Asymptotics of Time Harmonic Solutions to a Thin Ferroelectric Model

We introduce new model equations to describe the dynamics of the electric polarization in a ferroelectric material. We consider a thin cylinder representing the material with thickness ε and discuss the asymptotic behavior of the time harmonic solutions to the model when ε tends to 0. We obtain a reduced model settled in the cross-section of the cylinder describing the dynamics of the plane components of the polarization and electric fields.


Introduction
In this work, we are interested in the model equations of ferroelectric materials introduced in [1] and discussed in [1][2][3] for example.We consider time harmonic solutions to the model as studied in [4].We first rewrite the equations of the model given in [1] to precise the boundary conditions we will use.Let (E,H) be the electromagnetic field acting on the ferroelectric material Ω, which is a bounded and a regular domain of R 3 .Let P be the electric polarization induced in Ω.The electric displacement is then given by D = (E + P) where > 0 is the electric permittitivity of the vacuum.The Maxwell equations satisfied by the electromagnetic field are μ∂ t H − curlE = 0, ∂ t (E + P) + curlH + σE = 0, (1.1) where μ > 0 is the magnetic permeability of the vacuum and σ > 0 is the conductivity constant of the ferroelectric material.The behavior of the electric polarization P is driven where curl 2 P = curl(curlP), E eq (P) is the nonlinear equilibrium electric field which will be given later, and λ 2 = 1/ μ.The parameters a and b are some physical positive constants.This model is obtained as follow (see [1]).Denoting by m the internal magnetization and by j the current density which is driven by the difference between the equilibrium field E eq (P) and the electric field E, then with the internal polarization field P they satisfy the set of equations which reduces to the nonlinear Maxwell equation (1.2) satisfied by P. The internal magnetization m satisfies the boundary condition m × n = 0, then the second equation of (1.3) implies that P satisfies curlP × n = 0 on ∂Ω.(1.4)In this work, we consider, on ∂Ω, Leontovitch-type boundary conditions for E extending the one used in [1], that is, where β is some nonnegative function defined on ∂Ω and n is the unit outward normal to ∂Ω.The equilibrium field is assumed to be the gradient of a potential function φ(|P| 2 ).We have E eq (P) = 2Pφ (|P| 2 ) where φ : R + → R is a C 2 convex function satisfying the hypotheses given in [1], more precisely, we assume that there exist 0 < r 1 < r 0 and C 2 > 0 such that Hence, for all s ≥ R > r 1 , there exists C R > 0 such that φ (s) ≥ C R and for all s ≥ 0 where Examples of such potentials defined on R + satisfying the hypotheses are the following: N. Aïssa and K. Hamdache 3 With these hypotheses, the vector-valued function E eq (P) = 2Pφ (|P| 2 ) satisfies the estimate Let us mention other interesting models for ferroelectric materials, see [5][6][7] for example.In the first two papers, the authors consider deformable ferroelectric materials and give the evolution equation for the spontaneous polarization.The model obtained is different from the one given in [1], since it includes the deformation of the bodies.In the second one, a theoretical model is proposed explaining the lamellar morphology of domains of opposite polarization observed in ferroelectric crystals in their polar phases.The jump conditions for the electric field and polarization vector across domain walls play an important role in the characterization of the free energy.Many interesting mathematical problems, as the dimension reduction of domains, are contained in both papers.
In this paper, we are dealing with time harmonic solutions to the model (1.1)-(1.2).We write H(t,x) = e ıt H(x), E(t,x) = e ıωt E(x), P(t,x) = e ıωt P(x), and F(t,x) = e ıωt F(x) with ω > 0 fixed.The new complex field (E,P) satisfies the set of equations where The main difficulty in this problem is related to the lack of regularity of the polarization field P to prove the stability of the nonlinear equilibrium field E eq (P) with respect to the weak convergence of a sequence P m .It is easy to prove that a sequence of solutions (E m ,P m ) of (1.9) is such that P m , curlP m are bounded in L 2 (Ω).Even if we prove that div P m is also bounded in L 2 (Ω), the boundary condition curlP m × n = 0 satisfied by P m does not allow to deduce compactness in L 2 (Ω) of the sequence P m .Note that, in [8], the compactness of the sequence (P m ) is obtained in the case of the boundary condition curlP × n + βn × (P × n) = 0. To avoid this difficulty, we derive new model equations as follows.
