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Homogenization of Stiff Plates and Two-Dimensional High-Viscosity Stokes Equations
Marc Briane, Juan Casado-Diaz
To cite this version:
Marc Briane, Juan Casado-Diaz. Homogenization of Stiff Plates and Two-Dimensional High-Viscosity Stokes Equations. Archive for Rational Mechanics and Analysis, Springer Verlag, 2012, 205 (3), pp.753-794. �10.1007/s00205-012-0520-9�. �hal-00736565�
Homogenization of stiff plates
and two-dimensional high-viscosity Stokes equations
Marc BRIANE Juan CASADO-D´IAZ
Institut de Recherche Math´ematique de Rennes Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Universit´e Europ´eenne de Bretagne Universidad de Sevilla
[email protected] [email protected]
April 4, 2012
Abstract
The paper deals with the homogenization of rigid heterogeneous plates. Assuming that the coefficients are equi-bounded inL1, we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients. This compactness result is based on a div-curl lemma for fourth- order equations. On the other hand, using an intermediate stream function we deduce from the plates case a similar result for high-viscosity Stokes equations in dimension two, so that the nature of the Stokes equation is preserved in the homogenization process.
Finally, we show that the L1-boundedness assumption cannot be relaxed. Indeed, in the case of the Stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem, which violates the compactness result obtained under the L1-boundedness condition.
Keywords : Homogenization, plate, Stokes equation, div-curl lemma Mathematics Subject Classification : 35B27 – 74Q15 – 76M50
1 Introduction
Since the seminal work of Khruslov [17] extended in [1, 11, 13], it is known that the homogeniza- tion of sequences of scalar conductivity equations with high-contrast coefficients may induce nonlocal effects in dimension three. In fact, Mosco [21] showed that the limits – in the sense of Γ-convergence endowed with the L2 strong topology – of the associated diffusion energies are Dirichlet forms which can be represented thanks to the Beurling-Deny [2] formula involving nat- urally a nonlocal term. On the other hand, Camar-Eddine and Seppecher [13] obtained that the closure of the set of diffusion energies agrees with the set of regular Dirichlet forms. However, dimension two prevents from the appearance of nonlocal effects, since limits of equi-coercive diffusion energies are shown to be purely diffusive [5, 7].
The case of vector-valued problems is much more intricate. As a matter of fact, the lack of truncation principle implies that the Beurling-Deny representation formula does not apply.
Such important is the absence of constraints that Camar-Eddine and Seppecher [14] proved that in dimension three any objective (vanishing on rigid displacements) lower semi-continuous
case dimension two seems at first sight to be more favorable. Indeed, a sequence of two- dimensional equi-coercive elastic quadratic functionals is shown [10] to converge to a similar elastic functional provided that the sequence of rigidity tensors is bounded in L1. However, contrary to the scalar case this compactness result cannot be improved significantly, since the counter-example of [10] shows that very rigid strips may induce second gradient terms in the limit energy as in dimension three.
It is then natural to study other systems in dimension two. From the Mechanics point of view the asymptotic analysis of plates is relevant. The homogenization of heterogeneous plates with a periodic structure was performed rigorously in [16]. This work was extended in [15] to the non-periodic case including a 2D-3D dimension reduction approach, and using a variant of theH-convergence method [22]. In these papers the elastic coefficients are assumed to be equi- bounded from below and above. Up to our knowledge the homogenization of plates with non uniformly bounded coefficients has not be studied. It seems to be also interesting to compare the plate case to the above mentioned two-dimensional elasticity case under the high-contrast assumption. This is the aim of the present work.
Let Ω be a bounded open set of R2, and let An be a sequence of equi-coercive (in the weak sense of (2.3), see also Remark 2.2) symmetric fourth-order tensors in L∞ Ω;Ls(R2×2s )
. In section 2 we study the asymptotic behavior of the sequence of the plate equations
div2(AnD2un) =fn in Ω, (1.1)
where fn strongly converges to f in H−2(Ω), simultaneously with the asymptotic behavior of the associated energies
FnΩ(u) = Z
Ω
AnD2u:D2u dx, foru∈H2(Ω). (1.2) Assuming that the sequence |An| is bounded in L1(Ω) and converges weakly-∗ to a Radon measureµ, we prove (see Theorems 2.5, 2.6, and Remark 2.7) that the limit equation of (1.3) is still a plate equation of the type
div2 AΩD2u µ
=f in Ω, (1.3)
whereAΩ is a strongly local linear operator mapping the set of the admissible second gradients intoL1µ Ω;Ls(R2×2s )
. This limit behavior is actually independent of the boundary conditions imposed to (1.1). Moreover, whenuis regular, we get that AΩD2u=AD2u µ-a.e. in Ω, where the symmetric fourth-order tensor A satisfies: |A| ≤ 1 µ-a.e. in Ω, and A(nn) : (nn) = 0 µ-a.e. along any smooth curve with outer normal n (see Proposition 2.8). The tensor-valued measure Aµ is also proved to vanish on sets of zero capacity. These properties imply that not any equi-coercive symmetric fourth-order tensor can be attained. In terms of energy the sequenceFnΩ, with domain H02(Ω), is shown to Γ-converge for the strong topology ofL2(Ω) to a quadratic functionalFΩ which satisfies (see also Remark 2.9)
FΩ(u) = Z
Ω
AD2u:D2u dµ, for any u∈Cc1(Ω), piecewise C2 in Ω. (1.4) The proof of this compactness result uses two ingredients: a suitable div-curl lemma and the Γ-convergence approach. On the one hand, the div-curl lemma (see Theorem 2.11) allows us to pass to the limit in the product AnD2un :D2vn, when un solves the plate equation (1.1) and FnΩ(vn) is bounded. This result is an extension to fourth-order equations of the div-curl
lemmas for second-order equations established in [6] (scalar case) and in [10] (elasticity case).
