ANALYSIS OF THE INCOMPATIBILITY OPERATOR AND APPLICATION IN INTRINSIC ELASTICITY WITH DISLOCATIONS

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ANALYSIS OF THE INCOMPATIBILITY

OPERATOR AND APPLICATION IN INTRINSIC ELASTICITY WITH DISLOCATIONS

Samuel Amstutz, Nicolas van Goethem

To cite this version:

Samuel Amstutz, Nicolas van Goethem. ANALYSIS OF THE INCOMPATIBILITY OPERA-

TOR AND APPLICATION IN INTRINSIC ELASTICITY WITH DISLOCATIONS. 2015. �hal-

01150071v2�

IN INTRINSIC ELASTICITY WITH DISLOCATIONS

SAMUEL AMSTUTZ AND NICOLAS VAN GOETHEM

Abstract. The incompatibility operator arises in the modeling of elastic materials with dislocations and in the intrinsic approach to elasticity, where it is related to the Riemannian curvature of the elastic metric. It consists of applying successively the curl to the rows and the columns of a 2nd-rank tensor, usually chosen symmetric and divergence free. This paper presents a systematic analysis of boundary value problems associated with the incompatibility operator. It provides answers to such questions as existence and uniqueness of solutions, boundary traces lifting and transmission conditions. Physical interpretations in dislocation models are also discussed, but the application range of these results exceed by far any specific physical model.

1. Introduction

The incompatibility operator is a 2nd-order differential operator consisting of taking the curl of the rows and the columns of a 2nd-rank tensor , viz.,

inc= Curl Curl^t, (1.1)

the curl being taken row-wise. The incompatibility operator arises in physics, in the area of dislocation modeling, since the linear elastic strainis incompatible in the presence of disloca- tions, that is, cannot be written as a symmetric gradient, as soon as inc6= 0. Specifically, its incompatibility is related to the tensor-valued density of dislocations Λ as found by Kr¨oner [14]

and further discussed in, e.g. [20, 23], and shows ultimately as a macroscopic manifestation of plasticity (let us recall that plasticity is generated by dislocation motion). The insight of Kr¨oner was to understand the incompatibility as a genuine geometric property of the dislocated crystal related to the connection torsion and contortion (we refer to [9, 18, 20]), the crystallo- graphic evidence of the latter had been first identified by Nye [17]. In a recent contribution [21]

to this discussion, it was shown that the incompatible strain writes by virtue of the Beltrami decomposition [15] as

=∇^Su+ incF, (1.2)

where umay be given the meaning of a displacement field, here complemented with a tensor- valued symmetric and divergence-free fieldF which is related to the dislocation density by the formula

inc incF = inc = Curlκ, (1.3)

where the last equality is due to Kr¨oner [14], and with the contortion tensor κ:= Λ−I

2tr Λ. (1.4)

Here Λ is the macroscopic counterpart of the mesoscopic dislocation density tensor Λ_L :=

τ⊗bH¹_xL, whereτis the unit tangent vector to the dislocation lineL, andbits Burgers vector.

Moreover, Eq. (1.1) shows tensor Curl^t, called the Frank tensor [23] and from which the infinitesimal rotations and the displacement field are classically defined in linear elasticity.

In fact, in the presence of dislocation lines, the displacement and rotation jumps around the lines are explicitly given [15] by means of recursive line integration of linear combinations of the elastic strain and Frank tensors. Obviously, these jumps vanish if and only if the strain incompatibility vanishes.

2010Mathematics Subject Classification. 35J48,35J58,53A05,74A45,74B99.

Key words and phrases. incompatibility, dislocations, intrinsic elasticity, Sobolev spaces, lifting, transmission conditions.

1

So far, we can say that incompatibility is an important operator in Continuum mechanics, which is related to dislocations and which carries a clear geometric interpretation. About this latter point, it should be emphasized that incis another way of writing the curvature tensor associated to the elastic metric g := I−2, to the first order (the explicit relation between the 4th-rank curvature tensor and the 2nd-rank strain incompatibility can be found in [15]), whose properties in mathematical models of elasticity have been discussed in a series of recent works by Ph. Ciarlet (see for instance [6, 7]), in what he calls the intrinsic model. Let us here emphasize the deep impact of this point of view for modeling, since it consists in a change of paradigm: no more to consider the displacement as the main model variable, rather the strain, and from its knowledge (be it given, or be it deduced from a constitutive law from the stress field) to define an elastic metricg, the associated Riemann curvature tensor (and possibly, the Cartan torsion, as far as dislocations are considered, see, e.g. [14, 20]), and then in a second step to deduce a displacement field. In Ciarlet’s series of works, the displacement is found as soon as the curvature tensor associated to g vanishes. We also share this point of view, and consider the elastic strain as the primal variable, here defined from the stressσ by the linear homogeneous, isotropic and isothermal constitutive law

=A⁻¹σ,

where A is the elasticity tensor. However, for us, not only the Riemann curvature does not vanish because of (1.3), but it is the central model variable besides the elastic strain.

The main motivation and purpose of this work is indeed the mathematical study of the incompatibility operator, in particular in terms of function spaces used and their properties.