In what follows, we are interested only in the regular part U of the polarization field P and assume that the potential ϕ is constant in Ω.Hence, we have P = U, then div P = 0 in Ω, P • n = 0 on ∂Ω. (1.12) Next, we assume that the source term F satisfies in Ω the condition div By considering the equation satisfied by E in (1.9), we deduce the compatibility condition ζ 1 (ω)div E − ω 2 div P = 0 which implies div E = 0. Hence, under the divergence free condition for P, (1.9) shows that (E,P) satisfies in Ω the new problem where π is the Lagrange multiplier associated with the constraint div P = 0 and where we used the relation −Δ = curl 2 +∇div.Combining the condition div P = 0 with the compatibility condition div E = 0 and using the second equation of (1.14), we see that the equilibrium eelectric field should satisfy the condition div E eq (P In the remainder of the paper, we assume that the ferroelectric domain is the cylinder Ω ε = Ω T × (0,ε) with thickness ε > 0 and the cross-section Ω T which is an open, bounded, and regular set of R 2 .The generic point of Ω ε is denoted by x = (x T ,x 3 ) where x T = (x 1 ,x 2 ) ∈ Ω T and 0 ≤ x 3 ≤ ε.We also assume that the function β appearing in the boundary condition satisfied by the electric field E depends on ε and is given by where β, β k are positive constants.The boundary ∂Ω ε writes as (Ω Let us set some notations.We define the norm of the complex Lebesgue space L 2 (Ω ε ) by setting 3 where G * stands for the complex conjugate of G.We use the same notations for the Lebesgue space L 2 (∂Ω T × (0,ε)).If Ω = Ω T × (0,1), we write | • | for the norm of L 2 (Ω) and (•;•) for its scalar product.We denote by (u 1 ,u 2 ,u 3 ) the canonical basis of R 3 .
The paper is organized as follows.In Section 2, we prove uniform bounds for the solution of the model equations (1.14).In Section 3, we then introduce a change of variable with respect to the vertical variable to transform the thin domain Ω ε to the cylinder Ω with thickness 1.We deduce the uniform bounds for the scaled solutions satisfying the model equations (3.5).In Section 4, we pass to the limit in the weak formulation of (3.5) and deduce the reduced model.The last section is devoted to some remarks.
In order to obtain uniform estimates, we multiply the first equation of (1.14) by E * ε and the second one by P * ε and use the Green formula We get (notice that (∇π ε ,P ε ) ε = 0) The real parts of each equation write as and the imaginary parts give (2.4) 6 Abstract and Applied Analysis Adding the last equalities and using the property (P Using the fact that is independent of ε, then there exists c > 0 which is independent of ε such that ) for all η > 0. We obtain, for η small enough, the following result.
Lemma 2.1.There exists C > 0 which is independent of ε (depending on ω and F) such that Moreover, (2.7)

The scaled problem and convergences
We introduce the change of variable z = x 3 /ε for x T ∈ Ω T fixed.We define the cylinder Ω = Ω T × (0,1) with generic point (x T ,z).For a given vector-valued function G(x T ,x 3 ) defined on Denoting ∇ T the gradient with respect to the variable x T we have Let g be a scalar function and let G T = (G 1 ,G 2 ) be a vector-valued function both defined in Ω.We set With the change of variable, we have (1/ε) We rewrite curl ε G ε as follows: Notice that θ ε • u 3 = 0 a.e.Here we have identified the 2D vectors θ ε and curl T g ε with the vectors (θ ε ,0) of R 3 and (∂ 2 g ε ,−∂ 1 g ε ,0), respectively.This identification will be used throughout this manuscript.