It is based on the embedding of the set of matrix-valued measures which are div2 free, into H−1(Ω;R2×2), involving the Strauss inequality [23] (see Lemma 2.13), and the concentration compactness lemma of P.-L. Lions [20]. On the other hand, similarly to [6] the Γ-convergence approach provides a functional framework given by the domain of the Γ-limit of the energies (1.2). However, two difficulties arise in this approach. Firstly, contrary to the diffusion case studied in [7] there is no truncation principle. Secondly, the fourth-order character of the equations makes delicate the localization with a regular cut-off function ϕ. Indeed, to this end we should consider sequences of the typeAnD2(vnϕ) :D2(vnϕ), with FnΩ(vn) bounded, for which we cannot estimate the termAn(∇un ∇ϕ) : (∇un ∇ϕ), since|An|is only bounded in L1(Ω). So, the proof of Theorems 2.5 is split in nine steps using several fourth-order equations with Dirichlet or Neumann boundary conditions and various quadratic functionals, and needs several back and forth in order to overcome these difficulties.
Taking into account the compactness result for the plates case, in Section 3 we study the homogenization of the Stokes equations
( −Div (BnDun) +∇pn =gn in Ω
divun = 0 in Ω, (1.5)
where the sequencegn strongly converges tog inH−1(Ω), andBn is a sequence of equi-coercive (in the sense of (3.3)) matrix-valued viscosities in L∞ Ω;Ls(R2×20 )
. Assuming that |Bn| is bounded inL1(Ω) and converges weakly-∗ to a Radon measureν, we prove (see Theorems 3.5, 3.6, and Remark 3.7) that the homogenized equation of (1.5) is given by
( −Div (BΩDu ν) +∇p=g in Ω
divu= 0 in Ω, (1.6)
whereBΩis a local linear operator mapping the set of admissible gradients intoL1µ Ω;Ls(R2×20 ) . Moreover, when u is regular, we obtain that BΩDu =BDu ν-a.e. in Ω, where the symmetric fourth-order tensor B satisfies: |B| ≤ 1 ν-a.e. in Ω, and B(tn) : (tn) = 0 ν-a.e. along any smooth curve with tangentt and outer normaln.
DefiningAnbyAnξ =−J Bn(J ξ) for ξ∈R2×2s , whereJ is the rotation matrix of 90◦ angle, the previous compactness result is easily deduced from the plates case, while the convergence of the pressurepn in Hloc−1(Ω)/R is a consequence of [8].
Finally in Section 4, we show in the case of the Stokes equation that the L1-boundedness of the coefficients cannot be relaxed. Following [10] we build a counter-example with rigid strips, but here we consider one strip concentrated on a line. We obtain (see Theorem 4.1) a homogenized problem with second gradient terms, which thus does not belong to the class of equations (1.6). Due to the incompressibility condition, our approach differs radically from the one used in the elasticity case of [10]. In particular, the proof involves surprisingly a wave equation (see Lemma 4.2).
1.1 Notations
• I denotes the unit matrix of R2×2, andJ =
0 −1
1 0
.
• R2×2s denotes the set of the (2×2) symmetric real matrices, andLs(R2×2s ) the set of the fourth-order tensorsAmappingR2×2s into itself, which are symmetric, i.e. Aξ :η=Aη :ξ for any ξ, η ∈R2×2s .
• R2×20 denotes the set of (2×2) real matrices with zero trace, andLs(R2×20 ) the set of the symmetric fourth-order tensors mapping R2×20 into itself.
• denotes the symmetric tensorial product, i.e., for any x, y ∈R2, xy is the matrix of R2×2s with entries (xy)ij = 12(xiyj +xjyi), 1≤i, j ≤2.
• ∇u denotes the gradient of the scalar distribution u, and D2u is the symmetric Hessian matrix of the second-order derivatives of u.
• DU denotes the Jacobian matrix of the vector-valued distributionU = (U1, U2), the rows of which are ∇U1, ∇U2, ande(U) = 12 DU +DUT
is the symmetrized gradient of U.
• div denotes the classical divergence operator acting on the vector-valued distributions, Div denotes the vector-valued differential operator taking the divergence of each row of matrix-valued distributions, and div2 denotes the operator div (Div).