From a mathematical viewpoint, it should also be stressed that the incompatibility operator is related to the Laplacian in the sense that for symmetric and divergence-free fieldsE, one has tr incE= ∆ trE¹and inc incE = ∆². Thus, in some sense, it constitutes a tensor general- ization of the Laplacian, but as for the Curl Curl operator (recall that Curl Curl =−∆ for solenoidal vectors), any associated boundary value problem must be addressed carefully since it applies to solenoidal fields and hence might not be an elliptic operator (see [13] for the analysis of Curl Curl systems, the vector and 1st-order counterpart of our study). Indeed, the solution must satisfy the divergence-free condition in the domain, as well as specific boundary conditions (i.e., complementary in the sense of Agmon, Douglis and Nirenberg [1]) normal and/or tangen- tial components of its boundary traces. Note that inH¹-spaces, the study of divergence-free fields is of the utmost importance in fluid dynamics [10, 19]. In particular, boundary lifiting results can be found in [11].

A first natural question is to seek the appropriate boundary conditions (if one thinks of the strong form) in order to well define the BVP inc (Minc E) = G, or the appropriate function space (if one thinks of the weak form) to have existence of minimizers forR

Ω(¹₂MincE·incE− G·E)dx, whereGis some given symmetric and divergence-free force dual to E. A first issue is therefore the bilinear form coercivity and the function trace lifting properties, that is, given appropriate combinations ofEand its normal derivatives, is it possible to find a divergence-free fieldE inH² whose trace on the boundary corresponds to these values?

These issues are positively answered in this paper. As a first step, a study of the extension of boundary tangent and normal vectors will also be achieved. Such extensions and their differentiability properties can here be found in Theorem 2.2. We emphasize that our solution method is not very standard, since it is coordinate-free and based on the extension of an orthonormal basis on the boundary, which is thus viewed as locally Euclidean, though with a triad of local, non constant, basis vectors. Our core result is Theorem 3.10 and states that one can findF ∈H₀²(Ω) with prescribed divergence in Ω. Its main application for our purposes is about trace boundary lifting for solenoidal fields inH²(Ω). The exact statement can be found in Theorem 3.12. Then, in Section 4, combining this latter result with the bilinear form coercivity in H₀²(Ω) (namely, Theorem 3.9), existence and uniqueness of the nonhomogeneous boundary value problem inc (MincE) = G follow in a standard way. Lastly, with a view to perform

1trace tr here meaning the sum of the diagonal components of a 2nd-rank tensor.

topological sensitivity analysis in future work in the spirit of [2, 3], transmission conditions are identified by means of an appropriate Gauss-Green formula. They are given in Theorem 4.4.

To the knowledge of the authors, our most substantial auxiliary results, namely Theorems 3.10 and 3.12 were not found elsewhere in the literature, and seem of the utmost importance for a broad range of applications, exceeding by far our study of the incompatibility operator. Let us emphasize that the incompatibility operator and its physical interpretation in dislocation modeling must be considered here as one possible application, which was the motivation for addressing this problem, but Theorems 3.10 and 3.12 show a level of generality which we believe renders their study useful to a large community of mathematicians working in applied sciences.

Let us also stress the importance of boundary lifting in numerical analysis, in particular in the finite elements methods [5, 11].

To conclude, the proper boundary value problem is discussed in Section 4, while its applica- tion in dislocation models is proposed in Section 5.

Physical meaning of the model field E in Elasticity. The first physical interpretation of the variable E is the field F in Beltrami decomposition (1.2). Let be given the stress σ and define the elastic strain as :=A⁻¹σ, whereA is the assumed constant elasticity tensor, i.e., A= 2µI4+λI2⊗I2, withλandµthe Lam´e coefficicents. Equilibrium reads

−div σ = −div A= 0 in Ω,

σN = g on ∂Ω, (1.5)

and from (1.2) is rewritten as

−div (A∇^Su) = F_ΛL :=λ∇tr ( incF) in Ω,

(A∇^Su)N = g−λtr ( incF)N=g on ∂Ω, (1.6) recalling the solenoideal property of incF. Therefore,uis called the generalized displacement field. By virtue of (1.3) with a suitable choice of boundary conditions, one also has







inc incF = inc = Curlκ in Ω,

F = 0 on ∂Ω,

(∂NF×N)^t×N = 0 on ∂Ω.

(1.7) Systems (1.6) and (1.7) are also discussed in [21].

