a piezoelectric contact problem with slip dependent coefficient of ...

236 M. Sofonea, El-H. Essoufito L ∞ (Γ 3 ). This is the case when the coefficient of friction does not depend on theslip. Frictional contact problems involving this last assumption on the coefficient offriction were studied in [5, 17] in the case of purely elastic materials. Notice also thatassumption (3.12) is satisfied if µ(x, ·) : IR → IR + is an increasing function, a.e.x ∈ Γ 3 .We now turn to the variational formulation of Problem P and, to this end, weintroduce further notation. Let h : V × V −→ IR be the functional∫h(u, v) = µ(‖u τ ‖) |S| ‖v τ ‖da, ∀ u, v ∈ V (3.17)Γ 3and, using Riesz’s representation theorem, consider the elements f ∈ V and q ∈ Wgiven by∫∫∫(f, v) V = f 0 · v dx +Ωf 2 · v da +Γ 2S v ν daΓ 3∀ v ∈ V, (3.18)∫ ∫(q, ψ) W = q 0 ψ dx + q 2 ψ daΓ b∀ ψ ∈ W. (3.19)ΩKeeping in mind assumptions (3.8) – (3.11) it follows that the integrals in (3.17) –(3.19) are well-defined.Using integration by parts, it is straightforward to see that if (u, σ, ϕ, D) aresufficiently regular functions which satisfy (2.3) – (2.10) then(σ, ε(v) − ε(u)) H + h(u, v) − h(u, u) ≥ (f, v − u) V ∀ v ∈ V, (3.20)(D, ψ) L 2 (Ω) d = (q, ψ) W ∀ψ ∈ W. (3.21)We plug (2.1) in (3.20), (2.2) in (3.21) and use the notation E = −∇ϕ to obtainthe following variational formulation of Problem P , in the terms of displacementfield and electric potential.Problem P V . Find a displacement field u ∈ V and an electric potential ϕ ∈ Wsuch that(Fε(u), ε(v) − ε(u)) H + (E T ∇ϕ, v − u) L 2 (Ω) d (3.22)+h(u, v) − h(u, u) ≥ (f, v − u) V ∀v ∈ V,(β∇ϕ, ∇ψ) L2 (Ω) d − (Eε(u), ∇ψ) L 2 (Ω) d = (q, ψ) W ∀ ψ ∈ W. (3.23)Our main existence and uniqueness result which we establish in Section 5 is thefollowing.Theorem 1. Assume that (3.5)–(3.10) hold. Then :1) Under the assumption (3.11), Problem P V has at least one solution.2) Under the assumptions (3.11) and (3.12), there exists L 0 , which depends only onΩ, Γ 1 , Γ 3 , Γ a , F, β, S, such that if L µ < L 0 then Problem P V has unique solution(u, ϕ) which depends Lipschitz continuously on f ∈ V and q ∈ W .