• For a bounded open setω ofR2,Hdiv1 (ω) andH0,div1 (ω) denote the space of the divergence free vector-valued functions in H1(ω;R2) and H01(ω;R2) respectively.
• M(Ω) denotes the space of bounded Radon measures on Ω, and −∗*denotes the weak-∗
convergence in M(Ω), considered as the dual of the space C00(Ω) of continuous functions vanishing on ∂Ω.
• A curve of class C2 in R2 is the range of a mapping η ∈C2 0,1;R2
such that η0(t) 6= 0 for anyt∈(0,1). Sometimes, we will also considerC2curves as mapping inC2([0,1];R2).
• For a bounded set O of R2 and a bounded open set ˜Ω such that O ⊂Ω, the capacity of˜ O relating to ˜Ω is defined by
Cap(O,Ω) = inf˜ Z
Ω˜
|∇ϕ|2dx: ϕ∈H01( ˜Ω), ϕ≥1 a.e. in a neighborhood of O
. Clearly, with this definition, the capacity ofOdepends on the set Ω, but the zero capacity does not depend on ˜Ω. Recall that the elements of the Sobolev space H1(Ω) have a continuous representative defined up to a set of zero capacity. Moreover, any Radon measure in H−1(Ω) vanishes on sets of zero capacity.
1.2 Recalls of Γ-convergence
In this section we recall the definition of the De Giorgi Γ-convergence and some of its properties which will be used in the sequel. We refer to [12] for an exhaustive presentation of Γ-convergence (see also [4] for an elementary approach).
Definition 1.1. Letd, N be two positive integers, and let ω be a bounded open set of Rd. A sequence of functionals Fn :L2(ω)N →[0,∞] is said to Γ-converge to F :L2(ω)N →[0,∞] for the strong topology of L2(ω)N, if for any u inL2(ω)N,
i) the Γ-liminf inequality holds
∀un →u in L2(ω)N, F(u)≤lim inf
n→∞ Fn(un), (1.7)
ii) the Γ-limsup inequality holds
∃un →u in L2(ω)N, F(u) = lim
n→∞Fn(un). (1.8)
Any sequence satisfying (1.8) will be called a recovery sequence (forFn) of limit u.
Properties 1.2. The following properties hold:
a) Since L2(ω)N is separable, any sequence of functionals Fn : L2(ω)N → [0,∞] has a subsequence which Γ-converges with respect to the strong topology of L2(ω)N.
b) Let Fn : L2(ω)N → [0,∞] be a sequence of quadratic forms which Γ-converges to F. Then, F is a quadratic form on L2(ω)N which is lower semi-continuous with respect to the topology of L2(ω)N. Moreover, the domain D(F)of F is a Hilbert space endowed with the norm
kukD(F)=
kuk2L2(ω)N +F(u)12
. (1.9)
c) Let Fn : L2(ω)N → [0,∞] be a sequence of quadratic forms which Γ-converges to F. Let Φn be the bilinear form associated with Fn on its domain. Then, for any u∈L2(ω)N, with F(u) < ∞, a sequence un converging to u in L2(ω)N, with Fn(un) bounded, is a recovery sequence (1.8) (for Fn), if and only if
∀vn→0 in L2(ω)N, with Fn(vn)≤c, lim
n→∞ Φn(un, vn) = 0. (1.10) In the sequel, we will always consider the Γ-convergence for the strong topology ofL2(ω)N. This topology will be not necessarily mentioned. Indeed, all the functionals Fn will have the property that any sequence un bounded in L2(ω)N, with Fn(un) bounded, is compact in L2loc(ω)N. In this framework the weak convergence inL2(ω)N agrees with the weak convergence inL2(ω)N combined with the strong convergence in L2loc(ω)N.
2 Homogenization of plates
2.1 Statement of the results
Consider a bounded connected open set Ω of R2 and a sequence An ∈ L∞ Ω;Ls(R2×2s ) such that
|An| −∗* µ inM(Ω), (2.1)
An(x)ξ :ξ≥0, ∀ξ ∈R2×2s , a.e. x∈Ω (2.2) and such that for any open setsO,ωwithO ⊂ω ⊂Ω, there exists a constantα(O, ω) satisfying
α(O, ω) Z
O
|D2u|2dx ≤ Z
ω
|u|2dx+ Z
ω
AnD2u:D2u dx, ∀u∈H2(ω). (2.3) The purpose of the present section is to study the asymptotic behavior of a sequence of functions un ∈H2(ω), for an open set ω⊂Ω, which solve the plate equation
div2(AnD2un) =fn in ω, (2.4)
for some compact sequencefn∈H−2(ω), and which satisfy the limits un →u in L2(ω), lim sup
n→∞
Z
ω
AnD2un :D2undx <∞, (2.5)
Remark 2.1. Observe that no boundary condition is imposed in (2.4). We will show actually that the limit equation for (2.4) does not depend on the boundary conditions.