N. Aïssa and K. Hamdache 7 Let (E ε ,P ε ) be a solution to problem (1.14) associated with the source term F satisfying the hypothesis Using the previous notations, let E ε = (E ε T ,e ε ) and P ε = (P ε T , p ε ) be the scaled solution to (1.14) and let Π ε be the scaled function associated with π ε .Then (E ε ,P ε ) satisfies in Ω the system of equations where n = (n T ,n 3 ) is the unit outward normal to Ω.We have n = u 3 for z = 1, n = −u 3 for z = 0, and n = n T = (n 1 ,n 2 ) on ∂Ω T for 0 ≤ z ≤ 1.
Let θ ε be the 2D vector appearing in the definition of curl ε E ε .We have The boundary conditions satisfied by (E ε ,P ε ) are rewritten as follows.On z = 0 and z = 1, we have and on ∂Ω T × (0,1), we have where Applying the uniform bounds of Lemma 2.1 to the scaled solution (E ε ,P ε ) and using (3.7), we get the following.Lemma 3.1.There exists C > 0 which is independent of ε such that (3.9) 8 Abstract and Applied Analysis Moreover, the traces of the solution satisfy the estimates (3.10) We will prove the following general result which is useful in the sequel.
Since curl ε ϕ = −∂ z (φ × u 3 ) + ε(Curl T φ T )u 3 , then passing to the limit in (3.12), we get which implies that ∂ z U T = 0 in the sense of distributions so, U T is independent of the variable z.Next, let A j be the weak limit in L 2 (Ω T ) of a subsequence of the traces (U ε × u 3 ) |z= j for j = 0,1.To identify A 1 , we choose in the Green formula ϕ = εzφ with φ = (φ 1 (x T ),φ 2 (x T ),0) ∈ (Ᏸ(Ω T )) 3 .Passing to the limit in (3.12), we get N. Aïssa and K. Hamdache 9 which shows that A 1 = U T × u 3 .Secondly, we use the test function ϕ 2 in the Green formula (3.12) and pass to the limit, we get Thus, we get A 0 = U T × u 3 and A 0 = A 1 .Finally, let g be the weak limit in L 2 (∂Ω T × (0,1)) of a subsequence of the traces U ε × n T|∂Ω T ×(0,1) .To characterize g, we consider the test function ϕ=(0,0,φ 3 (x T )) with φ 3 ∈ Ᏸ(Ω T ).Observing that curl ε ϕ=curl T φ 3 =(∂ 2 φ 3 ,−∂ 1 φ 3 ,0) and passing to the limit in (3.12), since U is independent of the variable z, we deduce that , we get the following.Lemma 3.3.There exist subsequences, still denoted, E ε and θ ε such that the following weak convergences in L 2 (Ω) hold: and E T is independent of z.Moreover, the traces satisfy the convergences Proof.Lemma 3.1 implies the strong convergence of θ ε |z= j to 0. Next, set satisfies the conditions of the previous proposition, then θ ε θ weakly in L 2 (Ω) and θ is independent of the variable z.Using again Proposition 3.2, we get (θ ε × u 3 ) |z=k θ × u 3 weakly in L 2 (Ω T ) for k = 1,2.Since (θ ε × u 3 ) |z=k → 0 strongly, then θ ≡ 0 in Ω.Now, we consider the convergences for P ε .We have the following.Lemma 3.4.There exists a subsequence, still denoted by P ε such that in L 2 (Ω).Moreover, P T and Π are independent of the variable z and e = 0. Finally, P T satisfies on ∂Ω T the boundary condition P T • n T = 0.
