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Acta Appl Math (2006) 94: 251–276DOI 10.1007/s10440-006-9079-5

Existence and Uniqueness to a Dynamic BilateralFrictional Contact Problem in Viscoelasticity

Zdzisław Denkowski · Stanisław Migórski ·Anna Ochal

Received: 22 September 2005 / Accepted: 20 November 2006 /Published online: 19 December 2006© Springer Science + Business Media B.V. 2006

Abstract In this paper we examine an evolution problem which describes thedynamic bilateral contact of a viscoelastic body and a foundation. The contact ismodeled by a friction multivalued subdifferential boundary condition which incor-porates the Coulomb law of friction, the SJK model and the orthotropic frictionlaw. The main result concerns the existence and uniqueness of weak solutions to thehyperbolic variational inequality when the friction coefficient is sufficiently small.The proof is based on a surjectivity result for multivalued operators and a fixed pointargument.

Key words contact problem · variational inequality · subdifferential · Coulomb law ·friction · hyperbolic · viscoelasticity · evolution inclusion.

Mathematics Subject Classifications (2000) 74M15 · 34G25 · 35L85 · 74H20 · 74H25.

1 Introduction

The paper is a contribution to the mathematical theory of contact mechanics whichis a growing field in engineering and scientific computing. In the recent years aconsiderable progress has been achieved in the modeling and analysis of mechanicalprocesses involved in contact between deformable bodies.

We deal with a model for a mechanical problem describing bilateral frictionalcontact between a viscoelastic body and a rigid foundation. The model consists of

Research supported in part by the State Committee for Scientific Researchof the Republic of Poland (KBN) under Grants no. 2 P03A 003 25 and 4 T07A 027 26.

Z. Denkowski · S. Migórski (B) · A. OchalFaculty of Mathematics and Computer Science, Institute of Computer Science,Jagiellonian University, ul. Nawojki 11, 30072 Krakow, Polande-mail: [email protected]

252 Z. Denkowski, et al.

a hyperbolic system of equations of motion, representing momentum conservation,considered in a bounded domain subjected to mixed boundary conditions. Our maininterest lies in the boundary conditions on the contact surface. The bilateral contactcondition describes the situation when contact between the body and the foundationis maintained at all times. This is the case in many machines and in moving partsand components of mechanical equipment such as the contact between the pistonrings and the engine block in a car and the frictional contact of the wheels withthe rail when a train is braking. Mathematically there is no separation (no gap)between the body and the foundation, and the normal component of the displace-ment on the contact surface vanishes. We model the friction with a multivaluedsubdifferential boundary condition which incorporates the Coulomb law of frictionand the Strömberg, Johansson and Klarbring (SJK for short) generalization of it(cf. [23]). The latter seems to be more appropriate when the wear and hardness ofthe contacting surfaces is taken into account (cf. [22]). Such boundary conditionslead to the coupling of the contact conditions in the tangential and in the normaldirections. This friction law appears in realistic situations since it takes into accountthe dependence of the friction yield limit on the normal stress. On the other hand,it is known (cf. [22]) that it leads to considerable mathematical difficulties with thenormal value of the stress |σN| which is either a distribution or does not make senseas a trace on the boundary of a function in L2(�).

In the present paper we set the problem as a hyperbolic variational inequality andwe prove the existence and uniqueness of a solution under the hypothesis that thefriction coefficient is sufficiently small. The general outline of the proof is as follows.We begin by considering a problem in which the normal stress field on the contactboundary is assumed to be known. We formulate it equivalently as an evolutioninclusion. This inclusion possesses a unique solution so that in the next step we areable to show that a map which for a boundary value of the stress assigns the stresstensor in the whole domain has a unique fixed point. The new ingredients of the paperare as follows. The problem is formulated as a time dependent variational problemand it is solved by exploiting the surjectivity result for multivalued pseudomonotoneoperators. Our main existence and uniqueness result is proved under the hypothesisthat the L∞-norm of the time dependent friction coefficient is sufficiently small.However, we remark that this hypothesis is not needed in the first part of theproof when the assumptions H(p) and H( j) hold and the normal stress field onthe boundary is known, cf. Theorem 18. The method of the proof of Theorem 18is similar to the one employed in [18] for a dynamic hemivariational inequality.

The first attempts to deal with the dynamic contact problem with friction was car-ried out in [9]. Recent existence results for dynamic problems involving the Coulombfriction law can be found in [10] (with the Signorini condition for a homogeneousand isotropic viscosity in two dimensions), [14] (with the Signorini condition byusing regularization and penalization methods) and [15] (bilateral contact with slipdependent friction coefficient). Dynamic contact with the Signorini condition andslip rate dependent friction coefficient was considered in [16], where the existence ofweak solutions was obtained by applying the theory of multivalued pseudomonotonemaps. Quasistatic problems were treated e.g. in [3] by a normal compliance pe-nalization technique, [2] by a discretization and fixed point property and [11] bystandard arguments of time dependent elliptic variational inequalities. An exampleof nonuniqueness for the static unilateral contact problem with Coulomb friction in

Dynamic bilateral contact 253

linear elasticity can be found in [13]. We also mention the recent paper [19] whichtreats models with nonmonotone multivalued relations of the subdifferential form.

The paper is organized as follows. In Section 2 we recall some notation andpresent some auxiliary material. In Section 3 we state the mechanical problemand describe the classical model for the process. We also derive the variationalinequality formulation of the model and state the hypotheses. We also comment onthe Coulomb law of friction and the orthotropic friction. The statement of the mainexistence and uniqueness result is given in Section 4. The two steps of the proof areprovided in Sections 5 and 6. Section 7 contains the proofs of some auxiliary results.

2 Preliminaries and Notation

In this section we introduce the notation and recall some definitions needed in thesequel.

We denote by Sd the linear space of second order symmetric tensors on Rd (d =

2, 3), or equivalently, the space Rd×ds of symmetric matrices of order d. We define the

inner products and the corresponding norms on Rd and Sd by

u · v = uivi, ‖v‖ = (v · v)1/2 for all u, v ∈ Rd,

σ : τ = σijτij, ‖τ‖Sd = (τ : τ)1/2 for all σ, τ ∈ Sd.

The summation convention over repeated indices is used.Let � ⊂ R

d be a bounded domain with a Lipschitz boundary � and let n denotethe outward unit normal vector to �. The assumption that � is Lipschitz ensures thatn is defined a.e. on �. We use the following spaces

H = L2(�; Rd), H = {

τ = {τij} : τij = τ ji ∈ L2(�)} = L2(�;Sd),

H1 = {u ∈ H : ε(u) ∈ H} = H1(�; Rd), H1 = {τ ∈ H : div τ ∈ H} ,

where ε : H1(�; Rd) → L2(�;Sd) and div : H1 → L2(�; R

d) denote the deforma-tion and the divergence operators, respectively, given by

ε(u) = {εij(u)}, εij(u) = 1

2(ui, j + u j,i), div σ = {σij, j}

and the index following a comma indicates a partial derivative. The spaces H, H, H1

and H1 are Hilbert spaces equipped with the inner products

〈u, v〉H =∫

�

u · v dx, 〈σ, τ 〉H =∫

�

σ : τ dx,

〈u, v〉H1 = 〈u, v〉H + 〈ε(u), ε(v)〉H, 〈σ, τ 〉H1 = 〈σ, τ 〉H + 〈div σ, div τ 〉H.