Remark 2.2. We have preferred assuming that An satisfies the local integral ellipticity con- dition (2.3) since several classical elliptic operators for the plate equations (as for example the heterogeneous bi-Laplacian operator ∆ (an∆)) do not usually satisfy a strong pointwise ellipticity condition of the type
An(x)ξ:ξ ≥α|ξ|2, ∀ξ ∈R2×2s ,a.e. x∈ω, ∀n∈N, (2.6) for some constant α >0. Thanks to the local H2-estimates for the second-order elliptic equa- tions (see, e.g., [18] Theorem 9.11), in order to have (2.3) it is enough to assume that there exists a relatively compact sequence of matrix functionsMn inC0(Ω;R2×2s ), with Mn≥αI for some positive constantα, such thatAn satisfies
An(x)ξ :ξ≥ |Mn(x) :ξ|2, ∀ξ ∈R2×2s , a.e. x∈ω, ∀n∈N. (2.7) So, the bi-Laplacian ∆ (an∆) satisfies (2.7) with Mn=anI, whenan ≥α a.e. in ω.
Remark 2.3. Thanks to estimate (2.3) it is possible to solve some partial differential problems associated with the diffusion operators div2(AnD2u). In this sense observe that for any open setω⊂Ω and any h∈L2(ω), there exists a unique solution u of the Neumann problem
( div2(AnD2u) +u=h inω
(AnD2u)n = 0, Div(AnD2u)·n= 0 on ∂ω, or equivalently, of the minimization problem
min 1
2 Z
ω
AnD2v :D2v+|v|2 dx−
Z
ω
hv: v ∈Hloc2 (ω)∩L2(ω)
.
On the other hand, for an open setω strictly contained in Ω, extending by zero outside Ω the functions of H02(ω) and applying inequality (2.3), with ω replaced by Ω and O replaced by ω, we deduce thatAn satisfies
α(ω,Ω) Z
ω
|D2u|2dx≤ Z
ω
AnD2u:D2u dx+ Z
ω
|u|2, ∀u∈H02(ω), (2.8) Therefore, for anyf ∈H−2(ω), we can solve the Dirichlet problem
( div2(AnD2u) +u=f in ω u∈H02(ω).
We will use the following:
Definition 2.4. For any open setω ⊂Ω, denote byH(ω) the space of the functionsu∈H2(ω) such that there exists a sequence un ∈ H2(ω) satisfying (2.5), and by Hc(ω) the space of the functionsu∈H(ω) such that there exists a sequenceun ∈H2(ω) satisfying (2.5), with supp (un) contained in a compact subset of ω independent of n. Also denote by D2H(ω) and D2Hc(ω) the spaces defined by
D2H(ω) =
D2u: u∈H(ω) and D2Hc(ω) =
D2u: u∈Hc(ω) .
Our first result on the asymptotic behavior of the solutions of (2.4) is the following:
Theorem 2.5. There exists a subsequence of n, still denoted by n, and a family of symmetric bilinear mappingsaω :D2H(ω)×D2H(ω)→M(ω),for any open set ω⊂Ω, with the following properties:
• The operator aω is strongly local with respect to ω in the following sense: If ω1, ω2 are two open sets of ω and u1, v1 ∈ H(ω1), u2, v2 ∈ H(ω2) are such that D2u1 = D2u2, D2v1 =D2v2 a.e. in ω1∩ω2, then we have
aω1(D2u1, D2v1) = aω2(D2u2, D2v2) in ω1∩ω2. (2.9)
• For any open set ω ⊂Ω, and un, u∈H2(ω) satisfying (2.4) and (2.5) for some sequence fn converging strongly to f in H−2(ω), the function u is solution of
Z
ω
daω(D2u, D2v) =hf, ui, ∀v ∈Hc(ω). (2.10)
• Consider an open set ω⊂Ω, and un, u∈H2(ω) satisfying (2.5) and the convergences Z
ω
AnD2un :D2vndx−→0, (2.11) for any vn∈H2(ω), with support in a fixed compact of ω, such that
vn→0 in L2(ω), lim sup
n→∞
Z
ω
AnD2vn :D2vndx <∞. (2.12) Then, we have
AnD2un :D2vn
−∗* aω(D2u, D2v) in M(ω), (2.13) for any vn, v in H2(ω) such that
vn→v in L2(ω), lim sup
n→∞
Z
ω
AnD2vn:D2vndx <∞. (2.14)
• For any open set ω ⊂Ω, the sequence of functionals Gωn defined by
Gωn(u) =
Z
ω
AnD2u:D2u dx if u∈H02(ω)
∞ if u∈L2(ω)\H02(ω).
