PERIODIC UNFOLDING FOR LATTICE STRUCTURES

. This paper deals with the periodic unfolding for sequences deﬁned on one dimensional 3 lattices in R N . In order to port the known results of the periodic unfolding in R N to lattices, the 4 investigation of functions deﬁned as interpolation on lattice nodes play the main role. The asymptotic 5 behavior for sequences deﬁned on periodic lattices with information until the ﬁrst and until the second 6 order derivatives are shown. In the end, a direct application of the results is given by homogenizing 7 a 4th order Dirichlet problem deﬁned on a periodic lattice.

1. Introduction.In the present, starting from the results obtained in [8,14] about the periodic unfolding method for sequences defined on bounded domains in R N , we show in detail how to port such results to one-dimensional periodic lattice structures, spotting the obstacles we encountered and the tools we came up with to overcome them.
Given a small parameter ε and a bounded domain Ω ⊂ R N with Lipschitz boundary, we consider the periodic paving of Ω made with cells of size ε.In [8,Section 1.4] it is extensively investigated the asymptotic behavior of sequences {φ ε } ε uniformly bounded in W 1,p (Ω) and in W 2,p (Ω), while the entirety of [14] is devoted to sequences anisotropically bounded on W 1,p (Ω).In this paper, we first introduce the periodic lattice S ε ⊂ R N as one-dimensional arbitrary grids defined on the ε cells and periodically repeated for each cell of Ω.The main idea to port the periodic unfolding results from Ω ⊂ R N to S ε is based on extending the sequences bounded on S ε by different interpolation on the lattice nodes, applying the unfolding results in R N and then restricting the convergences to the lattice itself.Specifically, the Q 1 interpolation on lattice nodes (already introduced in [12,13]) allows to show the asymptotic behavior of sequences uniformly bounded in W 1,p (S ε ) and anisotropically on W 1,p (S ε ), while for sequences bounded in W 2,p (S ε ) some more work is involved due to the lack of mixed derivatives.Starting from different assumption strengths and leading to different regularity of the unfolded limit fields, two methods are developed (one involving extensions by a special Q 3 interpolation and another involving the obtained results for sequences bounded in W 1,p (S ε )).The sufficient assumptions on the sequences to ensure weak convergence in the space, as well as the rescaling factors for the unfolding operator for lattices according to the space dimension N and the L p norm are proved.
In the end, a direct application of such lemmas is done by homogenizing via unfolding the fourth order homogeneous Dirichlet problem defined on a lattice structure The homogenization via unfolding method, an equivalent to the two-scale convergence, has been exhaustively explained in [8] and it is a constant reference throughout this work.The method itself has, among many others, found application in the homogenization for thin periodic structures like periodically perforated shells (see [9]) and textiles made of long curved beams (see [10,11]).About the homogenization in the frame of lattice structures one can look, for an instance, into [1,2,4,5,6,7].
The present provides the main tools concerning the unfolding for lattice structures and gives a rigorous base for up-coming papers dealing with thin structures made from lattices.Among them, we would like to cite the homogenization via unfolding for stable lattice structures made of beams (see [12,13]) and the upcoming unstable case [15], where it is additionally taken into consideration the problem of an anisotropically bounded sequence.More generally, such tools can be applied to many other problems related to partial differential equations on domains involving periodic grids, lattices, thin frames and glued fiber structures.
The paper is organized as follows.In section 2, the standard notation and tools for the classical homogenization via unfolding method in periodic domains Ω ⊂ R N are listed.In section 3, we recall the main results concerning the periodic unfolding for sequences defined as Q 1 interpolated on the vertexes of the ε cells paving Ω and bounded uniformly and anisotropically on W 1,p (Ω), whose properties will be needed in the next sections.In section 4, we give a rigorous definition of one-dimensional lattice structure S ε ⊂ Ω, build the unfolding operator for lattices and give its main properties.In section 5, we show the asymptotic behavior of sequences asymptotically and uniformly bounded in W 1,p (S ε ).We first do it for functions defined as Q 1 interpolated on lattice nodes, showing that for such sequences, the unfolding for lattices is the mere extension of the functions from S ε to R N by Q 1 interpolation, application of the known results in section 3 and then restriction of the convergences to the lattice itself.Later, we extend the results to Sobolev spaces by first decomposing them into Q 1 interpolated part and reminder term.In section 6, we show the asymptotic behavior of sequences asymptotically and uniformly bounded in W 2,p (S ε ).