The associated norms in H, H, H1 and H1 are denoted by | · |H , ‖ · ‖H, ‖ · ‖H1 and‖ · ‖H1 , respectively.

For every v ∈ H1 we denote by v its trace γ v on �, where γ : H1(�; Rd) →

H1/2(�; Rd) ⊂ L2(�; R

d) is the trace map. Given v ∈ H1/2(�; Rd) we denote by vN

and vT the usual normal and the tangential components of v on the boundary �

vN = v · n, vT = v − vNn.


Similarily, for a regular (say C1) tensor field σ : � → Sd, we define its normal andtangential components by

σN = (σn) · n, σT = σn − σNn.

With no confusion the letter T will appear in the time interval and will be used asthe subscript in the tangential components of vectors and tensors. We also recall thatthe following Green formula holds

〈σ, ε(v)〉H + 〈div σ, v〉H =∫

�

σn · v d� for v ∈ H1, σ ∈ H1.

For a subset U of a Banach space Y, we write ‖U‖Y = sup{‖u‖Y : u ∈ U}. Given areflexive Banach space Y, we denote by 〈·, ·〉Y the pairing between Y and its dual Y∗.We recall some definitions for a multivalued operator T : Y → 2Y∗

(see e.g. [7, 20]).An operator T is said to be pseudomonotone if it satisfies

(a) For every y ∈ Y, Ty is a nonempty, convex and weakly compact set in Y∗;(b) T is upper semicontinuous from every finite dimensional subspace of Y into Y∗

endowed with the weak topology; and(c) If yn → y weakly in Y, y∗

n ∈ Tyn and lim sup 〈y∗n, yn − y〉Y ≤ 0, then for each

z ∈ Y there exists y∗(z) ∈ Ty such that 〈y∗(z), y − z〉Y ≤ lim inf 〈y∗n, yn − z〉Y .

Let L : D(L) ⊂ Y → Y∗ be a linear densely defined maximal monotone operator.An operator T is said to be pseudomonotone with respect to D(L) (shortly L-pseudomonotone) if and only if (a) and (b) hold and

(d) If {yn} ⊂ D(L) is such that yn → y weakly in Y, Lyn → Ly weakly in Y∗, y∗n ∈

T(yn), y∗n → y∗ weakly in Y∗ and lim sup 〈y∗

n, yn〉Y ≤ 〈y∗, y〉Y , then (y, y∗) ∈Graph(T) and 〈y∗

n, yn〉Y → 〈y∗, y〉Y .

An operator T is said to be coercive if there exists a function c : R+ → R with

c(r) → ∞ as r → ∞ such that⟨y∗, y

⟩Y ≥ c (||y||Y) ||y||Y for every (y, y∗) ∈ Graph(T).

A single-valued operator T : Y → Y∗ is pseudomonotone if for each sequence{yn} ⊆ Y such that it converges weakly to y0 ∈ Y and lim sup〈Tyn, yn − y0〉Y ≤ 0, wehave 〈Ty0, y0 − y〉Y ≤ lim inf〈Tyn, yn − y〉Y for all y ∈ Y.

The following surjectivity result (see [7]) for L-pseudomonotone operators will beused in our existence theorem.

Proposition 1 If Y is a reflexive, strictly convex Banach space, L : D(L) ⊂ Y → Y∗is a linear densely defined maximal monotone operator and T : Y → 2Y∗ \ {∅} isbounded coercive and pseudomonotone with respect to D(L), then L + T is surjective.

Finally, we recall the following properties of convex functions (cf. Proposition5.2.10 and Theorem 5.3.33 in [6]).

Proposition 2 (1) If X is a finite dimensional Banach space and f : X → R is convex,then f is locally Lipschitz on int dom f . (2) Let X and Y be Banach spaces,l ∈ L(Y, X) (a space of linear bounded operators) and let f : X → R be a convexfunction. Then ∂( f ◦ l)(x) = l∗∂ f (lx) for x ∈ Y, where l∗ ∈ L(X∗, Y∗) denotes theadjoint operator to l and ∂ is the subdifferential in the sense of convex analysis.


3 Problem Formulation

In this section we describe the classical model and then we give its variationalformulation.

We consider a deformable viscoelastic body which occupies the reference config-uration � ⊂ R

d, d = 2, 3. We suppose that � is a bounded domain with Lipschitzboundary � and � is divided into three mutually disjoint measurable parts �D, �N

and �C such that meas(�D) > 0. The body is held fixed on �D, so the displacementfield vanishes there and we use the homogeneous Dirichlet condition on �D. Volumeforces of density f1 act in � and the surface tractions of density f2 are applied on�N so we use the Neumann condition on �N . The body may come in contact with afoundation over the potential contact surface �C.

We denote by u(x, t) = (u1(x, t), . . . , ud(x, t)) the displacement vector for (x, t) ∈Q = � × (0, T) with 0 < T < +∞, by σ = {σij} the stress tensor and by ε(u) ={εij(u)} the linearized (small) strain tensor, where i, j = 1, . . . , d. We suppose theKelvin–Voigt viscoelastic constitutive relation

σ(u, u′) = C(ε(u′)) + G(ε(u)),

where C and G are given nonlinear and linear constitutive functions, respectively. Weremark that in linear viscoelasticity the above law takes of the form

σij = cijklεkl(u′) + gijklεkl(u),

where C = {cijkl} and G = {gijkl}, i, j, k, l = 1, . . . , d are the viscosity and elasticitytensors, respectively.

The classical model for dynamic bilateral contact with friction is as follows: find adisplacement u : Q → R

d and a stress field σ : Q → Sd such that

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

u′′ − div σ(u, u′) = f1 in Qσ(u, u′) = C(ε(u′)) + G(ε(u)) in Qu = 0 on �D × (0, T)

σn = f2 on �N × (0, T)

uN = 0, −σT ∈ μ p(|RσN|)∂ j(u′T) on �C × (0, T)

u(0) = u0, u′(0) = u1 in �.

(P)

Here, for the sake of simplicity, the material density is assumed constant and setequal to one. We set the problem (P) in a variational form. To this end we introducethe closed subspace of H1 defined by

V = {v ∈ H1 : v = 0 on �D, vN = 0 on �C} .

This is a Hilbert space with the inner product and the corresponding norm given by

(u, v)V = (ε(u), ε(v))H, ‖v‖ = ‖ε(v)‖H for u, v ∈ V.

From the Korn inequality ‖v‖H1 ≤ c‖ε(v)‖H for v ∈ V with c > 0, it follows that‖ · ‖H1 and ‖ · ‖ are the equivalent norms on V. Identifying H with its dual, we obtainan evolution triple of spaces V ⊂ H ⊂ V∗ (cf. e.g. [7, 17, 24]) with dense, continuous


and compact embeddings. We denote by 〈·, ·〉 the duality of V and its dual V∗, by‖ · ‖V∗ the norm in V∗. We have 〈u, v〉 = 〈u, v〉H for all u ∈ H and v ∈ V.

