,2 = min max s,2 (R n ) u 2, λ C D k u C ρ 1 2 u 2 N k,2 = min max s,2 (), (2.4) L 2 C () D k u C ρ 1 2 u 2 L 2 () where D k = { X s,2 0 () : dim = k} and D k = { s,2 () : dim = k}. The variational characterization of eigenvalues lays a fundamental role in our analysis and the roof of our results since it allows to reduce the eigenvalues convergence to the study of oscillating integrals. Bounds for the eigenvalues (2.3) are also necessary for our arguments. hen the weight function ρ satisfy (1.2), it is easy to see that (ρ + ) 1 λ k, λ Dk, (ρ ) 1 λ k,, (2.5) where λ k, is the k-th eigenvalue of the Dirichlet fractional Lalacian, i.e., it satisfies the following equation ( ) s u = λ u 2 u in, u = 0 in R n \. (2.6) Moreover, observe that for any u, u 0, (1.2) gives u s, () ρ 1/ u L () 1 ρ ( 1 + [u] u L () and using that X s, 0 () s, () it is straightforward to see that Neumann eigenvalues can be bounded with the Dirichlet ones, i.e., Λ N k, (ρ ) 1 (1 + λ k, ). (2.7) Since the sequences (2.3) are bounded in terms of the eigenvalues of (2.6), it is desirable to estimate such eigenvalues. ),

7 EJDE-2016/312 EIGENVALUES HOMOGENIZATION 7 For > 1, s (0, 1) and s > n, the authors in [10] roved that for k large the following bounds hold c 1 s s n n k n λ k, c 2 s n k n n+s n (2.8) for some ositive constants c 1 and c 2 deending on s, and n. Nevertheless, they susect that these estimates are not otimal. However when = 2, recise estimates for this sequences are known. In 1959, Blumenthal and Getoor [2] roved a eyl s formula for λ k,2 in the context of s stable symmetric rocesses, whose generators are the fractional Lalacians, more recisely, they roved the asymtotic formula λ k,2 (4π) s ( k 1 Γ(1 + n 2 ) ) 2s/n, k +. Moreover, in [6] it was roved that there exists some constant c indeendent on k such that c( µ k,2 ) s λ k,2 ( µ k,2 ) s, where µ k,2 is the k-th eigenvalue of the Lalacian with Dirichlet boundary conditions on. Since it is well-known that there exist constants c 1 and c 2 indeendent on k such that c 1 k 2 n µ k,2 c 2 k 2 n (see for instance [5]), for the case = 2, inequality (2.5) reads as C 1 k 2s/n λ k,2 C 2 k 2s/n (2.9) where C 1 and C 2 are two constant indeendent on k and s. 3. Proof of main results The convergence of the sequence of Dirichlet and Neumann eigenvalues is a consequence of the following simle lemma concerning oscillating integrals. hen eriodicity is not assumed on the weight functions, the result does not rovide any information about the order of the convergence. Lemma 3.1. Let R n be a bounded domain. Let {g ε } ε>0 be a set functions such that 0 < g g ε g + < + for g ± constants and g ε g weakly* in L (). Then lim (g ε g) u = 0 ε 0 for every u s, (), 0 < s < 1. Proof. The weak* convergence of g ε in L () says that g εϕ gϕ for all ϕ L 1 (). In articular, since u s, (), we have that u L 1 () and the result is roved. Proof of Theorem 1.1. Let δ > 0 and C k,δ X s, 0 () be a set of index greater or equal than k such that λ 0 k, = [u] inf su [u] C D k u C ρ = su 0 u u C k,δ ρ 0 u + O(δ) where D k = { X s, 0 () : i() k}. e use now the set C k,δ, which is admissible in the variational characterization of the kth eigenvalue of (1.1), to find a bound for it as follows, [u] u C k,δ [u] λ ε k, su u C k,δ ρ = su ε u ρ 0 u ρ 0 u ρ ε u. (3.1)