(2.15)
Γ-converges (up to a subsequence) to a functional Gω which satisfies Gω(u) =
Z
ω
daω(D2u, D2u), ∀u∈Hc(ω). (2.16)
• The family aω satisfies the following ellipticity condition: For any open sets O, U, ω with O ⊂U ⊂U ⊂ω⊂Ω, we have
α(O, U) Z
|D2u|2dx≤ Z
|u|2dx+ Z
daω(D2u:D2u), ∀u∈H2(ω). (2.17)
• For any open set ω ⊂Ω and any u∈W2,∞(ω), we have Z
ω
daω(D2u, D2u)≤µ(ω)kD2uk2L∞(ω;R2×2s ). (2.18)
• The family aω satisfies the following continuity property: For any open setω ⊂Ωand any sequence uk ∈ H(ω) which converges to some u ∈ H(ω) in L2(ω), with aω(D2uk, D2uk) bounded in M(ω), we have
aω(D2uk, D2v) −*∗ aω(D2u, D2v) in M(ω), ∀v ∈H(ω). (2.19) For the case of smooth functions, it is possible to obtain an integral representation of aω. This is given by the following result:
Theorem 2.6. For any open set ω ⊂ Ω, there exists a linear operator Aω : D2H(ω) → L1µ(ω;R2×2s ), with the following properties:
• The operator Aω is strongly local with respect to ω in the following sense: If ω1, ω2 are two open subsets of Ω and u1 ∈H(ω1), u2 ∈ H(ω2) are such that D2u1 = D2u2 a.e. in ω1∩ω2, then
Aω1D2u1 =Aω2D2u2 µ-a.e. in ω1∩ω2. (2.20)
• The operators aω and Aω are related by
aω(D2u, D2v) =AωD2u:D2v µ, ∀u, v ∈H(ω), v∈C2(ω). (2.21)
• AωD2u µ vanishes on sets of zero capacity and satisfies Z
ω
|AωD2u|ϕ dµ ≤ Z
ω
ϕ daω(D2u, D2u) 12 Z
ω
ϕ dµ 12
, ∀ϕ∈C00(ω), ϕ≥0. (2.22) Moreover, there exists a non-negative tensor A∈L∞µ Ω;Ls(R2×2s )
, with kAkL∞
µ(Ω;Ls(R2×2s ))≤1. (2.23) such that for any open set ω⊂Ω and any u∈C2(ω)∩H2(ω), we have
AωD2u=AD2u µ-a.e. in ω. (2.24) Remark 2.7. A straightforward consequence of Theorems 2.5 and 2.6 is that for any open set ω ⊂ Ω, and for any un, u ∈ H2(ω) satisfying (2.4), (2.5) for some sequence fn converging strongly to f in H−2(ω), the limit u solves the equation
div2(AωD2u µ) = f in ω.
In the case where ubelongs to C2(ω), this equation can be written as div2(AD2u µ) = f in ω.
Equations (2.24) and (2.21) give a representation of the operatorsaωfor functions of classC2. This representation formula holds true actually for functions which are less smooth. A result in this sense is the following one:
Proposition 2.8. For any open curve γ ⊂ Ω of class C2, the tensor A ∈ L∞µ (ω;Ls(R2×2s )) given by Theorem 2.6 satisfies
A(nn) : (nn) = 0 µ-a.e. in γ, (2.25) where n is the unitary normal to γ.
Let ω be an open set of Ω, and let O1, O2 be two disjoint open sets of class C2 such that
ω=ω∩O1∪ω∩O2. (2.26)
Then, for any u∈C1(ω), with u∈C2(ω∩O1)∩C2(ω∩O2), equality (2.24) holds true.
Remark 2.9. For any function u∈C1(ω), withu∈C2(ω∩O1)∩C2(ω∩O2), the derivatives
∂2u
∂t2 and ∂2u
∂t∂n,
where t is a tangent vector to ∂O1∩∂O2, are uniquely defined on ∂O1 ∩∂O2. Thus, thanks to (2.25)AD2uis well defined for the measure µ. Moreover, such a function u with a compact support inω belongs clearly to Hc(ω) due to the boundedness of |An| inL1(ω). Therefore, the representation formula of the Γ-limit (2.16) holds true for anyu∈Cc1(ω), which is piecewiseC2 inω.
Remark 2.10. Since for any u ∈ H2(ω), AωD2u µ vanishes on sets of zero capacity, the measureAωD2u µdoes not charge single points.
The proof of our results will be based on the following div-curl type result:
Theorem 2.11. Consider an open set ω ⊂R2, and sequences Λn ∈ L2(ω;R2×2s ), vn ∈ H2(ω).
Assume the existence of measures λ, ν ∈M(ω), Λ∈M(ω;R2×2s ), and of a function v ∈H2(ω), satisfying the convergences
Λn
−∗* Λ in M(ω;R2×2s ), div2Λn →div2Λ in H−2(ω), |Λn−Λ| −∗* λ in M(ω), (2.27) vn * v in H2(ω), |D2(vn−v)| −∗* ν in M(ω). (2.28) Then, there exist a constant C >0 independent of n, and sequences (xi)i∈N in ω, (ai)i∈N in R2, satisfying
|ai| ≤C λ({xi})ν({xi})12, ∀i∈N, (2.29) such that for any ϕ∈Cc∞(ω), the following limit holds
n→∞lim Z
ω
Λn :D2vnϕ dx=
div2Λ, v ϕ
H−2(ω),H02(ω)
−2
Λ,∇v ∇ϕ
H−1(ω;R2×2s ),H01(ω;R2×2s )− Z
ω
v D2ϕ:dΛ +
∞
X
i=0
ai· ∇ϕ(xi).