In what follows we need the spaces V = L2(0, T; V), H = L2(0, T; H) andW = {w ∈ V : w′ ∈ V∗}, where the time derivative involved in the definition ofW is understood in the sense of vector valued distributions. Endowed with thenorm ‖v‖W = ‖v‖V + ‖v′‖V∗ , the space W becomes a separable reflexive Banachspace. We also have W ⊂ V ⊂ H ⊂ V∗. The duality for the pair (V,V∗) is denotedby 〈〈z, w〉〉V∗×V = ∫ T

0 〈z(s), w(s)〉 ds for z ∈ V∗, w ∈ V . It is well known (cf. [7, 17,24]) that the embeddings W ⊂ C(0, T; H) and {w ∈ V : w′ ∈ W} ⊂ C(0, T; V) arecontinuous.

We need the following hypotheses on the data. We suppose that the coefficientof friction μ, the force and traction densities f1, f2, the initial displacement andvelocity u0 and u1 and the regularization operator R satisfy the following conditions,respectively.

H(μ) : μ ∈ L∞(�C × (0, T)), μ ≥ 0 a.e. on �C × (0, T);H( f ) : f1 ∈ L2(0, T; H), f2 ∈ L2(0, T; L2(�N; R

d)), u0 ∈ V and u1 ∈ H;H(R) : R ∈ L(H−1/2(�); L2(�)).

The assumptions on the friction function p (cf. (8.5.8) in [22]) and the contact(superpotential) function j are as follows.

H(p) : p : �C × R → R+ satisfies

(1) p(·, r) is measurable on �C for all r ∈ R;(2) |p(x, r1) − p(x, r2)| ≤ Lp|r1 − r2| for all r1, r2 ∈ R, a.e. x ∈ �C with Lp > 0;(3) p(·, 0) ∈ L2(�C).

H( j) : j: �C × (0, T) × Rd → R satisfies

(1) j(·, ·, ξ) is measurable on �C × (0, T) for all ξ ∈ Rd and j(·, ·, 0) ∈ L1(�C ×

(0, T));(2) j(x, t, ·) is convex on R

d for a.e. (x, t) ∈ �C × (0, T);(3) ‖∂ j(x, t, ξ)‖R d ≤ c0 for all ξ ∈ R

d, a.e. (x, t) ∈ �C × (0, T) with c0 ≥ 0.

Remark 3 If j satisfies H( j)(2) and (3), then j(x, t, ·) is Lipschitz continuous on Rd.

This follows from the fact that if the effective domain of a convex function coincideswith R

d, then j(x, t, ·) is locally Lipschitz (cf. Proposition 2(1)). Next, it is enough touse H( j)(3) and apply the Lebourg mean value theorem (cf. Theorem 5.6.25 of [6]).

Moreover, we need also the hypotheses.

H(p)1 : p : �C × R → R+ satisfies H(p)(1),(2) and p(x, r) ≤ p0 for all r ∈ R, a.e.x ∈ �C with p0 ≥ 0;

H( j )1 : j: �C × (0, T) × Rd → R satisfies H( j)(1),(2), it is Lipschitz continuous in

ξ and ‖∂ j(x, t, ξ)‖R d ≤ c1(1 + ‖ξ‖R d) for all ξ ∈ Rd, a.e. (x, t) ∈ �C × (0, T)

with c1 ≥ 0.

Remark 4 If the function p satisfies H(p)1, then H(p) holds. Every function j sat-isfying H( j) satisfies also H( j)1.


The viscosity operator satisfies the following condition.

H(C) : C : Q × Sd → Sd satisfies the properties

(1) C(·, ·, ε) is measurable on Q for all ε ∈ Sd;(2) C(·, ·, 0) ∈ L2(Q;Sd);(3) ‖C(x, t, ε1) − C(x, t, ε2)‖Sd ≤ LC‖ε1 − ε2‖Sd for all ε1, ε2 ∈ Sd, a.e. (x, t) ∈ Q

with LC > 0;(4) (C(x, t, ε1) − C(x, t, ε2)) : (ε1 − ε2) ≥ 0 for all ε1, ε2 ∈ Sd, a.e. (x, t) ∈ Q;(5) C(x, t, ε) : ε ≥ α‖ε‖2

Sd− α1(x, t)‖ε‖Sd for all ε ∈ Sd, a.e. (x, t) ∈ Q with α > 0,

α1 ∈ L2(Q), α1 ≥ 0.

Remark 5 It should be observed that the hypothesis H(C) is more general than theones considered in the literature, cf. e.g. conditions (6.34) in Chapter 6.3 of [12]and assumption (6.4.4) in Chapter 6.4 of [22]. The Lipschitz condition H(C)(3) is asubstantial assumption, it excludes terms with power greater than one, cf. Remark6, but is satisfied within linearized viscoelasticity, and is satisfied by truncatedoperators. The condition H(C)(4) means that the viscosity operator is monotone.This assumption is quite natural and together with the coercivity condition H(C)(5)

is satisfied for strongly monotone operators, cf. Remark 7. Conditions H(C)(1) and(2) are needed for mathematical reasons.

Remark 6 If the conditions H(C)(2) and (3) hold, then ‖C(x, t, ε)‖Sd ≤ LC‖ε‖Sd +b(x, t) for all ε ∈ Sd, a.e. (x, t) ∈ Q, where b(x, t) = ‖C(x, t, 0)‖Sd , b ∈ L2(Q), b ≥ 0.

In Section 6, in the second part of the proof of Theorem 12 we need the followinghypothesis:

H(C)1 : C : Q × Sd → Sd satisfies H(C)(1), (2), (3) and the strong monotonicitycondition (C(x, t, ε1) − C(x, t, ε2)) : (ε1 − ε2) ≥ m‖ε1 − ε2‖2

Sdfor all ε1, ε2 ∈

Sd, a.e. (x, t) ∈ Q with m > 0.

Remark 7 If the operator C satisfies H(C)1, then it satisfies H(C). To see this, it isenough to observe that under H(C)1, we have

m‖ε‖2Sd

≤ (C(x, t, ε) − C(x, t, 0)) : ε ≤ C(x, t, ε) : ε + ‖C(x, t, 0)‖Sd‖ε‖Sd

for all ε ∈ Sd, a.e. (x, t) ∈ Q, i.e. H(C)(5) holds with α = m and α1(x, t) =‖C(x, t, 0)‖Sd .

The hypothesis on the elasticity operator is as follows.

H(G) : G : � × Sd → Sd is of the form G(x, ε) = E(x)ε (the Hooke law) with a sym-metric and nonnegative elasticity tensor E, i.e. E = {Gijkl}, i, j, k, l =1, . . . , d with Gijkl ∈ L∞(�), Gijkl = G jikl = Glkij and Gijkl(x)χijχkl ≥ 0 fora.e. x ∈ � and for all symmetric tensors χ = {χij}.

Next, for f1 and f2 satisfying the regularity in H( f ), we define f ∈ V∗ by

〈 f (t), v〉 = ( f1(t), v)H + ( f2(t), v)L2(�N;R d) for v ∈ V and a.e. t ∈ (0, T).