8 8 A. M. SALORT EJDE-2016/312 To bound λ ε k, we look for bounds of the two quotients in (3.1). For every function u C k,δ we have [u] [v] ρ su 0 u v C k,δ ρ = λ 0 0 v k, + O(δ). (3.2) Since u C k,δ X s, 0 (), by Lemma 3.1 we obtain ρ 0 u ρ 1 + O(ε). (3.3) ε u Then, combining (3.1), (3.2) and (3.3) we find that λ ε k, (λ0 k, + O(δ))( 1 + O(ε)), from where it follows that λ ε k, λ 0 k, O(ε, δ). In a similar way, interchanging the roles of λ 0 k, and λε k,, we obtain that λ0 k, λε k, O(ε, δ). Gathering both inequalities and letting δ 0 and ε 0 it is obtained the desired result. Proof of Theorem 1.2. The roof of the Neumann case follows with an analogous argument to that of Theorem 1.1 by considering the Rayleigh quotients related to Λ 0 k, and Λε k, and by alying Lemma 3.1. hen eriodicity assumtions are made on the weight functions, besides the convergence of the eigenvalues, estimates on the rates of the convergence are obtained. The roofs of Theorems 1.3 and 1.4 follow the ideas introduced by Oleĭnik et al. in [16], where the roblem of obtaining rates on the eigenvalues is reduced to the study of the convergence rates of oscillating integrals. First we rove the Dirichlet case. Later, since the Neumann case involves estimates on the boundary of the domain, it will be necessary to assume some additional hyothesis; nevertheless the main idea in the roof still being the same. The following inequality will be useful to rove our next lemma. e refer to [14] for a roof. Lemma 3.2. For > 1 and x, y R n, x y, x y x 2 x (x y). Lemma 3.3. Let R n be a bounded domain and denote by Q the unit cube in R n. Let g L (R n ) be a Q-eriodic function such that ḡ = 0. Then the inequality g( x ε ) v cε s [v] v 1 L () holds for every v X s, 0 () with s (0, 1). The constant c deends only on, n, and the bounds of g. Proof. Denote by I ε the set of all z Z n such that Q z,ε, Q z,ε := ε(z + Q). Given v X s, 0 () we consider the function v ε given by v ε (x) = 1 ε n v(y) dy Q z,ε for x Q z,ε. e denote by 1 = z I εq z,ε. Thus, we can write g ε v = g ε ( v v ε ) + g ε v ε, 1 1

9 EJDE-2016/312 EIGENVALUES HOMOGENIZATION 9 and we can bound the revious exression as g ε v g + v v ε + g ε v ε. (3.4) 1 1 The first integral in (3.4) can be slit as v v ε = v v ε + v ε v (3.5) 1 I 1 I 2 where I 1 = {x : v v ε 0} and I 2 = {x : v v ε < 0}. Then, by using Lemma 3.2 we can bound (3.5) as v 1 v v ε + v ε 1 v v ε. (3.6) 1 1 To bound this exression, first we observe that by using Lemma 2.2 we have v v ε = v v ε dx 1 z I ε Q z,ε c ε s z I ε [v] s, (Q z,ε) = c ε s z I ε c ε s z I ε Q z,ε Q z,ε z I ε v(x) v(y) x y n+s dxdy Qz,ε Q z,ε v(x) v(y) x y n+s dxdy (3.7) = c ε s [v] s, ( 1) c ε s [v]. Secondly, if = 1 we have v ( 1) 1 = v. 1 (3.8) Moreover, since v ε ( ) 1/ Q v C 1 v we obtain v ε ( 1) C v. (3.9) 1 1 Then, combining (3.7), (3.8) and (3.9) we can bound (3.6) as ( ) ( 1/ ( ) v v ε v ( 1) 1/ ( ) ) + v ε ( 1) 1/ (3.10) cε s [v] v 1 L (). Finally, since ḡ = 0 and since g is Q eriodic, we obtain g ε v ε = v ε g ε = 0. (3.11) 1 z I ε Q z,ε Now, combining (3.10) and (3.11) we can bound (3.4) by g ε v Cε s [v] v 1 L (), and the roof is comlete.