(2.30)
When v belongs to C2(ω), (2.30) can be written as Λn:D2vn −∗* Λ :D2v− div
∞
X
i=0
aiδxi
!
in D0(ω). (2.31)
Corollary 2.12. If in lemma 2.11 the sequenceΛn :D2vn is bounded inL1(ω)and the function v belongs to C2(ω), then we have
Λn :D2vn −∗* Λ :D2v in M(ω). (2.32) Theorem 2.11 is a consequence of the following two lemmas. The first one is based on results of [8] (which are themselves based on [3] and [23]). In particular it gives a sense to the second term of the right-hand side of (2.30). The second one is based on the previous one and the P.L.
Lions concentration compactness [20].
Lemma 2.13. For any smooth bounded open set ω ⊂ R2, the space Mdiv2(ω;R2×2s ) of matrix- valued functions Λ∈M(ω;R2×2s ) satisfying
div2(Λ) = 0 in ω, (2.33)
is continuously embedded inH−1(ω;R2×2s ).
Lemma 2.14.Consider a smooth bounded open setω ⊂R2, and measuresΛn ∈Mdiv2(ω;R2×2s ), λ∈M(ω) satisfying the convergences
Λn −∗* 0 in M(ω;R2×2s ), |Λn| −* λ∗ in M(ω). (2.34) Define Zn as the solution of
( −∆Zn= Λn in ω
Zn ∈H01(ω;R2×2s ). (2.35)
Then, there exist a constantC > 0, and sequences (xi)i∈N in ω, (ai)i∈N in R2, satisfying
|ai| ≤C λ({xi})2, ∀i∈N, (2.36) such that, up to extract a subsequence, the following limit holds
|∇Zn|2 −*∗
∞
X
i=0
aiδxi in M(ω). (2.37)
2.2 Proof of the results
Proof of Lemma 2.13. It is easy to check that for any Λ∈Mdiv2(ω;R2×2s ), with entries Λij, i, j ∈ {1,2}, the matrix-valued measureQΛ defined as
QΛ =
−Λ12 −Λ22 Λ11 Λ12
has zero trace and satisfies curl Div(QΛ)
= 0 in ω. The result then follows from [8] in which it is proved that the space of matrix-valued measures Υ with zero trace and satisfy- ing curl Div(Υ)
= 0 in ω, is continuously imbedded in H−1(ω;R2×2s ). In the present two- dimensional case this can be deduced from the Strauss theorem [23]. In higher dimension it is
based on the work of Bourgain, Brezis [3].
Proof of Lemma 2.14. First we remark that since Λntends to zero in Mdiv2(ω), Lemma 2.13 implies thatZn converges weakly to zero in H01(ω;R2×2s ).
Fixϕ∈Cc1(ω), putting Znϕ2 as test function in (2.35), we have
2
X
i,j=1
Z
ω
|∇(Zn)ij|2ϕ2dx+ 2 Z
ω
(Zn)ijϕ∇(Zn)ij · ∇ϕ dx
=
Λnϕ, Znϕ
H−1(ω;R2×2s ),H01(ω;R2×2s )≤ kΛnϕkH−1(ω;R2×2s )kZnϕkH1
0(ω;R2×2s ).
(2.38)
In this inequality the weak convergence to zero of ∇(Zn)ij in L2(ω;RN) and the Rellich- Kondrachov compactness theorem imply that
2
X
i,j=1
Z
ω
(Zn)ijϕ∇(Zn)ij · ∇ϕ dx −→ 0, (2.39)
kZnϕkH1
0(ω;R2×2s )−
2
X
i,j=1
Z
ω
|∇(Zn)ij|2ϕ2dx
!12
−→ 0. (2.40)
Now, fixq ∈(1,2). Since Λnconverges weakly-∗to zero in the measures sense, Λnalso converges strongly to zero inW−1,q(ω). Hence, the sequence
ζn= div2(Λnϕ) =
2
X
i,j=1
2∂i(Λn)ij∂jϕ+ (Λn)ij∂ij2 ϕ
converges strongly to zero in W−2,q(ω). Therefore, there exists Hn ∈ Lq(ω;R2×2s ) converging strongly to 0 inLq(ω;R2×2s ) such that div2(Hn) =ζn. Using that
div2(Λnϕ−Hn) = 0 inω, we can apply Lemma 2.13 to deduce
kΛnϕ−HnkH−1(ω;R2×2s )≤CkΛnϕ−HnkM(ω;
R2×2s ).
