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234 M. Sofonea, El-H. Essoufi{ (a) E = (eijk ) : Ω × S d → R d .(b) e ijk = e ikj ∈ L ∞ (Ω).(3.6)⎧(a) β = (β ij ) : Ω × R d → R d .⎪⎨(b) β ij = β ji ∈ L ∞ (Ω).⎪⎩(c) There exists m β > 0 such that β ij (x)E i E j ≥ m β ‖E‖ 2 ∀ E ∈ R d ,a.e. x ∈ Ω.(3.7)f 0 ∈ L 2 (Ω) d , f 2 ∈ L 2 (Γ 3 ) d (3.8)q 0 ∈ L 2 (Ω), q 2 ∈ L 2 (Γ b ), (3.9)S ∈ L ∞ (Γ 3 ) and ‖S‖ L ∞ (Γ 3) > 0· (3.10)⎧(a) µ : Γ 3 × IR → IR + .⎪⎨ (b) There exist c µ 1 ≥ 0 and cµ 2 ≥ 0 such thatµ(x, r) ≤ c µ 1 |r| + cµ 2 ∀ r ∈ IR + , a.e. x ∈ Γ 3 .(c) The mapping x ↦→ µ(x, r) is Lebesgue measurable on Γ 3 for any r ∈ IR.⎪⎩(d) The mapping r ↦→ µ(x, r) is continuous on IR + , a.e. x ∈ Γ 3 .(3.11){There exists Lµ > 0 such that(µ(x, r 2 ) − µ(x, r 1 )) · (r 1 − r 2 ) ≤ L µ |r 1 − r 2 | 2 ∀ r 1 , r 2 ∈ IR, a.e. x ∈ Γ 3 .(3.12)We make in what follows some comments on the assumptions (3.5) – (3.12). Asstated in Section 2, below we suppress the dependence of various functions on thespatial variable x ∈ Ω ∪ Γ .First, we note that the condition (3.5) is satisfied in the case of the linear elasticconstitutive law σ = Fε(u) in whichFξ = (f ijkl ξ kl ), (3.13)provided that f ijkl ∈ L ∞ (Ω) and there exists α > 0 such thatf ijkl (x)ξ k ξ l ≥ α‖ξ‖ 2 ∀ ξ ∈ S d , a.e. x ∈ Ω.To provide examples of nonlinear constitutive laws which satisfy (3.5), for everytensor ξ ∈ S d we denote by tr ξ the trace of ξ and let ξ D be the deviatoric part of ξgiven bytr ξ = ξ ii , ξ D = ξ − 1 d (tr ξ)I d,
A Piezoelectric Contact Problem with Slip Dependent Coefficient 235where I d ∈ S d represents the identity tensor. Let K denotes a nonempty closed convexset in S d and let P K represents the projection mapping on K. We also considera forth order symmetric and positively defined tensor E : S d → S d and takeF(ξ) = Eξ + 1 λ (ξ − P Kξ) ∀ξ ∈ S d , (3.14)where λ is a strictly positive constant. Using the properties of the projection mappingit is straightforward to see that the elasticity operator F defined by (3.14) satisfiescondition (3.5). Constitutive laws of the form σ = Fε(u)) with F given by (3.14)have been considered by many authors, see. e.g. [8], [17] p. 97 and [20] p. 68. Mostof them have defined the convex K by the relationship K = { ξ ∈ S d | G(ξ) ≤ k}where G : S d → IR is a convex continuous function such that G(0) = 0 and k > 0.A second example of nonlinear elastic equations is provided by nonlinear Henckymaterials (see [25] for details). For a Hencky material, the stress-strain relation isgiven byσ = K 0 (tr ε(u)) I d + ψ(‖ε D (u)‖ 2 ) ε D (u),so that the elasticity operator isF(ξ) = K 0 (tr ξ) I d + ψ(‖ξ D ‖ 2 ) ξ D ∀ ξ ∈ S d . (3.15)Here, K 0 > 0 is a material coefficient, the function ψ is assumed to be piecewisecontinuously differentiable, and there exist positive constants c 1 , c 2 , d 1 and d 2 , suchthat for s ≥ 0ψ(s) ≤ d 1 , −c 1 ≤ ψ ′ (s) ≤ 0, c 2 ≤ ψ(s) + 2 ψ ′ (s) s ≤ d 2 .Under these assumption it can be shown that the elasticity operator F defined in(3.15) satisfies condition (3.5).Next, as it is shown in (3.6) and (3.7), we see that the piezoelectric operator E aswell as the electric permitivitty operator β are assumed to be linear and, moreover, βis symmetric and positive definite. Recall also that the transposite tensor E T is givenby E T = (e T ijk ) where eT ijk = e kij, and the following equality holds:Eσ · v = σ · E ∗ v ∀σ ∈ S d , v ∈ R d . (3.16)We also remark that (3.8) represent regularity assumptions on the densities ofvolume forces and surface tractions while (3.9) represent regularity assumptions onthe densities of volume and surface electric charges. Condition ‖S‖ L∞ (Γ 3) > 0 in(3.10) is imposed here in order to obtain a genuine frictional contact problem. Indeed,if S = 0 a.e. on Γ 3 then by (2.7) and (2.8) it follows that the Cauchy stress vector σνvanishes on Γ 3 and therefore problem (2.1) – (2.10) becomes a purely displacementtractionproblem for electroelastic materials.Finally, we observe that the assumptions (3.11) on the coefficient of friction µare pretty general. Clearly, these assumptions are satisfied if µ is a bounded functionwhich is continuously differentiable with respect to the second variable, as it wasconsidered in [6]. We also remark that assumptions (3.11) and (3.12) are satisfied ifµ does not depend on the second argument and is a positive function which belongs
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