Moreover, recall (1.2) and takeE=⁰:= incF. By (1.3), the symmetric and solenoidal field ⁰ is the part of the elastic strain which plays a role with dislocations. Thus it may be called the dislocation-induced elastic strain. Let us address now the question of a thermodynamical setting in our elastic body with dislocations in which the free energy would depend on internal defect variables such as ⁰ and κ. Considering a high-order model in the spirit of [4], which involves the derivatives of κin the form of its curl, then by (1.3) the free energy is a function of ⁰ and inc⁰. If now a quadratic model is proposed (we refer again to [4]) one would naturally consider terms such as ¹₂Minc⁰·inc⁰in the free energy, withMa positive-definite 4th-rank tensor whose components are related to material properties of the crystal and of the dislocations. Therefore, one is lead to study variational problems of form

inf

=∇^Su+⁰

Z

Ω

1

2A∇^Su.∇^Su+1

2Minc ⁰· inc⁰

dx− Z

Ω

G·dx,

where Gis a body force that works against the total strain . Of course, surface forces could also be incorporated, as discussed in Section 5. If now Gis decomposed as G= ∇^Sw+G⁰, ones has

Z

Ω

G·dx= Z

Ω

−div∇^Sw.u+G⁰.⁰ dx,

where all boundary terms have again been dropped for simplicity. Thus, f := −div ∇^Sw is recognized as a body force that works against the displacement, while G⁰ works against the solenoidal part of the strain. Moreover, the minimizations with respect to u and ⁰ become uncoupled. The former one provides the standard linear elasticity equations, and the latter one formally yields

inc (Minc ⁰) =G⁰.

With these two examples of physical fieldsE, whose incompatibility plays a central role, let us now begin the mathematical analysis.

Notations and conventions. Let Ω be a bounded domain of R³ with smooth boundary

∂Ω. By smooth we mean C^∞, but this assumption could be considerably weakened. Let M³ denote the space of square 3-matrices, and S³ of symmetric 3-matrices. Divergence, curl, incompatibility and cross product with 2nd order tensors are defined componentwise as follows with the summation convention on repeated indices. Here,E andT are 2nd rank tensors,N is a vector, andis the Levi-Civita 3rd rank tensor. One has:

( divE)_i := ∂_jE_ij,

( Curl T)ij := (∇ ×T)ij=jkm∂kTim,

( inc E)_ij := ( Curl Curl^tE)_ij=_ikmjln∂_k∂_lE_mn, (N×T)ij := −(T×N)ij=jkmNkTim,

(E×T)_ijk := _jmnE_imT_kn, tr (E×T)j := jmnEpmTpn.

2. Extension and differentiation of the normal and tangent vectors to a surface

2.1. Signed distance function and extended unit normal. We denote byN∂Ωthe outward unit normal to∂Ω, and bybthe signed distance to∂Ω, i.e.,

b(x) =

dist(x, ∂Ω) ifx /∈Ω,

−dist(x, ∂Ω) ifx∈Ω.

We recall the following results ([8], Chap. 5, Thms 3.1 and 4.3).

Theorem 2.1. There exists an open neighborhoodW of∂Ωsuch that (1) bis smooth in W;

(2) everyx∈W admits a unique projectionp_∂Ω(x)onto∂Ω;

(3) this projection satisfies

p∂Ω(x) =x−1

2∇b²(x), x∈W; (2.1)

(4) it holds

∇b(x) =N∂Ω(p∂Ω(x)), x∈W.

In particular, this latter property shows that∇b(x) =N_∂Ω(x) for allx∈∂Ω and|∇b(x)|= 1 for allx∈W. Therefore, we define the extended unit normal by

N(x) :=∇b(x) =N_∂Ω(p_∂Ω(x)), x∈W. (2.2) 2.2. Tangent vectors on ∂Ω. For allx∈∂Ω, we denote by T_∂Ω(x) the tangent plane to∂Ω atx, that is, the orthogonal complement ofN_∂Ω(x). As∂Ω is smooth, there exists a covering of

∂Ω by open ballsB1, ..., BM ofR³such that, for each indexk, two smooth vector fieldsτ_∂^A_Ω, τ_∂Ω^B can be constructed on∂Ω∩B_k where, for allx∈∂Ω∩B_k, (τ_∂Ω^A(x), τ_∂Ω^B (x)) is an orthonormal basis of T_∂Ω(x). In all the sequel, the index k will be implicitly considered as fixed and the restriction to B_k will be omitted. In fact, for our needs, global properties and constructions will be easily obtained from local ones through a partition of unity subordinate to the covering.

Using that the Jacobian matrixDN(x) =D²b(x) of N(x) is symmetric, differentiating the equality|N(x)|²= 1 entails

∂_NN(x) =DN(x)N(x) = 0, x∈W. (2.3)

In other words, N(x) is an eigenvector of DN(x) for the eigenvalue 0. For all x ∈ ∂Ω, the system (τ_∂Ω^A (x), τ_∂Ω^B (x), N_∂Ω(x)) is an orthonormal basis ofR³. In this basis, DN(x) takes the form

DN(x) =





κ^A_∂Ω(x) ξ_∂Ω(x) 0 ξ_∂Ω(x) κ^B_∂Ω(x) 0

0 0 0



, x∈∂Ω, (2.4)

whereκ^A_∂Ω, κ^B_∂Ωandξ are smooth scalar fields defined on∂Ω.

If R∈ {A, B}, we denote by R^∗ the complementary index of R, that is,R^∗ =B ifR =A andR^∗=AifR=B.

