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

240 M. Sofonea, El-H. Essoufifor all y = (v, ψ) ∈ V . We use (5.1) and the previous inequality to obtain(Ax 1 − Ax 2 , y) X ≤ 4c 2 ‖x 1 − x 2 ‖ V ‖y‖ V∀y ∈ Xand, taking y = Ax 1 − Ax 2 ∈ X, we find‖Ax 1 − Ax 2 ‖ X ≤ 4c 2 ‖x 1 − x 2 ‖ V . (5.7)Lemma 2 is now a consequence of inequalities (5.6) and (5.7). Next we investigate the properties of the functional j given by (5.3), (3.17). Wefirst remark that j satisfies condition (4.3). Moreover, we have the following results.Lemma 3. The functional j satisfies conditions (4.5), (4.6) and (4.7).Proof. Let η = (w, ξ), x = (u, ϕ) ∈ X and let λ ∈]0, 1]. Using (5.3) and (3.17) itresults that∫j(η, x − λx) − j(η, x) = −λ µ(‖w τ ‖) |S| ‖u τ ‖ daΓ 3and, keeping in mind (4.4), we deduce thatj ′ 2(η, x; −x) ≤ 0 ∀η, x ∈ X. (5.8)We conclude by (5.8) that the functional j satisfies conditions (4.5) and (4.6).Let now consider two sequences {x n } = {(u n , ϕ n )} ⊂ X and {η n } ={(w n , ξ n )} ⊂ X such that x n ⇀ x = (u, ϕ) ∈ X, η n ⇀ η = (w, ξ) ∈ X. Usingthe compactness property of the trace map it follows that u n → u and w n → w inL 2 (Γ 3 ) d , which imply that‖u nτ ‖ → ‖u τ ‖ in L 2 (Γ 3 ), (5.9)‖w nτ ‖ → ‖w τ ‖ in L 2 (Γ 3 ). (5.10)Moreover, (3.12), (5.10) and Kranoselski’s theorem (see for instance [7]) yieldµ(‖w nτ ‖) → µ(‖w τ ‖) in L 2 (Γ 3 ). (5.11)Therefore, we use the definition of j, (5.9) and (5.11) to deduce thatj(η n , y) → j(η, y) ∀y ∈ X and j(η n , x n ) → j(η, x), as n → ∞.We conclude that the functional j satisfies the condition (4.7). Lemma 4. If (3.12) holds, then the functional j satisfies the inequalityj(x, y) − j(x, x) + j(y, x) − j(y, y) ≤ c 2 0 L µ‖S‖ L∞ (Γ 3)‖x − y‖ 2 X∀x, y ∈ X.(5.12)