Remark 8 The introduction of the normal regularizing operator R is due to Duvaut[8]. As mentioned in [22] the main reason for this is mathematical since in theweak formulation of the problem the stress is only square-integrable over � and,therefore, its values or trace on the contact surface are not well defined mathematicalfunctions. As an example of such an operator one may use the convolution of σ withan infinitely differentiable function with support in a small ball. For some physicaljustification we refer to [22]. Let us observe that if R satisfies H(R), then

‖RgN‖L2(�C) ≤ cR‖g‖H for g ∈ H with cR > 0.

In fact, using the continuity of the normal trace mapping H � g �→ gN = (gn) · n ∈H−1/2(�), we have

‖gN‖H−1/2(�) ≤ c‖g‖H with c > 0.

Combining the latter with the bound ‖Rw‖L2(�) ≤ c‖w‖H−1/2(�) for w ∈ H−1/2(�) withc > 0, we deduce the desired inequality.

We now comment on the hypotheses of the problem (P). We consider two casesof the contact condition on �C. In the first case under quite general hypothesis H(p)

we consider the situation when ‖∂ j(x, t, ξ)‖R d ≤ c0 with c0 ≥ 0. In the second case weallow the subdifferential ∂ j to have a sublinear growth ‖∂ j(x, t, ξ)‖R d ≤ c1(1 + ‖ξ‖R d)

with c1 ≥ 0 but we put more restrictive assumption on the friction function, that is,p(x, r) ≤ p0 with p0 ≥ 0. Moreover, we observe that in both cases the function j isLipschitz continuous. In what follows we treat the problem (P) under the hypothesesH(p) and H( j) or H(p)1 and H( j)1, respectively.

Example 9 (The Coulomb law of friction) Let j: �C × (0, T) × Rd → R be given by

j(x, t, ξ) = ‖ξ‖R d for ξ ∈ Rd. Since the subdifferential of the function j(x, t, ·) is the

unit vector in the direction of ξ when ξ �= 0 and otherwise it is the unit ball B1 = {ξ ∈R

d : ‖ξ‖R d ≤ 1}, that is, ∂ j(x, t, ξ) equals ξ/‖ξ‖ if ξ �= 0 and equals B1 if ξ = 0, weeasily observe that j satisfies H( j) with c0 = 1. Then the contact boundary condition

−σT(x, t) ∈ μ(x, t)p(x, |RσN(x, t)|) ∂ ‖u′T(x, t)‖R d on �C × (0, T) (1)

is equivalent to⎧⎨

⎩

‖σT‖ ≤ μp(|RσN|) with‖σT‖ < μp(|RσN|) =⇒ u′

T = 0,

‖σT‖ = μp(|RσN|) =⇒ ∃ λ ≥ 0 : σT = −λu′T on �C × (0, T).

(2)

In the case when p is a known function which is independent of |RσN|, i.e. p(x, r) =h(x) with h ∈ L∞(�C), h ≥ 0, p satisfies H(p) and the conditions (2) become theTresca friction law (cf. Section 2.6 of [22] for a detailed discussion). We refer alsoto [1] for a contact problem with time dependent friction yield limit. If p(x, r) = |r|,then H(p) holds and (2) reduces to the usual regularized Coulomb friction boundarycondition

⎧⎨

⎩

‖σT‖ ≤ μ|RσN| with‖σT‖ < μ|RσN| =⇒ u′

T = 0,

‖σT‖ = μ|RσN| =⇒ ∃ λ ≥ 0 : σT = −λu′T on �C × (0, T)


which was extensively used in the literature (cf. e.g. [2, 3, 9, 12, 21, 22]). If p(x, r) =|r|(1 − δ|r|)+ with (·)+ = max{·, 0}, where δ is a small positive coefficient related tothe wear and hardness of the surface, then H(p)1 holds and we obtain a modificationof the Coulomb law of friction. Such a modification, called the SJK model, consistsof the factor (1 − δ| · |)+ and was derived in [23] from the thermodynamical consid-erations. It leads to the condition

⎧⎨

⎩

‖σT‖ ≤ μ|RσN|(1 − δ|RσN|)+ with‖σT‖ < μ|RσN|(1 − δ|RσN|)+ =⇒ u′

T = 0,

‖σT‖ = μ|RσN|(1 − δ|RσN|)+ =⇒ ∃ λ ≥ 0 : σT = −λu′T on �C × (0, T).

For the discussion of the SJK generalization of the Coulomb law, we refer to [21–23].

Example 10 (The orthotropic friction) Let j: �C × (0, T) × Rd → R be given by

j(x, t, ξ) = α1|ξ1| + . . . + αd|ξd|, where αi ≥ 0, i = 1, . . . , d. In this case we can in-terpret the contact boundary condition (1) as follows. Since j satisfies H( j), byProposition 5.6.33 of [6], we know that

∂ j(u′T) ⊂ α1∂|u′

T1| × . . . × αd∂|u′

Td|.

This leads to the relations −σTi ∈ μp(|RσN|) αi ∂|u′Ti

| for i = 1, . . . , d, which give

⎧⎨

⎩

|σTi | ≤ μi p(|RσN|) with‖σTi‖ < μi p(|RσN|) =⇒ u′

Ti= 0,

‖σTi‖ = μi p(|RσN|) =⇒ σTi = −λiu′Ti

with λi on �C × (0, T)

for μi = μαi, i = 1, . . . , d.An analogous interpretation can be given if j(x, t, ξ) = ∑d

i=1 αi ji(ξi), whereji : R → R are convex, Lipschitz continuous and |∂ ji(s)| ≤ ci

0 for s ∈ R with ci0 > 0,

i = 1, . . . , d.

In order to obtain the variational formulation of the problem (P), we use thedynamic equations of motion in (P), multiply them by v − u′(t) with v ∈ V and applythe Green formula (assuming the regularity of the functions involved) and we have

〈u′′(t), v−u′(t)〉+(σ (t), ε(v)−ε(u′(t)))H = ( f1(t), v−u′(t))H +∫

�

σ (t)n · (v−u′(t)) d�

for v ∈ V and t ∈ (0, T). Taking into account the boundary conditions on � we obtain∫

�

σ (t)n · (v − u′(t)) d� =∫

�C

σT(t) · (vT − u′T(t)) d� + ( f2(t), v − u′(t))L2(�N;R d).

Recall that for a test function v ∈ V, we denote by vN and vT the normal and the tan-gential components of v, respectively, in the sense of traces, i.e. vN = (γ v)N = (γ v)ini

and vT = (γ v)T = v − vNn. Using the boundary condition on �C, the definition of thesubdifferential and the fact that μ and p are nonnegative, we have∫

�C

σT(t) · (vT −u′T(t)) d� ≥ −

(∫

�C

μp(|RσN|) j(vT) d�−∫

�C

μp(|RσN|) j(u′T(t)) d�

).


Next, we introduce the contact functional J : (0, T) × H × V → R given by

J(t, g, z) =∫

�C

μ(x, t)p(x, |RgN(x)|) j(x, t, zT) d� for (t, g, z) ∈ (0, T) × H × V. (3)

We obtain the following variational formulation of (P): find a displacement fieldu : (0, T) → V such that

⎧⎪⎪⎨

⎪⎪⎩

〈u′′(t), v − u′(t)〉 + (σ (t), ε(v) − ε(u′(t)))H + J(t, σ (t), v) − J(t, σ (t), u′(t)) ≥≥ 〈 f (t), v − u′(t)〉 for all v ∈ V and a.e. t ∈ (0, T)

σ (t) = C(ε(u′(t))) + G(ε(u(t))) for a.e. t ∈ (0, T)

u(0) = u0, u′(0) = u1.