2.3. Extended tangent vectors. Letdbe defined inW by d= 1 +b κ^A_∂Ω◦p_∂Ω

1 +b κ^B_∂Ω◦p_∂Ω

−(b ξ_∂Ω◦p_∂Ω)². Possibly adjustingW so thatd(x)>0 for allx∈W, we define inW:

τ^R=τ_∂Ω^R ◦p∂Ω, R=A, B, (2.5) κ^R=d⁻¹

(1 +b κ^R_∂Ω^∗◦p∂Ω)(κ^R_∂Ω◦p∂Ω)−b(ξ∂Ω◦p∂Ω)²

, R=A, B, ξ=d⁻¹ξ∂Ω◦p∂Ω,

κ=κ^A+κ^B,

γ^R= divτ^R, R=A, B.

Obviously, for each x ∈ W, the triple τ^A(x), τ^B(x), N(x)

forms an orthonormal basis of R³. Next, we compute the normal and tangential derivatives of these vectors. We denote the tangential derivative ∂_τR by ∂R for simplicity. Specifically ∂Rg(x) is the differential of the vectorg atxin the directionτ^R, viz.,

∂_Rg(x) :=Dg(x)τ^R(x).

Theorem 2.2. It holds in W:

∂Nτ^R = 0, (2.6)

∂RN = κ^Rτ^R+ξτ^R∗, (2.7)

∂_Rτ^R = −κ^RN−γ^R∗τ^R∗, (2.8)

∂_R^∗τ^R = γ^Rτ^R∗−ξN, (2.9)

div N = trDN= ∆b=κ. (2.10)

Proof. Differentiating (2.1) in the directionh∈R³and using (2.2) yields Dp∂Ω(x)h=h−(N(x).h)N(x)−b(x)DN(x)h.

Choosing h = N(x) and using (2.3) entails Dp∂Ω(x)N(x) = 0, then (2.6) in view of (2.5).

Choosing now h=τ^R(x) gives

Dp_∂Ω(x)τ^R(x) =τ^R(x)−b(x)DN(x)τ^R(x). (2.11) Differentiating N(x) =N(p_∂Ω(x)) in the directionτ^R(x), one obtains using (2.11)

(I+b(x)DN(p∂Ω(x)))∂RN(x) =DN(p∂Ω(x))τ^R(x). (2.12) Plugging (2.4) into (2.12) yields (2.7). Differentiating the relationsτ^R(x)·N(x) = 0,|τ^R(x)|²= 1 and τ^R(x)·τ^R∗(x) = 0 in the direction τ^R(x) yields∂_Rτ^R(x)·N(x) = −κ^R(x), ∂_Rτ^R(x)· τ^R(x) = 0 and∂Rτ^R(x)·τ^R∗(x) +∂Rτ^R∗(x)·τ^R(x) = 0, respectively. From

divτ^R(x) =∂_Rτ^R(x)·τ^R(x) +∂_R^∗τ^R(x)·τ^R∗(x) +∂_Nτ^R(x).N(x) and the preceding relations one infers

γ^R(x) =∂R^∗τ^R(x)·τ^R∗(x) =−∂R^∗τ^R∗(x)·τ^R(x). (2.13) This leads to (2.8). Differentiating τ^R(x).N(x) = 0 and |τ^R(x)|² = 1 in the directionτ^R∗(x) yields∂_R^∗τ^R(x).N(x) =−ξ(x) and∂_R^∗τ^R(x)·τ^R(x) = 0. With the help of (2.13) one arrives at (2.9). Finally, (2.10) is a straightforward consequence of (2.7) and the definitions.

Corollary 2.3. If f is twice differentiable inΩ it holds

∂R∂Nf =∂N∂Rf+κ^R∂Rf +ξ∂R^∗f. (2.14)

Proof. We have on one hand

∂R∂Nf = ∂R(∇f·N)

= ∂R∇f·N+∇f·(κ^Rτ^R+ξτ^R∗)

= D²f τ^R·N+κ^R∂Rf+ξ∂R^∗f, and on the other hand

∂N∂Rf =∂N(∇f·τ^R) =∂N∇f·τ^R=D²f N·τ^R.

One concludes using the standard Schwarz lemma.

2.4. Divergence expression in the local basis. Let us decompose a 3×3 symmetric matrix E in the local basis (τ^A, τ^B, N):

E_ij =E_{N N}N_iN_j+ X

R=A,B

E_{N R}(N_iτ_j^R+N_jτ_i^R)+ X

R=A,B

E_RRτ_i^Rτ_j^R+E_AB(τ_i^Aτ_j^B+τ_j^Aτ_i^B). (2.15)

Using Theorem 2.2 we obtain:

∂jEij = ∂jEN NNiNj+EN N(∂NNi+κNi)

+X

R

∂jEN R(Niτ_j^R+Njτ_i^R) +EN R(∂RNi+Ni∂jτ_j^R+∂jNjτ_i^R+∂Nτ_i^R)

+X

R

∂jERRτ_i^Rτ_j^R+ERR(∂Rτ_i^R+τ_i^R∂jτ_j^R)

+∂_jE_AB(τ_i^Aτ_j^B+τ_j^Aτ_i^B) +E_AB(∂_Bτ_i^A+τ_i^A∂_jτ_j^B+τ_i^B∂_jτ_j^A+∂_Aτ_i^B)