Taking into account that the strong convergence of Hn to 0 in Lq(ω;R2×2s ) implies strong convergence inM(ω;R2×2s ) and in H−1(ω;R2×2s ), the previous estimate yields
lim sup
n→∞
kΛnϕkH−1(ω;R2×2s ) ≤C lim
n→∞kΛnϕkM(ω;
R2×2s ) =C Z
ω
|ϕ|dλ. (2.41) Using (2.39), (2.40) and (2.41) in (2.38), and denoting by z the limit of |∇Zn|2 in the weak-∗
sense of the measures (which exists at least for a subsequence), we have just proved that Z
ω
|ϕ|2dz ≤C Z
ω
|ϕ|dλ 2
, ∀ϕ∈Cc1(ω).
Finally, thanks to Lemma 1.3 in [20], there exist two sequences (xi)i∈N in ω and (ai)i∈N inR2, satisfying (2.36), such that
z =
∞
X
i=0
aiδxi,
which implies the desired limit (2.37).
Proof of Theorem 2.11. For any ϕ∈Cc∞(ω), we have div2Λn, vnϕ
H−2(ω),H02(Ω)
= Z
ω
ϕΛn:D2vndx+ 2 Z
ω
Λn: (∇vn ∇ϕ) dx+ Z
ω
vnΛn:D2ϕ dx.
(2.42)
In this equality the strong convergence of div2Λn to div2Λ in H−2(ω), the weak-∗ convergence of Λn to Λ in M(ω;R2×2s ), and the weak convergence of vn to v in H2(ω), together with the compact embedding of H2(ω) into C00(ω) imply that
Z
ω
vnΛn :D2ϕ dx −→
Z
ω
vD2ϕ:dΛ (2.43)
div2Λn, vnϕ
H−2(ω),H02(ω) −→
div2Λ, v ϕ
H−2(ω),H02(ω). (2.44) In order to pass to the limit in the second term of the right-hand side of (2.42), we remark that since div2(Λn−Λ) converges strongly to zero inH−2(ω), there exists a sequenceHn converging strongly to zero in L2(ω;R2×2s ) such that div2(Λn−Λ−Hn) = 0 in ω. Consider the solution Zn of
( −∆Zn= Λn−Λ−Hn in ω
Zn ∈H01(ω;R2×2s ). (2.45)
Then, we have 2
Z
ω
Λn: (∇vn ∇ϕ) dx
=
2
X
i,j=1
Z
ω
∇(Zn)ij·(∇∂ivn∂jϕ+∇∂jvn∂iϕ) +∇(Zn)ij·(∂ivn∇∂jϕ+∂jvn∇∂iϕ) dx + 2
Λ +Hn,∇vn ∇ϕ
H−1(ω;R2s),H01(R2s), .
Since Zn converges weakly to zero in H01(ω;R2×2s ) by Lemma 2.13, ∇vn converges weakly to
∇v inH1(ω;R2) and thus strongly in L2(ω;R2×2s ), and sinceHn converges strongly to zero in L2(ω;R2×2s ), we have
2
X
i,j=1
Z
ω
∇(Zn)ij ·(∂ivn∇∂jϕ+∂jvn∇∂jϕ)dx −→ 0
2
Λ +Hn,∇vn ∇ϕ
H−1(ω;R2s),H01(R2s) −→ 2
Λ,∇v ∇ϕ
H−1(ω;R2s),H01(R2s).
On the other hand, from Lemma 2.14 above combined with Lemma 2.11 of [9], we easily deduce
that 2
X
i,j=1
Z
ω
∇(Zn)ij ·(∇∂ivn∂jϕ+∇∂jvn∂jϕ)dx −→ −
∞
X
i=0
ai· ∇ϕ(xi).
for some sequences (xi)i∈Ninω and (ai)i∈N inR2, which do not depend onϕand satisfy (2.29).
Therefore, we have 2
Z
ω
Λn: (∇vn ∇ϕ) dx −→ 2
Λ,∇v ∇ϕ
H−1(ω;R2s),H01(R2s)−
∞
X
i=0
ai· ∇ϕ(xi). (2.46)
Finally, substituting (2.43), (2.44) and (2.46) in (2.42) we get (2.30).
Proof of Corollary 2.12. The boundness of Λn :D2vn inL1(ω) implies that the convergence (2.31) holds actually in the weak-∗ sense of the measures on ω. As Λ :D2v is a measure on ω, so is the distribution
div
∞
X
i=0
aiδxi
! .
This implies thatai = 0, for any i∈N. Hence, (2.31) gives (2.32).
Proof of Theorem 2.5. To obtain the representation formula for the limit of the operator div2(AnD2), we need to make several back and forth, since we cannot use directly trial functions asϕ un, forϕ∈Cc∞(Ω), in the Γ-convergence process (see the Introduction). In particular, we are led to first study the case of the open sets which are strictly contained in Ω. The proof is divided in nine steps:
• In Step 1 we prove that for any open ball B ⊂Ω, there exists a subsequence still denoted by n, such that for any recovery sequence un of limit u associated with the sequence of energies with density AnD2un : D2un in B, we get that AnD2un : D2un converges weakly-∗ in M(B) to some Radon measure νu,uB which is quadratic in u.
