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

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238 M. Sofonea, El-H. Essoufi⎧For every sequence {x ⎪⎨n } ⊂ X with ‖x n ‖ X → ∞and every sequence {t n } ⊂ [0, 1] one has[1](4.5)⎪⎩ lim infn→∞ ‖x n ‖ 2 j ′ (t n x n , x n ; −x n ) < m.X⎧For every sequence {x ⎪⎨n } ⊂ X with ‖x n ‖ X → ∞and every bounded sequence {η n } ⊂ X one has[1](4.6)⎪⎩ lim infn→∞ ‖x n ‖ 2 j ′ (η n , x n ; −x n ) < m.X⎧⎪⎨ For every sequences {x n } ⊂ X and {η n } ⊂ X such thatx n ⇀ x[∈ X, η n ⇀ η ∈ X and for every y ∈ X one has⎪⎩ lim sup j(ηn , y) − j(η n , x n ) ] (4.7)≤ j(η, y) − j(η, x).n→∞{ There exists α < m such thatj(x, y) − j(x, x) + j(y, x) − j(y, y) ≤ α ‖x − y‖ 2 X ∀x, y ∈ X. (4.8)In the study of the quasivariational inequality (4.1) we have the following result.Theorem 2. Let conditions (4.2) – (4.3) hold. Then :1) Under the assumptions (4.5) – (4.7) there exists at least one element x ∈ X whichsolves (4.1).2) Under the assumptions (4.5) – (4.8), problem (4.1) has unique solution x = x fwhich depends Lipschitz continuously on f with the Lipschitz constant (m − α) −1 .Theorem 2 has been obtained in [14] and therefore we do not provide here thedetails of the proof. We just specify that the proof was obtained in several steps andit is based on standard arguments of elliptic variational inequalities and topologicaldegree theory.5. Proof of Theorem 1The proof of Theorem 1 will be carried out in several steps. To present it we considerthe product space X = V × W together with the inner product(x, y) X = (u, v) V + (ϕ, ψ) W ∀x = (u, ψ), y = (v, ψ) ∈ X (5.1)and the associated norm ‖ · ‖ X . Everywhere below we assume that (3.5) – (3.11)hold.We introduce the operator A : X → X defined by(Ax, y) = (Fε(u), ε(v)) H + (β∇ϕ, ∇ψ) L2 (Ω) d + (ET ∇ϕ, ε(v)) H (5.2)− (Eε(u), ∇ψ) L 2 (Ω) d∀x = (u, ψ), y = (v, ψ) ∈ Xand we extend the functional h defined by (3.17) to a functional j defined on X ×X,that isj(x, y) = h(u, v) ∀x = (u, ψ), y = (v, ψ) ∈ X. (5.3)Finally, we consider the element f ∈ X given byWe start with the following equivalence result.f = (f, q) ∈ X. (5.4)
A Piezoelectric Contact Problem with Slip Dependent Coefficient 239Lemma 1. The couple x = (u, ϕ) is a solution to Problem P V if and only if(Ax, y − x) X + j(x, y) − j(x, x) ≥ (f, y − x) X ∀ y ∈ X. (5.5)Proof. Let x = (u, ϕ) ∈ X be a solution to Problem P V and let y = (v, ψ) ∈ Y .We use the test function ψ − ϕ in (3.21), add the corresponding inequality to (3.20)and use (5.1) – (5.4) to obtain (5.5). Conversely, let x = (u, ϕ) ∈ X be a solutionto the quasivariational inequality (5.5). We take y = (v, ϕ) in (5.5) where v is anarbitrary element of V and obtain (3.22); then we take successively y = (v, ϕ + ψ)and y = (v, ϕ − ψ) in (5.5), where ψ is an arbitrary element of W ; as a result weobtain (3.23), which concludes the proof. Notice that the quasivariational inequality (5.5) derived in Lemma1 is of the form(4.1). Therefore, in order to apply the abstract result provided by Theorem 2, we startwith the study of the the properties of the operator A given by (5.2).Lemma 2. The operator A : X → X is strongly monotone and Lipschitz continuous.Proof.haveConsider two elements x 1 = (u 1 , ϕ 1 ), x 2 = (u 2 , ϕ 2 ) ∈ X. Using (5.2) we(Ax 1 − Ax 2 , x 1 − x 2 ) X = (Fε(u 1 ) − Fε(u 2 ), ε(u 1 ) − ε(u 2 )) H+ (β∇ϕ 1 − β∇ϕ 2 , ∇ϕ 1 − ∇ϕ 2 ) L2 (Ω) d + (ET ∇ϕ 1− E T ∇ϕ 2 , ε(u 1 ) − ε(u 2 )) H − (Eε(u 1 ) − Eε(u 1 ), ∇ϕ 1 − ∇ϕ 2 ) L2 (Ω) dand, since it follows by (3.16) that (E T ∇ϕ, ε(u)) H = (Eε(u), ∇ϕ) L 2 (Ω) dx = (u, ϕ) ∈ X, we find(Ax 1 − Ax 2 , x 1 − x 2 ) X =for all(Fε(u 1 ) − Fε(u 2 ), ε(u 1 ) − ε(u 2 )) H + (β∇ϕ 1 − β∇ϕ 2 , ∇ϕ 1 − ∇ϕ 2 ) L 2 (Ω) d.We use now (3.5), (3.7) and Friedrichs-Poincaré inequality (3.4) to see that thereexists c 1 > 0 which depends only on F, β, Ω and Γ a such that(Ax 1 − Ax 2 , x 1 − x 2 ) X ≥ c 1 (‖u 1 − u 2 ‖ 2 V + ‖ϕ 1 − ϕ 2 ‖ 2 W )and, keeping in mind (5.1), we obtain(Ax 1 − Ax 2 , x 1 − x 2 ) X ≥ c 1 ‖x 1 − x 2 ‖ 2 X. (5.6)In the same way, using (3.5) – (3.7), after some algebra it follows that there existsc 2 > 0 which depends only on F, β and E such that(Ax 1 − Ax 2 , y) X ≥c 2 (‖u 1 − u 2 ‖ V ‖v‖ V + ‖ϕ 1 − ϕ 2 ‖ W ‖v‖ V+ ‖u 1 − u 2 ‖ V ‖ψ‖ W + ‖ϕ 1 − ϕ 2 ‖ W ‖ψ‖ W )
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