= ∂NEN NNi+EN NκNi

+X

R

h

∂_RE_{N R}N_i+∂_NE_{N R}τ_i^R+E_{N R}(κ^Rτ_i^R+ξτ_i^R∗+γ^RN_i+κτ_i^R)i

+X

R

h

∂RERRτ_i^R+ERR(−κ^RNi−γ^R∗τ_i^R∗+γ^Rτ_i^R)i

+∂_BE_ABτ_i^A+∂_AE_ABτ_i^B+E_AB(γ^Aτ_i^B−ξN_i+γ^Bτ_i^A+γ^Aτ_i^B+γ^Bτ_i^A−ξN_i)

= Ni

∂NEN N+κEN N+X

R

∂REN R+γ^REN R−κ^RERR

−2ξEAB

+X

R

τ_i^R ∂NEN R+ (κ+κ^R)EN R+ξEN R^∗+∂RERR

+γ^RERR−γ^RER^∗R^∗+∂R^∗EAB+ 2γ^R∗EAB

.

Hence

div E=

∂NEN N+κEN N+X

R

∂REN R+γ^REN R−κ^RERR

−2ξEAB

N

+X

R

∂NEN R+ (κ+κ^R)EN R+ξEN R^∗+∂RERR

+γ^RERR−γ^RER^∗R^∗+∂R^∗EAB+ 2γ^R∗EAB

τ^R. (2.16)

3. Function spaces

3.1. Definitions and basic properties. Let Γ₀ be a subset of∂Ω which is not everywhere flat and has nonzero 2-dimensional Hausdorff measure. Define

H_curl(Ω,M³) := {E∈L²(Ω,M³) : CurlE∈L²(Ω,M³)}, Hinc(Ω,S³) := {E∈L²(Ω,S³) : incE∈L²(Ω,S³)},

H(Ω) := {E∈H²(Ω,S³) : divE= 0},

H0(Ω) := {E∈ H(Ω) : E= (∂NE×N)^t×N = 0 on∂Ω}, HΓ0(Ω) := {E∈ H(Ω) : E= (∂NE×N)^t×N = 0 on Γ0}, H˜₀¹(Ω,R³) :=

u∈H₀¹(Ω,R³) : Z

Ω

udx= 0

,

H˜^3/2(∂Ω,S³) :=

E∈H^3/2(∂Ω,S³) : Z

∂Ω

EN dS(x) = 0

.

GivenE∈H˜^3/2(∂Ω;S³) andF∈H^1/2(∂Ω;S³) such thatR

∂ΩEN dS(x) = 0 andFN= 0, we define the affine spaces

H_E,F(Ω) :={E∈ H(Ω) : E =E, (∂NE×N)^t×N =Fon∂Ω}, (3.1) and

H_E,F;Γ₀(Ω) :={E∈ H(Ω) : E=E, (∂NE×N)^t×N =Fon Γ0}. (3.2) Obviously, in this latter case, it suffices that E and F be defined on Γ0, and the condition R

∂ΩEN dS(x) = 0 is not restrictive whenever Γ0 ⊂⊂ ∂Ω. The spaces H(Ω), H0(Ω) and the above affine spaces are naturally endowed with the Hilbertian structure of H²(Ω,S³).

Lemma 3.1. For all E∈H²(Ω,S³) it holds inW Curl^tE×N=−(∂_NE×N)^t×N+ X

R

τ^R×∂_RE

!^t

×N on∂Ω.

Proof. We compute componentwise

−[ Curl^tE×N]mq = jqvNvmln∂lEjn

= jqvNvmlnNl∂NEjn+jqvNvmln

X

R

τ_l^R∂REjn

=

(∂NE×N)^t×N

mq−



 X

R

τ^R×∂RE

!t

×N





mq

,

proving the result.

Lemma 3.2. For all V ∈H¹(Ω,R³)it holds in W

CurlV ·N =∂_AV_B−∂_BV_A−γ^BV_A+γ^AV_B. Proof. We have

Curl V.N = ijkNi∂jVk

= _ijkN_i ∂_NV_kN_j+∂_AV_kτ_j^A+∂_BV_kτ_j^B

= ∂_AV ·τ^B−∂_BV ·τ^A

= ∂A(VAτ^A+VBτ^B+VNN)·τ^B−∂B(VAτ^A+VBτ^B+VNN)·τ^A

= ∂AVB+ (VA∂Aτ^A+VB∂Aτ^B+VN∂AN)·τ^B

−∂BVA−(VA∂Bτ^A+VB∂Bτ^B+VN∂BN)·τ^A.

Then one concludes using Theorem 2.2.

Lemma 3.3. Every E∈ H0(Ω) satisfies

div Curl^t E = 0 inΩ, (3.3)

Curl^t E×N = 0 on∂Ω, (3.4)

∂NE = 0 on∂Ω. (3.5)

Proof. One has

[ div Curl^tE]_i=_ikm∂_j∂_kE_jm=_ikm∂_k∂_jE_mj= 0 and Curl^t E×N = 0 by Lemma 3.1.

From (2.16) one infers on∂Ω

∂NEN N = 0, ∂NEN R= 0.

Therefore (2.15) entails

∂NEij =X

R

∂NERRτ_i^Rτ_j^R+∂NEAB(τ_i^Aτ_j^B+τ_j^Aτ_i^B).