The nature of a sequence bounded on a lattice leads to the lack of mixed derivatives, since the derivation only makes sense in the lattice directions.To overcome such deficiency, two approach are considered, one by a procedure analogous to section 5 but with a decomposition on Q 3 interpolation on lattice nodes and reminder term, and one by using twice (on the sequence and on the sequence gradient) the proved for functions bounded in W 1,p (S ε ).At last, in section 7 we consider the fourth order Dirichlet problem shown above.Using the results in the previous sections, existence and uniqueness of the limit problem are shown and through the homogenization via unfolding, the cell problems and the macroscopic limit problem are found.
2. Preliminaries and notation.Let R N be the euclidean space with usual basis (e 1 , . . ., e N ) and Y = (0, 1) N the open unit parallelotope associated with this basis.For a.e.z ∈ R N , we set the unique decomposition Let {ε} be a sequence of strictly positive parameters going to 0. We scale our paving by ε writing (2.1) This manuscript is for review purposes only.
Let now Ω be a bounded domain in R N with Lipschitz boundary.We consider We recall the definitions of classical unfolding operator and mean value operator.
Definition 2.1.(see [8,Definition 1.2])For every measurable function φ on Ω, the unfolding operator T ε is defined as follows: Note that such an operator acts on functions defined in Ω by operating on their restriction to Ω ε .
Definition 2.2.(see [8,Definition 1.10])For every measurable function φ on L 1 (Ω × Y ), the mean value operator M Y is defined as follows: Since we will deal with Sobolev spaces, we give hereafter some definitions: We split the space by setting This manuscript is for review purposes only. and For every x ∈ R N and y ∈ Y , we write From now on, however, we find easier to refer to such partition with the vectorial notation Similarly to (2.1), we apply the paving to a.e.x ∈ R N1 and x ∈ R N2 setting Definition 2.3.For every φ ∈ L 1 (Ω × Y ), the partial mean value operators are defined as follows: We endow these spaces with the respective norms: This manuscript is for review purposes only.
3. Periodic unfolding in R N for sequences defined as Q 1 interpolates.
The periodic unfolding for this class of functions has two main advantages.The first is that less hypothesis are required for the sequences to ensure weak convergence.The second is that the convergences can be restricted to subspaces with lower dimension and it will be fundamental in the next sections, where lattice structures are taken into account.
Define the spaces Note that the covering Ω ε is now a connected open set and from (2.2) we have Hence, we need to extend the definition of the classical unfolding operator (2.1) to functions defined in the following neighborhood of Ω: Definition 3.1.For every measurable function φ on Ω ε , the unfolding operator is defined as follows: Every measurable function defined in Ω can be extend to Ω ε by setting it to 0 in Ω ε ∩ (R N \ Ω).Now, assume {Φ ε } ε to be a sequence uniformly bounded in L p ( Ω ε ), p ∈ (1, +∞).Then, the sequence Hence, there exists a subsequence of {ε}, still denoted {ε}, and Φ ∈ L p (Ω × Y ) such that For simplicity, we will omit the restriction and always write the above convergence as In this sense, all the results obtained in [8,14] are easily transposed to this operator.
Define the space of Q 1 interpolated functions on Ω ε by This manuscript is for review purposes only.
Due to the Q 1 interpolation character, for every function Φ ∈ Q 1 ε ( Ω ε ) we remind that there exist a constant depending only on p such that We have the following.
where the constant does not depend on ε.
The same results hold for p = +∞ with weak topology replaced by weak-* topology in the corresponding spaces.
Proof.First, since the sequence The constant does not depend on ε.The statement follows by [14,Lemma 4.3] and As a direct consequence, we have the following corollary.
where the constant does not depend on ε.
The same results hold for p = +∞ with weak topology replaced by weak-* topology in the corresponding spaces.
Proof.The proof directly follow from Lemma 3.2 in the particular case N 1 = N and N 2 = 0.As an equivalent proof, the statement follows by [8, Corollary 1.37 and Theorem 1.41] and the fact that This manuscript is for review purposes only.
4. The periodic lattice structure.We start by giving a rigorous definition of 1-dimensional periodic lattice structure in R N .Let i ∈ {1, . . ., N } and let K 1 , . . ., K N ∈ N * .Set We denote K the set of points in the closure of the unit cell Y by In this sense, the whole unit cell Y has the following split where Y K is the cell defined by We denote S (i) the set of all segments whose direction is e i by Hence, the lattice structure in the unit cell Y is defined by Given Ω ⊂ R N , we cover it as in (3.1) by a union of ε cells.The periodic lattice structure is therefore defined by Denote S the running point of S and s that of S ε .That gives ( i ∈ {1, . . ., N }) This manuscript is for review purposes only.
Let C(S) and C(S ε ) be the spaces of continuous functions defined on S and S ε respectively.For p ∈ [1, +∞], we denote the spaces of functions defined on the lattice by 4.1.The unfolding operator for periodic lattices.We are now in the position to define an equivalent formulation of the unfolding operator and mean value operator (see Definition 2.1 and 2.2) for lattice structures.