(4)

The existence and uniqueness result for this problem will be proved in the nextsections.

4 Main Result

The goal of this section is to state the main existence and uniqueness result for thevariational inequality (4). We also provide the properties of the data involved in thisproblem.

Definition 11 A function u ∈ V is said to be weak solution of the problem (P) if andonly if u′ ∈ W and (4) holds.

Theorem 12 If the hypotheses H(μ), H( f ), H(R), H(p), H( j), H(G), H(C)1 holdand ‖μ‖L∞(�C×(0,T)) is sufficiently small, then there is the unique weak solution tothe problem (P). The same assertion holds provided H(p) and H( j) are replaced byH(p)1 and H( j)1, respectively. Moreover, the stress field satisfies σ ∈ L2(0, T;H) anddiv σ ∈ V∗.

The proof of this theorem will be carried out in two main steps, in Sections 5 and6, respectively. In the first step we study the problem (4) when the stress field σ onthe contact boundary �C is supposed to be known. The existence and uniqueness ofsolutions to this auxiliary problem is obtained by employing the surjectivity result formultivalued L-pseudomonotone operators (cf. Theorem 1.3.73 in [7]). In the secondstep of the proof we use the Banach fixed point theorem and obtain existence anduniqueness result for (4).

In the first step of the proof of Theorem 12, for every fixed g ∈ L2(0, T;H), weconsider the following problem: find u ∈ V with u′ ∈ W such that⎧⎪⎪⎨

⎪⎪⎩

〈u′′(t), v − u′(t)〉 + (σ (t), ε(v) − ε(u′(t)))H + J(t, g(t), v) − J(t, g(t), u′(t)) ≥≥ 〈 f (t), v − u′(t)〉 for all v ∈ V and a.e. t ∈ (0, T)

σ (t) = C(ε(u′(t))) + G(ε(u(t))) for a.e. t ∈ (0, T)

u(0) = u0, u′(0) = u1.

(Pg)

Since in this problem, the stress on the contact boundary is known, by insertingσ(t) into the inequality, we have

〈u′′(t) + A(t, u′(t)) + Bu(t) − f (t), v − u′(t)〉 + J(t, g(t), v) − J(t, g(t), u′(t)) ≥ 0


for all v ∈ V and a.e. t ∈ (0, T), where the operators A : (0, T) × V → V∗ andB : V → V∗ are defined by

〈A(t, u), v〉 = (C(x, t, ε(u)), ε(v))H , (5)

〈Bu, v〉 = (G(x, ε(u)), ε(v))H (6)

for u, v ∈ V and t ∈ (0, T). Hence, by the definition of the subdifferential, we obtainthe following equivalent form of the problem (Pg): find u ∈ V with u′ ∈ W such that

{u′′(t) + A(t, u′(t)) + Bu(t) + ∂ J(t, g(t), u′(t)) � f (t) a.e. t ∈ (0, T)

u(0) = u0, u′(0) = u1.(Ig)

Here the symbol ∂ J denotes the subdifferential of J with respect to the third variable.Thus, in order to establish the existence and uniqueness to (Pg), it is enough to

study (Ig). The result on the existence and uniqueness of solutions to the evolutioninclusion (Ig) will be given in the next section (cf. Theorem 18).

Now, we study the properties of the operators A, B and the functional J which areneeded in the sequel. We start with the introduction of the space Z = Hθ (�; R

d)

with a fixed θ ∈ ( 12 , 1). Denoting by i : V → Z the embedding injection and by

γ : Z → L2(�; Rd) the trace operator, for all v ∈ V we have γ v = γ (iv). For simplic-

ity we omit the notation of the embedding and we write γ v = γ v for v ∈ V. So wehave V ⊂ Z ⊂ H ⊂ Z ∗ ⊂ V∗ with all embeddings being compact. This also impliesthat W ⊂ V ⊂ Z ⊂ H ⊂ Z∗ ⊂ V∗, where Z = L2(0, T; Z ) and Z∗ = L2(0, T; Z ∗)denotes its dual.

We establish the properties of the operators A and B defined (5) and (6),respectively. The properties of A easily follow from the corresponding ones of theoperator C.

Lemma 13 Under the hypothesis H(C), the operator A given by (5) satisfies

H(A) : A : (0, T) × V → V∗ is such that

(1) A(·, v) is measurable on (0, T) for all v ∈ V;(2) A(·, 0) ∈ V∗;(3) ‖A(t, u1) − A(t, u2)‖V∗ ≤ LC‖u1 − u2‖ for all u1, u2 ∈ V, a.e. t ∈ (0, T) with

LC > 0;(4) A(t, ·) is monotone;(5) 〈A(t, v), v〉 ≥ α||v||2 − a(t)‖v‖ for all v ∈ V a.e. t ∈ (0, T), where α > 0, a ≥ 0,

a ∈ L2(0, T).

Under the hypothesis H(C)1, the operator A satisfies

H(A)1 : A : (0, T) × V → V∗ satisfies H(A)(1), (2), (3) and the strong monotonicitycondition 〈A(t, u1) − A(t, u2), u1 − u2〉 ≥ m‖u1 − u2‖2 for all u1, u2 ∈ V,a.e. t ∈ (0, T) with m > 0.

Remark 14 If the conditions H(A)(2) and (3) hold, then ‖A(t, v)‖V∗ ≤ LC‖v‖ + a1(t)for all v ∈ V, a.e. t ∈ (0, T), where a1(t) = ‖A(t, 0)‖V∗ , a1 ∈ L2(0, T). Furthermore,if the operator A satisfies H(A)1, then H(A) holds. In fact, if A(t, ·) is stronglymonotone and H(A)(2) is satisfied, then H(A)(5) holds.


Lemma 15 Under the assumption H(G), the operator B : V → V∗ defined by (6)satisfies

H(B) : B : V → V∗ is a bounded, linear, monotone and symmetric operator,(i.e. B ∈ L(V, V∗), 〈Bv, v〉 ≥ 0 for all v ∈ V, 〈Bv, w〉 = 〈Bw, v〉 for all v,w ∈ V).

We also remark that if H( f ) is satisfied, then (H0) holds, where

(H0) : f ∈ V∗, u0 ∈ V, u1 ∈ H.

We conclude this section by establishing the properties of the contact functionalgiven by (3).

Lemma 16 Assume that H(μ) holds. If H(p) and H( j) hold, then the functional Jdefined by (3) satisfies

H(J) : J : (0, T) × H × V → R is such that

(1) J(·, g, z) is measurable on (0, T) for all g ∈ H, z ∈ V;(2) J(t, g, ·) is well defined and convex for t ∈ (0, T), g ∈ H;(3) ‖∂ J(t, g, z)‖Z ∗ ≤c‖μ‖ for (t, g, z)∈(0, T)×H×V, where c =2c0‖γ ‖‖pg‖L2(�C),

pg(x) = p(x, |RgN(x)|), ‖γ ‖ = ‖γ ‖L(Z ,L2(�;Rd)) and ‖μ‖ = ‖μ‖L∞(�C×(0,T)).