• In Step 2 we prove that the measure νu,uB only depends on the second gradient D2u.
• Step 3 allows us to define in any open set ω⊂ Ω, the limit energy aω(D2u, D2u), and to prove simultaneously the strong locality property (2.9) satisfied by the operator aω.
• Step 4 is devoted to the proof of the continuity of the linear mapping u ∈ H(ω) 7→
aω(D2u, D2v), for any v in the set H(ω) of the admissible trial functions (see Defini- tion 2.4).
• In Step 5 we obtain the limit variational formulation (2.10) associated with the sequence of equations div2(AnD2) =fn, for any strongly convergent sequence fn in H−2(ω).
• In Step 6 we derive the representation formula (2.16) for the Γ-limit of the sequence Gωn defined by (2.15), in any open set ω strictly contained in Ω.
• In Step 7 we establish for any open set O ⊂O⊂Ω, the convergence ofAnD2un :D2vn to aω(D2u, D2v) weakly-∗inM(ω), for suitable bounded energy sequencesun,vnconverging respectively to u, v. In Step 8 this convergence is extended to the case of an arbitrary open set ω ⊂Ω.
• Finally, Step 9 is devoted to the proofs of the bound from below (2.17), and the bound from above (2.18) satisfied by the limit density energy aω(D2u, D2u), for any open set ω ⊂Ω.
Step 1. Consider an open ball B ⊂ Ω, and for any n ∈ N, the functional FnB : L2(B) → R defined by
FnB(u) =
Z
B
AnD2u:D2u dx if u∈H2(B)
∞ if u∈L2(B)\H2(B).
(2.47)
By Properties 1.2 of Γ-convergence, we know that, for a subsequence ofn, the Γ-limitFB ofFnB does exist. It is a quadratic functional the domain of which D(FB) is a Hilbert space endowed with the norm given by (1.9).
Since by (2.1),C2( ¯B) is contained inD(FB), the domainD(FB) is continuously and densely embedded inL2(B). Then, by a duality argumentL2(B) is continuously and densely embedded in the dual D(FB)0. Thus, denoting by ΦB the bilinear form associated with FB and taking a countable dense subset {hk} of L2(B), the functions wk ∈D(FB) defined through the Riesz theorem by
wk∈D(FB) ΦB(wk, v) +
Z
B
wkv dx= Z
B
hkv dx, ∀v ∈D(FB), (2.48) form a dense subset ofD(FB). Moreover, the solution wkn∈H2(B) of the Neumann problem
( div2(AnD2wkn) +wnk =hk inB,
(AnD2wkn)n= 0, Div(AnD2wnk)·n= 0 on∂B, (2.49) with n the unitary outer normal to B on ∂B, is a recovery sequence (for FnB) of limit wk for any k ∈ N. By a diagonal argument, we can extract another subsequence of n, such that the following limit holds
AnD2wnk :D2wnk −∗* νk inM(B), ∀k ∈N. (2.50) Now, consider u∈D(FB) and a recovery sequence un (for FnB) of limit u. Extracting another subsequence if necessary, we can assume that there exists a non-negative measureνB ∈M(B), such that AnD2un : D2un converges weakly-∗ to νB in M(B). On the other hand, taking a sequencewkj converging touinD(FB), and using thatun+wnkj,un−wknj are recovery sequences (forFnB) of limitsu+wkj, u−wkj respectively, we have
kνB−νkjkM(B) ≤lim inf
n→∞
Z
B
AnD2un:D2un−AnD2wnkj :D2wnkj dx
≤ lim
n→∞
Z
B
AnD2(un+wnkj) :D2(un+wnkj)dx 12 Z
B
AnD2(un−wnkj) :D2(un−wknj)dx 12
= FB(u+wkj)12
FB(u−wkj)12 .
Therefore, νkj converges strongly to νB in M(B). As a consequence, the whole sequence AnD2un :D2unconverges toνB, which only depends onu. Since, the mapu∈D(FB)7→νB ∈ M(B) is quadratic, we may denote the measure νB associated withu∈D(FB) asνu,uB . We also have
kνu,uB kM(B)≤FB(u). (2.51)
Hence, the mappingu∈D(FB)7→νu,uB ∈M(B) is continuous.
Step 2. Let (Bm)m∈N be a countable family of open balls strictly contained inω, such that any open subset of Ω contains a ball of the family. Using a diagonal procedure we can apply Step 1 to any ball of the family, with a subsequence ofn independent ofm.
Now, consider an open set ω ⊂ Ω, and un, u ∈ H2(ω), satisfying (2.4) and (2.5) for some compact sequence fn ∈H−2(B). Let ˆu ∈H2(Bm) be a function such that D2u=D2uˆ in Bm. Sinceun+ ˆu−u converges strongly to ˆu inL2(Bm), and
lim sup
n→∞
Z
Bm
AnD2(un+ ˆu−u) :D2(un+ ˆu−u)dx= lim sup
n→∞
Z
Bm
AnD2un :D2undx <∞,