Definition 4.1.For every measurable function φ on S ε , the unfolding operator T S ε is defined as follows: For every function φ on L 1 (S (i) ), i ∈ {1, . . ., N }, the mean value operator M S (i) on direction e i is defined as follows: Observe that in the above definition of T S ε , the map (x, S) into S ε is almost everywhere one to one.This is not the case if we replace S by S c .
Below, we give the main property of T S ε .
The case p ∈ (1, +∞) follows by definition of L p norm.The case p = +∞ is trivial.

Periodic unfolding for sequences defined on lattices with information
on the first order derivatives.
5.1.Asymptotic behavior of bounded sequences defined as Q 1 interpolated on lattice nodes.On S ε (resp.S) we define the space Q 1 (S ε ) (resp.Q 1 (S)) by Similarly we define the spaces A function belonging to Q 1 (S ε ) is determined only by its values on the set of nodes K ε and thus can be naturally extended to a function defined in Ω ε .
Definition 5.1.For every function Define the spaces Similarly we define the spaces By definition, the extension operator Q ε is both one to one and onto from Q 1 (S ε ) to Below, we show the main properties of this operator.
Lemma 5.2.For every where the constants do not depend on ε.
Proof.We will only consider the case p ∈ [1, +∞), since the case p = +∞ is trivial.First, remind that for every function φ defined as Q 1 interpolate of its values on the vertexes of the nodes in K, we have (i ∈ {1, . . ., N }) where the constants do not depend on p.
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We prove now (5.1) 2 .Let i be in {1, . . ., N }.From (5.2) 2 and an affine change of variables, we have And thus (5.1) Note now that for every ψ ∈ Q 1 (S ε ), the unfolding on the lattice is equivalent to first extending ψ to Ψ = Q ε (ψ) (see Definition 5.1), then applying the unfolding results in R N and lastly restricting the convergences to the lattice again, as the following commutative diagrams show (i ∈ {1, . . ., N }): (5.3) We can finally show the asymptotic behavior of sequences which belong to Q 1 (S ε ) and we start with the following.
where the constant does not depend on ε.
Then, there exist a subsequence of {ε}, denoted {ε}, and φ ∈ L p (Ω; W 1,p per (S)) such that The same results hold for p = +∞ with weak topology replaced by weak-* topology in the corresponding spaces.
Proof.The sequence {T S ε (φ ε )} ε satisfies 1 As for Tε ext , this convergence must be understood It will be the same for all convergences involving the unfolding operator T S ε .
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The constant does not depend on ε.
We consider now sequences whose gradient is anisotropically bounded on the lattice.
Accordingly to Section 2, we apply the decomposition of the space and define (5.5) We have the following.
where the constant does not depend on ε.
Then, there exist a subsequence of {ε}, denoted {ε}, and φ ∈ L p (Ω, weakly in L p (Ω × S (i) ), The same results hold for p = +∞ with weak topology replaced by weak-* topology in the corresponding spaces.
Proof.We extend the sequence {φ ε } ε to the sequence . By Lemma 5.2 and the Q 1 property (3.2), we get where the constant does not depend on ε.
where the constant does not depend on ε.
Then, there exist a subsequence of {ε}, still denoted {ε}, and functions φ ∈ W 1,p (Ω) The same results hold for p = +∞ with weak topology replaced by weak-* topology in the corresponding spaces.
Proof.The proof directly follows from Lemma 5.4 in the particular case S = S and S = ∅.

Asymptotic behavior of sequences bounded anisotropically and uniformly in
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Every function φ in W 1,p (S) (resp.ψ ∈ W 1,p (S ε )) is defined on the set of nodes K (resp.K ε ) and therefore can be decomposed as where φ a , ψ a are the affine function defined as Q 1 interpolation on the nodes, and φ 0 , ψ 0 the reminder term which is zero on every node.
Step 2. We prove the statements of the Lemma.
We start with (5.8) 1 .By construction, S (i) is the union of a finite number of segments whose extremities belong to K. Hence, inequality (5.9) 1 and an affine change of variables leads to (i ∈ {1, . . ., N }) and thus (5.8) 1 is proved.Estimate (5.8) 2 follows by (5.8) 1 and an affine change of variables, while (5.8) 3 follows by (5.8) 2 and again a change of variables.The constant does not depend on ε since S (i) has a finite number of segments.
We show now the asymptotic behavior of sequences that are anisotropically bounded.
where S c .= S − M S (i) (S) • e i .
The same results hold for p = +∞ with weak topology replaced by weak-* topology in the corresponding spaces.