10 10 A. M. SALORT EJDE-2016/312 Now we are ready to rove our main result. Proof of Theorem 1.3. Let δ > 0 and let C k,δ X s, 0 () be a set of index greater or equal then k such that [u] λ 0 k, = inf su C D k u C ρ [u] = su u u C k,δ ρ + O(δ) u where D k = { X s, 0 () : i() k}. e use now the set C k,δ, which is admissible in the variational characterization of the k-th eigenvalue of (1.1), in order to find a bound for it as follows, [u] λ ε k, su [u] u C k,δ ρ = su ε u u C k,δ ρ ρ u u ρ ε u. (3.12) To bound λ ε k, we look for bounds of the two quotients in (3.12). function u C k,δ we have that For every [u] ρ [v] su u v C k,δ ρ = λ 0 v k, + O(δ). (3.13) Since u C k,δ X s, 0 (), by Lemma 3.3 we obtain ρ u ρ ε u 1 + [u] s,(rn) u cεs ρ ε u 1 L () 1 + cε s ρ [u] + s,(rn) u ρ ρ u 1 + cε s ρ + ρ [u] s, (Rn ) ρ ( u ) 1/ 1 + Cε s (λ 0 k, + O(δ)) 1/. Then, combining (3.12), (3.13) and (3.14) we find that Letting δ 0 we obtain 1 L () λ ε k, (λ 0 k, + O(δ)) ( 1 + Cε s (λ 0 k,) 1/). (3.14) λ ε k, λ 0 k, Cε s (λ 0 k,) (3.15) In a similar way, interchanging the roles of λ 0 k, and λε k,, we obtain Hence, from (3.15) and (3.16), we arrive at λ 0 k, λ ε k, Cε s (λ ε k,) (3.16) λ ε k, λ 0 k, Cε s max{λ 0 k,, λ ε k,} 1+ 1, and by using the bounds given (2.5) and (2.9) the result follows. The following Lemma is necessary to deal with the convergence rates of functions in s, ().

11 EJDE-2016/312 EIGENVALUES HOMOGENIZATION 11 Lemma 3.4. Let R n be a bounded domain with C 1 boundary and, for δ > 0, let G δ be a tubular neighborhood of, i.e. G δ = {x : dist(x, ) < δ}. Then there exists δ 0 > 0 such that for every δ (0, δ 0 ) and every v s, () we have whenever 1/ < s < 1. u L (G δ ) cδ 1/ u s, (). Proof. Given G δ, the sets G δ are uniformly smooth surfaces. By the trace s inequality stated in Proosition 2.3 and the continuous inclusion given in Lemma 2.1 we have u L ( G δ ) u s 1, ( G δ ) c u s, (G δ ) c u s, (), δ (0, δ 0 ) rovided that 1/ < s < 1, where c is a constant indeendent on δ and u. Integrating this inequality with resect to δ we obtain δ ( ) u L (G δ ) = u ds dτ cδ u s, () G τ and the result is roved. 0 The roof of the next Lemma follows with a slight modification to that of Lemma 3.3, and it is essential for handling the convergence rates of eigenvalues of the Neumann roblem (1.5). Lemma 3.5. Let R n be a bounded domain with C 1 boundary and denote by Q the unit cube in R n. Let g L (R n ) be a Q-eriodic function such that ḡ = 0. Then the inequality g( x ε ) v cε s v s, () holds for every v s, () with 1/ < s < 1. The constant c deends only on,, n and the bounds of g. Proof. The roof is quite similar to that of Lemma 3.3, however there are some details to have into account. Denote by I ε the set of all z Z n such that Q z,ε and Q z,ε is comletely contained in, being Q z,ε := ε(z + Q). Given v s, () we consider the function v ε given by v ε (x) = 1 ε n Q z,ε v(y) dy for x Q z,ε. e denote by 1 = z I εq z,ε. Thus, we can write g ε v = g ε v + g ε ( v v ε ) + g ε v ε. (3.17) G 1 1 where G = \ 1. As in the Dirichlet Lemma we have g ε ( v v ε ) + g ε v ε cε s [v] s, () u 1 L (). (3.18) 1 1