In the basis (τ^A, τ^B, N) one has

∂NE =





∂NEAA ∂NEAB 0

∂NEAB ∂NEBB 0

0 0 0



, ∂NE×N =





∂NEAB −∂NEAA 0

∂NEBB −∂NEAB 0

0 0 0



, (3.6)

0 = (∂_NE×N)^t×N =





∂_NE_BB −∂NE_AB 0

−∂_NE_AB ∂_NE_AA 0

0 0 0



, (3.7)

whereby (3.5) follows.

Remark 3.4. In the same token, for a general symmetric tensor T, one has in the basis (τ^A, τ^B, N):

T =





TAA TAB TAN

TBA TBB TBN

TN A TN B TN N



, T×N =





TAB −TAA 0 TBB −TBA 0 TN B −TN A 0



,

(T ×N)^t×N =





T_BB −TAB 0

−TAB T_AA 0

0 0 0



. (3.8)

Remark 3.5. Let T = Curl^t E be such that T ×N = 0on ∂Ω. Then, by Remark 3.4, one hasT_QR= 0 forR=A, B andQ=A, B, N. For fixedi, let V_j=T_ij in Lemma 3.2. We infer ( CurlT)N = ( incE)N = 0on ∂Ω.

Lemma 3.6(Kozono-Yanagisawa-von Wahl [13, 24]). LetF ∈H_curl(Ω;M³)such that div F = 0 inΩandF×N = 0on ∂Ω. Then F ∈H¹(Ω,M³)and it holds

k∇Fk_L2(Ω)≤CkCurlFk_L2(Ω) (3.9) for some positive constantC independent of F.

Lemma 3.7. For allE∈ H0(Ω)it holds

kEkH²(Ω)≤C kEkL²(Ω)+kCurl EkL²(Ω)+kincEkL²(Ω)

for some positive constantC independent of E.

Proof. By Lemma 3.6 we have already

k∇Ek_L2(Ω)≤CkCurlEk_L2(Ω).

SetF = Curl^t E. We have CurlF ∈L²(Ω), and, by Lemma 3.3, divF = 0 in Ω andF×N = 0 on∂Ω. Hence Lemma 3.6 entails k∇FkL²(Ω)≤CkCurlFkL²(Ω), i.e.,

k∂iCurl^tEkL²(Ω)≤CkincEkL²(Ω). This implies

kCurl∂iEk_L2(Ω)≤CkincEk_L2(Ω). (3.10)

In addition, div∂_iE=∂_idiv E= 0 in Ω. By Lemma 3.3,∂_NE×N = 0 on∂Ω, whereby, since E= 0 on∂Ω,∂_iE×N = 0 on∂Ω. Therefore, by Lemma 3.6,

k∂j∂_iEkL²(Ω)≤CkCurl∂_iEkL²(Ω). Using (3.10) we derive

k∂_j∂_iEk_L2(Ω)≤CkincEk_L2(Ω), (3.11)

and the proof is achieved.

Theorem 3.8 (Poincar´e). There exists a constant C >0 such that, for each u∈H¹(Ω,R³), kuk_L2(Ω)≤C

k∇uk_L2(Ω)+ Z

Γ₀

|u×N|dS

. (3.12)

Proof. By contradiction, assume that for eachk∈N, there exists a uk ∈H¹(Ω;R³) such that kukkL²(Ω)> k

k∇ukkL²(Ω)+ Z

Γ₀

|uk×N|dS

.

Defining ˙u_k := u_k/kukkL²(Ω), one has ku˙_kkL²(Ω) = 1 and hence (i) k∇u˙_kkL²(Ω) → 0, (ii) R

Γ0|u˙_k ×N|dS → 0 as k → ∞. By (i) and Rellich’s theorem there exists v ∈ H^s(Ω,R³), 1/2< s <1, such that a nonrelabelled subsequence ˙uk →vin H^s(Ω,R³), and hence by virtue of (i) and for everyϕ∈ D(Ω,R³),

Z

Ω

vdivϕdx= lim

k→∞

Z

Ω

˙

ukdivϕdx=− lim

k→∞

Z

Ω

Du˙kϕdx= 0,

whereby∇v= 0, meaning thatvis a constant vector. Condition (ii) now implies thatR

Γ₀|u˙k× N|dS→R

Γ0|v×N|dS= 0 ask→ ∞, i.e.,v×N = 0 and thusv is parallel toN, which is not constant and of unit length, and hence v= 0, a contradiction, sincekvkL²(Ω)= 1.

Theorem 3.9 (Coercivity). There exists a positive constantCsuch that, for eachE∈ H0(Ω), kEk_H2(Ω)≤CkincEk_L2(Ω). (3.13) Proof. By Theorem 3.8, the tensor counterpart of (3.12) reads

kFk_L2(Ω)≤C

k∇Fk_L2(Ω)+ Z

Γ₀

|F×N|dS

,

for all F ∈ H¹(Ω,M³). By Lemma 3.3, div Curl^t E = 0 in Ω and Curl^tE ×N = 0 on

∂Ω. Hence by Lemma 3.6 and again by Theorem 3.8, one has (with the nonrelabeled constant C >0),

kEkL²(Ω)≤Ck∇EkL²(Ω)≤CkCurlEkL²(Ω)=CkCurl^tEkL²(Ω)

≤Ck∇Curl^tEk_L2(Ω)≤CkCurl Curl^tEk_L2(Ω)=CkincEk_L2(Ω).