If H(p)1 and H( j)1 hold, then J satisfies

H(J)1 : the conditions H(J)(1), (2) hold and ‖∂ J(t, g, z)‖Z ∗ ≤ c ‖μ‖(1 + ‖z‖Z ) for(t, g, z) ∈ (0, T) × H × V, where c = 4c1 p0‖γ ‖ max{1, ‖γ ‖}.

Proof We observe that

J(t, g, z) = �(t, g, γ z) for (t, g, z) ∈ (0, T) × H × V, (7)

where � : (0, T) × H × L2(�C; Rd) → R is given by

�(t, g, v) =∫

�C

ϕt,g(x, v(x)) d�

and ϕt,g : �C × Rd → R is of the form

ϕt,g(x, ξ) = μ(x, t) p(x, |RgN(x)|) j(x, t, ξT) for (x, ξ) ∈ �C × Rd.

Assume first that H(p) and H( j) hold. It is clear that ϕt,g(·, ξ) is measurable, ϕt,g(x, ·)is locally Lipschitz (being convex with its domain equal to R

d) and ϕt,g(·, 0) ∈ L1(�C).Using the chain rule theorem (cf. Proposition 2(2)) and the fact that the operatorl ∈ L(Rd, R

d) given by lξ = ξT for ξ ∈ Rd is self-adjoint, we have

∂ϕt,g(x, ξ)=μ(x, t) p(x, |RgN(x)|) ∂( j(x, t, l ξ))=μ(x, t) p(x, |RgN(x)|) [∂ j(x, t, ξT)]T.

Hence, for η ∈ ∂ϕt,g(x, ξ), η ∈ Rd, we have η = μ(x, t) p(x, |RgN(x)|) ρT with ρ ∈

∂ j(x, t, ξT). By H( j)(3), we deduce

‖η‖R d = μ(x, t) p(x, |RgN(x)|)‖ρT‖Rd ≤ 2c0‖μ‖pg(x), (8)

where pg(x) = p(x, |RgN(x)|). Using H(p)(2) and (3), and the inequality ofRemark 8, we easily check that pg ∈ L2(�C). Therefore, by Theorem 5.6.39 of [6],


we know that � is finitely defined, it is convex in the last variable (so regular in thesense of Clarke) and

∂�(t, g, v) =∫

�C

μ(x, t) p(x, |RgN(x)|) [∂ j(x, t, vT)]T d�.

This means that for (t, g, v) ∈ (0, T) × H×L2(�C; Rd) and ζ ∈∂� (t, g, v) ⊂

L2(�C; Rd), there is η(t) ∈ L2(�C; R

d) such that

η(x, t) ∈ μ(x, t) p(x, |RgN(x)|) [∂ j(x, t, vT)]T

and

〈ζ, y〉L2(�C;R d) =∫

�C

η(x, t) · y(x) d� for y ∈ L2(�C; Rd).

Using (8), we have

‖∂�(t, g, v)‖L2(�C;R d) ≤ 2c0‖μ‖‖pg‖L2(�C). (9)

Furthermore, applying again the chain rule to (7), we deduce ∂ J(t, g, z) =γ ∗∂�(t, g, γ z) for (t, g, z) ∈ (0, T) × H × V, where γ ∗ : L2(�C; R

d) → Z ∗ is theadjoint operator to γ . Hence we obtain

‖∂J(t, g, z)‖Z ∗ ≤ ‖γ ∗‖ ‖∂�(t, g, γ z)‖L2(�C;Rd) ≤ 2c0‖μ‖ ‖γ ∗‖ ‖pg‖L2(�C)

which implies H(J)(3).Assume now that H(p)1 and H( j)1 hold. In this case, instead of (8), we have

‖η‖R d = μ(x, t) p(x, |RgN(x)|)‖ρT‖Rd ≤≤ 2c1μ(x, t) p(x, |RgN(x)|)(1 + 2‖ξ‖Rd) ≤ 4c1 p0‖μ‖(1 + ‖ξ‖Rd)

for η ∈ ∂ϕt,g(x, ξ), ρ ∈ ∂ j(x, t, ξT). In consequence, in place of (9), we derive theestimate

‖∂�(t, g, v)‖L2(�C;Rd) ≤ 4c1 p0‖μ‖(1 + ‖v‖L2(�C;Rd))

for (t, g, v) ∈ (0, T) × H × L2(�C; Rd).

Similarily as before, we deduce

‖∂ J(t, g, z)‖Z ∗ ≤ 4c1 p0‖μ‖ ‖γ ‖ (1 + ‖γ ‖‖z‖Z ) ≤ c ‖μ‖ (1 + ‖z‖Z )

for (t, g, z) ∈ (0, T) × H × V with c = 4c1 p0‖γ ‖ max{1, ‖γ ‖}. This proves H(J)1 andcompletes the proof of the lemma. ��

5 The First Part of the Proof

In this section we establish the existence and uniqueness of the solution to theevolution inclusion (Ig). In what follows we assume that g ∈ L2(0, T;H) is fixed. Webegin the study of (Ig) with the a priori estimate for the solutions.


Lemma 17 Let g ∈ L2(0, T;H) be fixed, H(A), H(B), (H0) hold and let u be asolution to (Ig). If H(J) holds or H(J)1 is satisfied with ‖μ‖ sufficiently small, then

‖u‖C(0,T;V) + ‖u′‖W ≤ C (1 + ‖u0‖ + |u1|H + ‖ f‖V∗) (10)

with a positive constant C.

Proof Let g ∈ L2(0, T;H) and let u ∈ V with u′ ∈ W be a solution to (Ig). We have∫ t

0〈u′′(s), u′(s)〉 ds +

∫ t

0〈A(s, u′(s)), u′(s)〉 ds +

∫ t

0〈Bu(s), u′(s)〉 ds +

+∫ t

0〈ζ(s), u′(s)〉 ds =

∫ t

0〈 f (s), u′(s)〉 ds

with ζ(s) ∈ ∂ J(s, g(s), u′(s)) for a.e. s ∈ (0, t), for every t ∈ (0, T). From the integra-

tion by parts formula (Proposition 23.23(iv) of [24]), we have∫ t

0〈u′′(s), u′(s)〉 ds =

1

2|u′(t)|2H − 1

2|u1|2H . By the monotonicity and symmetry of B (cf. H(B)), it follows

that∫ t

0〈Bu(s), u′(s)〉 ds = 1

2

∫ t

0

dds

〈Bu(s), u(s)〉 ds =

= 1

2〈Bu(t), u(t)〉 − 1

2〈Bu0, u0〉 ≥ −1

2‖B‖L(V,V∗)‖u0‖2.