As a direct consequence, it follows the asymptotic behavior of the uniformly bounded sequences.
where the constant does not depend on ε.
The same results hold for p = +∞ with weak topology replaced by weak-* topology in the corresponding spaces.
Proof.The proof directly follows from Lemma 5.7 in the particular case S = S and S = ∅.
6. Periodic unfolding for sequences defined on lattices with information until the second order derivatives.The main problem that arises for functions in W 2,p (S ε ) is the lack of mixed derivatives.This comes from the fact that a function defined on the lattice segments can be derived twice, only in the segment directions.
We overcome the problem in two different ways.
6.1.Unfolding via special Q 3 interpolation.Analogously to the previous section, we decompose a function into a reminder term and a cubic polynomial, this latter is extended to a special Q 3 interpolation to the whole space.Then, we use the periodic unfolding results for open subset in R N and finally restrict these results to the lattice.However, to bound the extension, further assumptions on the original function must be applied.
First, we recall a basic result concerning the functions in W 2,p (0, 1).
Moreover, there exists a constant C > 0, such that Proof.Given φ be in W 2,p (0, 1), it is clear that the decomposition is unique.Indeed, condition (6.1) implies that the function φ p must satisfy and therefore the 4 coefficients of the cubic polynomial are uniquely determined.Now, we observe that Then, we easily obtain the estimates (6.2) 1,2,3 .Estimate (6.2) 4 follows by assumption (6.1), the Poincaré inequality applied twice and estimate (6.2) 1 .
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Lemma 6.2.For every φ ∈ W 2,p (S), there exist two functions Φ p ∈ W 2,p (Y ) and where Φ p|S is a cubic polynomial on every small segment of S.
Moreover, there exists a constant C > 0 such that (6.4) and that Proof.We will only prove the case N = 2, since the extension to higher dimension is done by an analogous argumentation.
In every small rectangle build on the nodes of S we extend φ as described in Step 1.That gives a function Φ p ∈ W 2,p (Y ) satisfying (6.4) for N = 2. Estimate (6.5) follows by applying the Poincaré inequality twice and the fact that (see (6.6)) The proof is complete.
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We can finally show the asymptotic behavior of sequences bounded in W 2,p (S ε ), whose derivatives of the gradient extension from the lattice to the whole space are also bounded.
6.2.Unfolding via known results for sequences of functions uniformly bounded in W 1,p .We consider the sequences in W 2,p (S ε ) as sequences in W 1,p (S ε ) with partial derivatives belonging to W 1,p (S (i) ε ), for each i ∈ {1, . . ., N }.In this sense, we can apply the results obtained in section 5.Even though no gradient extension is needed, the additional work must be done to show that the N different limit functions, one for each partial derivative, are in fact a unique function restricted to each line.
Since ψ (i) does not depend on S in S (i) and φ is periodic with respect to S in S (i) we have ∂ i φ = ψ (i) and ∂ S φ = 0 a.e.Ω × S (i) for every i ∈ {1, . . ., N }.
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A necessary and sufficient condition to get existence of the function φ is (remind that A(k) a.e. in Ω.
Convergence (6.12) gives (remind that ∂ 2 ii φ does not depends on S) This manuscript is for review purposes only.
Similarly, one has (j = i) A(k+ej +ei) and same kind of results for the other two quantities.
Hence, to get (6.13),we have to prove that both quantities (6.15) admit the same limit or equivalently that the limit of their difference is 0. First we note that Hence, and is constant on every line of S (i) .One has a.e. in Ω ε This manuscript is for review purposes only.
where k ∈ K i is such that k = k + k i e i .Hence, we get where k ∈ K i is such that k = k + k i e i .Now, we can apply Lemma 8.1 and claim that the limit of the difference of the quantities in (6.15) and (6.16) is equal to 0. This proves (6.14) for every k ∈ K.
7. Application: homogenization of a fourth 4th order homogeneous Dirichlet problem on a periodic lattice structure.We can now give a direct application of the periodic unfolding for sequences in H 2 (S ε ).From now on, let Ω be a bounded domain in R N with a C 1,1 boundary.Let {A ε } ε be the sequence of functions belonging to L ∞ (S ε ) defined by By the Poincaré and Poincaré−Wirtinger inequalities, we have , where the constants do not depend on the parameter ε (note that M S (i) (∂ s φ) = 0 for every i ∈ {1, . . ., N }).
Consider the 4th order homogeneous Dirichlet problem in variational formulation: The Lax−Milgram theorem implies that the problem (7.2) has a unique solution.
Moreover, one has The constant does not depend on ε.
Below, we give the periodic homogenization via unfolding.
The solution u ε of (7.2) satisfies (7.3).Due to the convergences (7.4) we have that The constant does not depend on ε.