Proof.Using the bounds in L 2 (Ω) of ∇ ε P ε and P ε , we deduce that there exists a subsequence such that P ε P = (P T , p) weakly in H 1 (Ω).Moreover, ∂ z P ε → 0 strongly in L 2 (Ω).It follows that P is independent of z.Furthermore, the pressure Π ε converges weakly to Π in L 2 (Ω).Next, the trace of P ε on ∂Ω converges weakly in H 1/2 (∂Ω) to the trace of P. Since we have p ε (x T ,1) = p ε (x T ,0) = 0 and p is independent of z, then p = 0.The trace P ε × n P × n weakly in L 2 (∂Ω T × (0,1)).We may pass to the limit in the boundary condition to get P • n = 0 on ∂Ω which gives P T • n T = 0. We rewrite the Neumann boundary condition in its original form curl ε P ε × n = 0. We write curl ε P ε = (θ ε 1 ,Curl T P ε T ), where θ ε 1 is defined as in Section 3, then the boundary condition becomes ( We apply Proposition 3.2 to U ε = curl ε P ε .Since we have Δ ε P ε = curl ε (curl ε P ε ) because we have div ε P ε = 0, then by Lemma 3.1, it follows that curl ε U ε is bounded in L 2 (Ω).Applying Proposition 3.2, we deduce that θ 1 , the weak limit of θ ε 1 , is independent of z where θ 1 is the weak limit of θ ε 1 .Finally, using the boundary condition satisfied by θ ε 1 at z = 0 and z = 1, we deduce that θ 1 = 0. Next, we use the bound in L 2 (Ω) of ∇ ε P ε T to deduce that Curl T P ε T Curl T P T and P ε T • n T P T • n T weakly in H 1/2 (∂Ω).To end the proof of the lemma, we will prove that Π is independent of z and e = 0. We set σ ε = ∂ z ((1/ε)p ε ).The condition div ε P ε = 0 is rewritten as div T P ε T + σ ε = 0 and from the equation satisfied by P ε we deduce that −λ 2 ∂ z σ ε + ∂ z Π ε = εR ε where the remainder term R ε is bounded in L 2 (Ω).Since σ ε is bounded in L 2 (Ω), then passing to the limit we get div P T + σ = 0 and −λ 2 ∂ z σ + ∂ z Π = 0. Since P T is independent of z, then so is σ which implies that ∂ z Π = 0. Let us consider the equation satisfied by E ε .We multiply the equation by the test function ϕ = (0,0,φ) with φ ∈ Ᏸ(Ω).Since we have curl ε ϕ = (∂ 2 φ,−∂ 1 φ,0), then curl ε E ε • curl ε ϕ = θ ε • curl T φ, then we get after an integration by parts (recall that the third component of F is 0) (3.21) Passing to the limit, we obtain ζ 1 (ω)e − ω 2 p = 0. Using that p = 0, we get e = 0.

The reduced problem
Let us introduce the Hilbert space H(Curl T ,Ω T ) = {U∈L 2 (Ω T ) 2 ,Curl T U ∈ L 2 (Ω T )}.We will prove the following main result describing the dimensional reduction of the thin ferroelectric cylinder.
Theorem 4.1.Let F ∈ (L 2 (Ω T )) 2 be such that div T F = 0 in Ω T .Then for ω > 0 fixed, there exists a unique solution (E,P) ∈ H(Curl T ,Ω T ) × H(Curl T ,Ω T ) of the reduced problem Furthermore, Curl T E,Curl T P ∈ H 1 (Ω T ) and the solution is obtained as the limit of the sequence (E ε ,P ε ) of the model problem (3.5).
Applying our convergence results proved in Section 3 and passing to the limit in the weak formulation, we obtain Observe that the condition div T P = 0 shows that curl T (Curl T P) = −ΔP.Our main result is then proved.
N. Aïssa and K. Hamdache 13 Now, let E ∈ L 2 (Ω), then there exists a unique U ∈ L 2 (Ω) solution to the equation a(|U| 2 )U = bE which is given by (5.5) Finally, E should satisfy the nonlinear Maxwell equation (5.7) The electric field e with p satisfies in R 3 × (0,T) the electrostatic equations div(ρp + e) = 0, curle = 0, ( with the natural jump conditions across the boundary ∂Ω × (0,T).The parameters appearing in the equation are defined in [5,6].It is important to notice that the system is coupled to some elasticity model describing the dynamic of the deformation F (e.g., when we assume that F = I + ∇u where u is the mechanical displacement, see [6,Section 3]).Next, the nonhomogeneous boundary condition satisfied by p takes into account the density of the electric dipoles.This model is more complete than the one introduced in [1].If we consider rigid body, then both models are essentially the same.An interesting question is to study the full model satisfied by (e,p,u).