Next from the Cauchy inequality, we obtain∫ t

0〈 f (s), u′(s)〉 ds ≤

∫ t

0‖ f (s)‖V∗‖u′(s)‖ ds ≤ α

2‖u′‖2

L2(0,t;V) + 1

2α‖ f‖2

V∗

for α > 0. Using the coercivity condition H(A)(5) and the above relations, we have

1

2|u′(t)|2H − 1

2|u1|2H + α‖u′‖2

L2(0,t;V)− ‖a‖L2(0,T)‖u′‖L2(0,t;V) − 1

2‖B‖‖u0‖2+

+∫ t

0〈ζ(s), u′(s)〉 ds ≤ α

2‖u′‖2

L2(0,t;V) + 1

2α‖ f‖2

V∗ (11)

for all t ∈ (0, T). If H(J) holds, then∫ t

0〈ζ(s), u′(s)〉 ds =

∫ t

0〈ζ(s), u′(s)〉Z ∗×Z ds ≤

∫ t

0‖ζ(s)‖Z ∗‖u′(s)‖Z ds ≤

≤ 2c0‖μ‖ ‖γ ‖ ‖pg‖L2(�C)

∫ t

0‖u′(s)‖Z ds

≤ 2c0

√Tβ‖μ‖ ‖γ ‖ ‖pg‖L2(�C)‖u′‖L2(0,t;V),

where β > 0 is such that ‖ · ‖Z ≤ β‖ · ‖. Inserting the latter to Eq. 11, we have

1

2|u′(t)|2H + α

2‖u′‖2

L2(0,t;V)≤ 1

2|u1|2H + 1

2‖B‖‖u0‖2 + 1

2α‖ f‖2

V∗+

+(‖a‖L2(0,T) + 2c0

√Tβ‖μ‖ ‖γ ‖ ‖pg‖L2(�C)

)‖u′‖L2(0,t;V)


for all t ∈ (0, T). Hence we deduce

‖u′‖L2(0,t;V) ≤ c2 (1 + ‖u0‖ + |u1|H + ‖ f‖V∗) with some c2 > 0. (12)

If H(J)1 holds, then∫ t

0〈ζ(s), u′(s)〉 ds ≤ c ‖μ‖

∫ t

0(1 + ‖u′(s)‖Z )‖u′(s)‖Z ds ≤

≤ c ‖μ‖(∫ t

0‖u′(s)‖Z ds +

∫ t

0‖u′(s)‖2

Z ds)

≤

≤ c ‖μ‖(β√

T‖u′‖L2(0,t;V) + β2‖u′‖2L2(0,t;V)

).

Combining this estimate with (11), we get

1

2|u′(t)|2H +

(α

2− cβ2‖μ‖

)‖u′‖2

L2(0,t;V)≤

≤ 1

2|u1|2H +

(‖a‖L2(0,T) + c ‖μ‖β√

T)

‖u′‖L2(0,t;V) +

+1

2‖B‖‖u0‖2 + 1

2α‖ f‖2

V∗

for all t ∈ (0, T). Using the hypothesis that ‖μ‖ is sufficiently small (for ‖μ‖ <

α/(2cβ2)), we again deduce (12).

From the representation formula u(t) = u0 +∫ t

0u′(s) ds and (12) we get

‖u(t)‖ ≤ ‖u0‖ +∫ t

0‖u′(s)‖ ds ≤ ‖u0‖ + c2

√T (1 + ‖u0‖ + |u1|H + ‖ f‖V∗)

for all t ∈ (0, T), which implies

‖u‖C(0,T;V) ≤ c3 (1 + ‖u0‖ + |u1|H + ‖ f‖V∗) with some c3 > 0. (13)

To finish the proof, it is enough to show the estimate on ‖u′′‖V∗ . Since u is a solution,from Remark 14, H(B) and H(J) or H(J)1, we have

‖u′′‖V∗ ≤ ‖ f‖V∗ + c4‖u′‖V + c5 + ‖B‖‖u‖V + c6(1 + ‖u′‖V

), (14)

with c4, c5 and c6 > 0. Combining (12), (13) and (14), we obtain (10), which completesthe proof of the lemma. ��

Theorem 18 Let g ∈ L2(0, T;H) and assume the hypotheses H(A), H(B) and (H0).If H(J) holds or H(J)1 is satisfied with ‖μ‖ sufficiently small, then the problem (Ig)has the unique solution.

Proof Let us define the operator K : V → C(0, T; V) by Kv(t) = ∫ t0 v(s) ds + u0 for

v ∈ V . The problem (Ig) can be written as follows: find z ∈ W such that{

z′(t) + A(t, z(t)) + BKz(t) + ∂ J(t, g(t), z(t)) � f (t) a.e. t ∈ (0, T)

z(0) = u1.(15)


We remark that z ∈ W solves (15) if and only if u = Kz satisfies (Ig). In the followingwe proceed in two steps, first suppose that u1 ∈ V and then we consider the moregeneral case u1 ∈ H.

Let us assume that u1 ∈ V and define the following operators A1 : V → V∗,B1 : V → V∗ and N1 : V → 2V

∗by

(A1v) (t) = A(t, v(t) + u1), (16)

(B1v) (t) = B(K(v(t) + u1)), (17)

N1v = {w ∈ Z∗ : w(t) ∈ ∂ J(t, g(t), v(t) + u1) a.e. t ∈ (0, T)} (18)

for v ∈ V . We observe that A1v = A(v + u1) and B1v = B(K(v + u1)), where A andB are the Nemytski operators corresponding to A and B, respectively, i.e.

(Av) (t) = A(t, v(t)), (Bv) (t) = B(v(t)) for v ∈ V . (19)

Using these operators, from (15) we obtain the following problem:{

z′ + A1z + B1z + N1z � fz(0) = 0.

(20)

We observe that z ∈ W solves (15) if and only if z − u1 ∈ W solves (20).Let the operator L : D(L) ⊂ V → V∗ be defined by Lv = v′ with D(L) = {v ∈

W : v(0) = 0}. Recall (cf. [24], Proposition 32.10) that L is a linear, densely definedand maximal monotone operator. The problem (20) can be now formulated asfollows

find z ∈ D(L) such that (L + F)z � f,

where F : V → 2V∗

is given by Fv = (A1 + B1 + N1) v for v ∈ V .In order to prove the existence of solutions to (20), we will show that the operator

F is bounded, coercive and pseudomonotone with respect to D(L), and applyProposition 1. The following three auxiliary results provide the properties of theoperators A1, B1 and N1, respectively. The proofs of Lemmas 19, 21 below are givenin Section 7. For the proof of Lemma 20 we refer to Lemma 12 of [18].

Lemma 19 If H(A) holds and u1 ∈ V, then the operator A1 defined by (16) satisfies:

(a) ‖A1v‖V∗ ≤ a1 + b 1‖v‖V for all v ∈ V with a1 ≥ 0 and b 1 > 0;(b) 〈〈A1v, v〉〉V∗×V ≥ α

2‖v‖2

V − β1‖v‖V − β2 for all v ∈ V with β1, β2 ≥ 0;(c) ‖A1v1 − A1v2‖V∗ ≤ LC‖v1 − v2‖V for all v1, v2 ∈ V;(d) A1 is pseudomonotone with respect to D(L).

If H(A) holds, then the operator A defined in (19) satisfies(e) For every {vn} ⊂ W with vn → v weakly in W and lim sup 〈〈Avn, vn − v〉〉V∗×V ≤

0, it follows that Avn → Av weakly in V∗ and 〈〈Avn, vn〉〉V∗×V → 〈〈Av, v〉〉V∗×V .