The proof is completed using Lemma 3.7.

3.2. Lifting of boundary traces.

Theorem 3.10. Let g∈H˜₀¹(Ω,R³). There existsU ∈H₀²(Ω,S³)such that div U =g.

Proof. Step 1. Letv∈H¹(Ω,R³) be a solution (unique up to a rigid motion) of −div∇^sv=g in Ω,

∇^svN = 0 on∂Ω,

and setV =∇^sv. By elliptic regularity, v∈H³(Ω), thusV ∈H²(Ω). We have divV =g in Ω andV N = 0 on∂Ω.

Step 2. We aim at definingU =V +W whereW = inc Ψ, Ψ∈H⁴(Ω,S³), must satisfy:

W N = 0 on∂Ω, (3.14)

W τ^R = −V τ^R on∂Ω, (3.15)

∂NW = −∂NV on∂Ω. (3.16)

We assume a priori that Ψ∈H⁴(Ω,S³) satisfies

Ψ =∂_NΨ = 0 on ∂Ω. (3.17)

Then (3.14) holds true by Remark 3.5 and Lemma 3.1. We are going to derive other conditions on Ψ such that (3.15)-(3.16) are also satisfied.

Step 3. Let us rewrite the traces ofV and∂_NV on∂Ω in the local basis (τ^A, τ^B, N) as V =





VAA VAB VAN

VAB VBB VBN

VAN VBN VN N



=





VAA VAB 0 VAB VBB 0

0 0 0



, ∂NV =





∂NVAA ∂NVAB ∂NVAN

∂NVAB ∂NVBB ∂NVBN

∂NVAN ∂NVBN ∂NVN N



.

Assume that

∂_N²Ψ =





−VBB VAB 0 VAB −VAA 0

0 0 0



, (3.18)

and

∂_N³Ψ =





(−∂N + 2κ^A)VBB−2ξVAB (∂N −κ)VAB+ξ(VAA+VBB) 0 (∂N −κ)VAB+ξ(VAA+VBB) (−∂N + 2κ^B)VAA−2ξVAB 0

0 0 0



. (3.19) Let us compute the components of the vectorW τ^R.

•ForW τ^R·N, it holds

W τ^R·N :W τ^R·N =W N ·τ^R= 0 =−VRN. (3.20)

•ForW τ^R·τ^R, we compute componentwise Wijτ_j^Aτ_i^A=ikmτ_i^Ajlnτ_j^A∂k∂lΨmn

=_ikmτ_i^A_jln∂_k(τ_j^A∂_lΨ_mn)−_ikmτ_i^A_jln(∂_kτ_j^A)∂_lΨ_mn. The last term of the right hand side vanishes by (3.17), hence

Wijτ_j^Aτ_i^A=ikmτ_i^A∂k(jlnτ_j^A∂lΨmn)

=ikmτ_i^A(τ_k^B∂B+Nk∂N) jlnτ_j^A(τ_l^B∂B+Nl∂N)Ψmn

=(N_m∂_B−τ_m^B∂_N)(N_n∂_B−τ_n^B∂_N)Ψ_mn

=Nm∂B(Nn∂BΨmn)−Nm∂B(τ_n^B∂NΨmn)−τ_m^B∂N(Nn∂BΨmn) +τ_m^B∂N(τ_n^B∂NΨmn), which again by (3.17) yields

W_ijτ_j^Aτ_i^A=−τ_m^BN_n∂_N(∂_BΨ_mn) +τ_m^Bτ_n^B∂_N(∂_NΨ_mn), that is, by (3.17) and (2.14),

Wijτ_j^Aτ_i^A=∂N∂N(τ_m^Bτ_n^BΨmn) =∂_N²ΨBB. (3.21) Similarly, it holdsWijτ_j^Bτ_i^B=∂_N²ΨAA.

•Now, consider W τ^R·τ^R∗ and compute componentwise W_ijτ_j^Aτ_i^B=_ikmτ_i^B_jlnτ_j^A∂_k∂_lΨ_mn=_ikmτ_i^B∂_k(_jlnτ_j^A∂_lΨ_mn)

=ikmτ_i^B(τ_k^A∂A+Nk∂N) jlnτ_j^A(τ_l^B∂B+Nl∂N)Ψmn

= (−Nm∂_A+τ_m^A∂_N)(N_n∂_B−τ_n^B∂_N)Ψ_mn

=−Nm∂A(Nn∂BΨmn) +Nm∂A(τ_n^B∂NΨmn) +τ_m^A∂N(Nn∂BΨmn)−τ_m^A∂N(τ_n^B∂NΨmn).

By (3.17) and (2.14), this reads

Wijτ_j^Aτ_i^B =−∂_N²ΨAB. (3.22) Thus (3.15) is satisfied by (3.18) and (3.20)-(3.22).

Step 4. Let us compute∂NW.