12 12 A. M. SALORT EJDE-2016/312 The set G is contained in a δ neighborhood of with δ = cε for some constant c, and therefore, according to Lemma 3.4 we have g ε v cε v s, (). (3.19) G Since ε and s are less than 1, gathering (3.17), (3.18) and (3.19) we obtain g ε v Cε s [v] s, () u 1 L () + Cε v s, () and the roof is comlete. Cε s v s, () Since Lemma 3.5 has been roved, the roof of Theorem 1.4 is analogous to that of Theorem 1.3, by using Lemma 3.5 instead of Lemma 3.3 with the bound in (2.7). Acknowledgements. This research was artially suorted by the Universidad de Buenos Aires under grant UBACyT BA and by ANPCyT under grant PICT The author is member of CONICET. References [1] A. Bensoussan, J.-L. Lions, G. Paanicolaou; Asymtotic analysis for eriodic structures, AMS Chelsea Publishing, Providence, RI, 2011, Corrected rerint of the 1978 original [MR ]. MR [2] R. M. Blumenthal, R. K. Getoor; The asymtotic distribution of the eigenvalues for a class of Markov oerators, Pacific J. Math. 9 (1959), MR (21 #6023) [3] L. Boccardo and P. Marcellini; Sulla convergenza delle soluzioni di disequazioni variazionali, Ann. Mat. Pura Al. (4) 110 (1976), MR (54 #13300) [4] T. Chamion, L. De Pascale; Asymtotic behaviour of nonlinear eigenvalue roblems involving -Lalacian-tye oerators, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 6, MR (2009b:35315) [5] Courant, R., Hilbert, D.; Methoden der mathematischen Physik, vol. 1 (Vol. 2,. 1937), (1931). Berlin. [6] Zhen-Qing Chen, Renming Song; Two-sided eigenvalue estimates for subordinate rocesses in domains, J. Funct. Anal. 226 (2005), no. 1, MR (2006d:60116) [7] Del Pezzo, L. M., Salort, A. M.; The first non-zero Neumann -fractional eigenvalue., (2015). Nonlinear Analysis: Theory, Methods & Alications, 118, [8] Demengel, F., Demengel, G.; Functional saces for the theory of ellitic artial differential equations., (2012). Sringer. [9] J. Fernández Bonder, J. P. Pinasco, A. M. Salort; Convergence rate for some quasilinear eigenvalues homogenization roblems, (2015) Journal of Mathematical Analysis and Alications, 423(2), [10] Iannizzotto, A., Squassina, M. eyl-tye laws for fractional -eigenvalue roblems. Asymtot. Anal, 88(4) (2014), [11] C. Kenig, F. Lin, Z. Shen; Estimates of eigenvalues and eigenfunctions in eriodic homogenization, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, MR [12] S. Kesavan; Homogenization of ellitic eigenvalue roblems. II, Al. Math. Otim. 5 (1979), no. 3, MR (80i:65110) [13] C. Kenig, F. Lin, Z. Shen; Convergence rates in L 2 for ellitic homogenization roblems, Arch. Ration. Mech. Anal. 203 (2012), no. 3, MR [14] P. Lindqvist; On the equation div ( u 2 u) + λ u 2 u = 0, Proc. Amer. Math. Soc. 109 (1990), no. 1, MR [15] D. Motreanu, V. V. Motreanu, N. S. Paageorgiou; Toological and variational methods with alications to nonlinear boundary value roblems, Sringer, New York, xi+459. (2014). [16] O. A. Oleĭnik, A. S. Shamaev, G. A. Yosifian; Mathematical roblems in elasticity and homogenization, Studies in Mathematics and its Alications, vol. 26, North-Holland Publishing Co., Amsterdam, MR (93k:35025)

13 EJDE-2016/312 EIGENVALUES HOMOGENIZATION 13 [17] E. Sánchez-Palencia; Équations aux dérivées artielles dans un tye de milieux hétérogènes, C. R. Acad. Sci. Paris Sér. A-B [18] C. Schneider; Trace oerators in Besov and Triebel-Lizorkin saces. Univ. Leizig, Fak. fur Mathematik U. Informatik (2008). [19] R. Servadei, E. Valdinoci; Variational methods for non-local oerators of ellitic tye (2013). Discrete Contin. Dyn. Syst, 33 (5), Ariel Martin Salort Deartamento de Matemática, FCEN - Universidad de Buenos Aires and IMAS - CON- ICET. Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n. Buenos Aires, Argentina address: asalort@dm.uba.ar URL: htt://mate.dm.uba.ar/~asalort