•We first compute∂NW τ^R·τ^R. Recall that, from Corollary 2.3, one has

∂N A=∂AN−κ^A∂A−ξ∂B. (3.23)

By mere projections, we have

∂NWijτ_j^Aτ_i^A=∂N

ikmjlnτ_i^Aτ_j^A∂k∂lΨmn

=∂_N

_ikmjlnτ_i^Aτ_j^A(τ_k^A∂_A+τ_k^B∂_B+N_k∂_N)(τ_l^A∂_A+τ_l^B∂_B+N_l∂_N)Ψ_mn

=∂N

ikmτ_i^A(τ_k^A∂A+τ_k^B∂B+Nk∂N) jlnτ_j^A(τ_l^A∂A+τ_l^B∂B+Nl∂N)Ψmn

−ikmjlnτ_i^A(τ_k^A∂A+τ_k^B∂B+Nk∂N)τ_j^A(τ_l^A∂A+τ_l^B∂B+Nl∂N)Ψmn

=∂_N

(N_m∂_B−τ_m^B∂_N)(N_n∂_B−τ_n^B∂_N)Ψ_mn

−jln (Nm∂B−τ_m^B∂N)τ_j^A

(τ_l^A∂A+τ_l^B∂B+Nl∂N)Ψmn

=(N_m∂_{N B}−τ_m^B∂_{N N})(N_n∂_B−τ_n^B∂_N)Ψ_mn

−jlnNm∂Bτ_j^A(τ_l^A∂N A+τ_l^B∂N B+Nl∂N N)Ψmn

−jln (Nm∂N B−τ_m^B∂N N)τ_j^A

(τ_l^A∂A+τ_l^B∂B+Nl∂N)Ψmn, (3.24) and hence from (3.17) and (3.23), the right hand side of (3.24) equals to

(N_m∂_{N B}−τ_m^B∂_{N N})(N_n∂_B−τ_n^B∂_N)Ψ_mn−_jlnN_m∂_Bτ_j^A(τ_l^A∂_{N A}+τ_l^B∂_{N B}+N_l∂_{N N})Ψ_mn. By (3.23), it follows that

∂NWijτ_j^Aτ_i^A= (Nm∂BN−κ^BNm∂B−ξNm∂A−τ_m^B∂N N)(Nn∂B−τ_n^B∂N)Ψmn

−_jlnN_m(γ^Aτ_j^B−ξN_j)(τ_l^A∂_{N A}+τ_l^B∂_{N B}+N_l∂_{N N})Ψ_mn, thus by virtue of (3.17) and (3.23),

∂_NW_ijτ_j^Aτ_i^A=(N_m∂_BN−τ_m^B∂_{N N})(N_n∂_B−τ_n^B∂_N)Ψ_mn−_jlnN_m(γ^Aτ_j^B−ξN_j)(N_l∂_{N N})Ψ_mn

=(Nm∂B−τ_m^B∂N)(Nn∂N B−τ_n^B∂N N)Ψmn−Nm(γ^Aτ_n^A)∂N NΨmn, and again by (3.23),

∂_NW_ijτ_j^Aτ_i^A= (N_m∂_B−τ_m^B∂_N)(N_n∂_BN−κ^BN_n∂_B−ξN_n∂_A−τ_n^B∂_{N N})Ψ_mn

−γ^ANmτ_n^A∂N NΨmn. By (3.17), this entails that

∂_NW_ijτ_j^Aτ_i^A=N_m∂_B(−τ_n^B∂_{N N})Ψ_mn−τ_m^B(N_n∂_{N BN}−κ^BN_n∂_{N B}−(∂_Nκ^B)N_n∂_B

−ξNn∂N A−(∂Nξ)Nn∂A−τ_n^B∂N N N)Ψmn−γ^ANmτ_n^A∂N NΨmn, whereby, using again (3.17) and (3.23),

∂NWijτ_j^Aτ_i^A=−Nm∂B(τ_n^B∂N N)Ψmn−τ_m^B(Nn∂N BN−τ_n^B∂N N N)Ψmn−γ^ANmτ_n^A∂N NΨmn

=−Nm(−κ^BNn∂N N −γ^Aτ_n^A∂N N+τ_n^B∂BN N))Ψmn

−τ_m^B(Nn∂BN N−τ_n^B∂N N N)Ψmn−γ^ANmτ_n^A∂N NΨmn. Therefore,

∂_NW_ijτ_j^Aτ_i^A=κ^BN_mN_n∂_{N N}Ψ_mn−(N_mτ_n^B+N_nτ_m^B)∂_{BN N}Ψ_mn+τ_m^Bτ_n^B∂_{N N N}Ψ_mn

=κ^BNmNn∂N NΨmn−2Nmτ_n^B∂BN NΨmn+τ_m^Bτ_n^B∂N N NΨmn. (3.25) Yet

∂_B(N_mτ_n^B) = (κ^Bτ_m^B+ξτ_m^A)τ_n^B+N_m(−κ^BN_n−γ^Aτ_n^A)

=κ^Bτ_m^Bτ_n^B+ξτ_m^Aτ_n^B−κ^BNmNn−γ^ANmτ_n^A. (3.26)