Lemma 20 If H(B) holds and u1 ∈ V, then the operator B1 defined by (17) satisfies:

(a) ‖B1v‖V∗ ≤ β3(1 + ‖v‖V ) for all v ∈ V with β3 > 0;(b) 〈〈B1v, v〉〉V∗×V ≥ −β4‖v‖V − β5 for all v ∈ V with β4, β5 ≥ 0;(c) ‖B1v1 − B1v2‖V∗ ≤ β6‖v1 − v2‖V for all v1, v2 ∈ V with β6 > 0;(d) B1 is monotone;


(e) B1 is weakly continuous, i.e. for any sequence {vn} ⊂ V with vn → v weakly in V ,we have B1vn → B1v weakly in V∗.If H(B) holds, then the operator B defined by (19) satisfies

(f) 〈〈Bv, v′〉〉V∗×V ≥ 0 for all v ∈ V such that v′ ∈ W and v(0) = 0.

Lemma 21 If H(J) holds and u1 ∈ V, then the operator N1 defined by (18) satisfies:

(a) N1v is a nonempty, convex and weakly compact subset of Z∗ for every v ∈ V ;(b) ‖z‖Z∗ ≤ c1‖μ‖ for all z ∈ N1v and v ∈ V , where c1 = 2c0

√T‖γ ‖‖pg‖L2(�C) and

pg(x) = p(x, |RgN(x)|);(c) 〈〈z, v〉〉V∗×V ≥ −c1β ‖μ‖‖v‖V for all z ∈ N1v and v ∈ V ;(d) for every vn, v ∈ V with vn → v in Z and every zn, z ∈ Z∗ with zn → z weakly

in Z∗, if zn ∈ N1vn, then z ∈ N1v.If H(J)1 holds and u1 ∈ V, then (a), (d) hold and

(b’) ‖z‖Z∗ ≤c2‖μ‖(1+‖v‖V) for all z ∈ N1v and v∈V with c2 = √2 c max{β,√

T(1 + β‖u1‖)} and β > 0 such that ‖ · ‖Z ≤ β‖ · ‖;(c’) 〈〈z, v〉〉V∗×V ≥ −c2β ‖μ‖‖v‖2

V − c2β ‖μ‖‖v‖V for all z ∈ N1v and v ∈ V .

We now continue the proof of Theorem 18.

Claim 1 F is a bounded operator. The fact that the operatorF maps bounded subsetsof V into bounded subsets of V∗ follows from Lemma 19(a), Lemma 20(a), Lemma21(b) and (b’), and the continuity of the embedding Z∗ ⊂ V∗.

Claim 2 F is coercive. Let v ∈ V and η ∈ Fv, i.e. η = A1v + B1v + ξ with ξ ∈ N1v.If H(J) holds, then from Lemma 19(b), Lemma 20(b) and Lemma 21(c), it follows

〈〈η, v〉〉V∗×V = 〈〈A1v, v〉〉V∗×V + 〈〈B1v, v〉〉V∗×V + 〈〈ξ, v〉〉Z∗×Z ≥

≥ α

2‖v‖2

V − β1‖v‖V − β2 − β4‖v‖V − β5 − c1β ‖μ‖‖v‖V

which implies the coercivity of F . If H(J)1 is satisfied, then analogously as above, byusing now Lemma 21(c’), we obtain

〈〈η, v〉〉V∗×V ≥(α

2− c2β‖μ‖

)‖v‖2

V − (β1 + β4 + c2β‖μ‖) ‖v‖V − β2 − β5.

Since ‖μ‖ is sufficiently small, we have α2 − c2β‖μ‖ > 0, which ensures that F is

coercive.

Claim 3 F is pseudomonotone with respect to D(L). By the property (a) of Lemma21, we know that for every v ∈ V , Fv is a nonempty, convex and weakly compactsubset of V∗. We will show that F is upper semicontinuous in V × V∗, where V∗ isendowed with its weak topology. To this end, we show that if a set D is weakly closedin V∗, then the set

F−(D) = {v ∈ V : Fv ∩ D �= ∅} is closed in V .


Let {vn} ⊂ F−(D), vn → v in V . We can find ηn ∈ Fvn ∩ D for all n ∈ N and bydefinition we have

ηn = A1vn + B1vn + ξn with ξn ∈ N1vn. (21)

Using the boundedness of F (see Claim 1), we obtain that the sequence {ηn} isbounded in V∗. Hence, by passing to a subsequence, if necessary, we may assumethat

ηn → η weakly in V∗ with η ∈ D (22)

since D is weakly closed in V∗. Furthermore, by Lemma 21(b) and (b’) we know that{ξn} remains in a bounded subset of Z∗. Again we may suppose that

ξn → ξ weakly in Z∗ with ξ ∈ Z∗. (23)

Hence and from the fact that vn → v in Z (recall that V ⊂ Z continuously), byLemma 21(d), we deduce ξ ∈ N1v. Next, from the continuity of A1 and B1 (cf.Lemma 19(c) and Lemma 20(c), respectively), we have

A1vn → A1v, B1vn → B1v in V∗.

From these convergences, (22) and (23), passing to the limit in (21), we get η = A1v +B1v + ξ with ξ ∈ N1v which yields that η ∈ Fv ∩ D, so v ∈ F−(D). This shows thatF−(D) is closed in V , hence F is upper semicontinuous from V into V∗ equippedwith the weak topology.

To conclude the proof that F is pseudomonotone with respect to D(L), it isenough to prove the condition (c) in the definition of pseudomonotonicity (seeSection 2). Let {zn} ⊂ D(L), zn → z weakly in W , ηn ∈ Fzn, ηn → η weakly in V∗and assume that

lim sup 〈〈ηn, zn − z〉〉V∗×V ≤ 0. (24)

Thus ηn = A1zn + B1zn + ξn, where ξn ∈ N1zn for all n ∈ N. From the boundednessof N1 (cf. Lemma 21(b) and (b’)), we infer that {ξn} lies in a bounded subset of Z∗.By passing to a subsequence, if necessary, we may assume that

ξn → ξ weakly in Z∗. (25)

Since the embedding W ⊂ Z is compact (cf. Theorem 5.1 in Chapter 1 of [17]), wemay also suppose that

zn → z in Z . (26)

Exploiting Lemma 21(d), from (25) and (26) we deduce that ξ ∈ N1z. FromLemma 21(b) or (b’) and (26), we have

|〈〈ξn, zn − z〉〉Z∗×Z | ≤ ‖ξn‖Z∗‖zn − z‖Z ≤ c (1 + ‖zn‖V ) ‖zn − z‖Z → 0 (27)

with some c > 0. On the other hand, by the monotonicity of B1 (cf. Lemma 20(d))and from the convergence zn → z weakly in V , we obtain

lim sup 〈〈B1zn, z − zn〉〉V∗×V ≤ lim sup 〈〈B1z, z − zn〉〉V∗×V = 0. (28)


Combining the condition (24) with (27) and (28), we have

lim sup 〈〈A1zn, zn − z〉〉V∗×V ≤ lim sup 〈〈ηn, zn − z〉〉V∗×V+

+ lim sup 〈〈B1zn, z − zn〉〉V∗×V + lim sup 〈〈ξn, z − zn〉〉V∗×V ≤ 0.

From the L-pseudomonotonicity of A1 (cf. Lemma 19(d)), we infer that

A1zn → A1z weakly in V∗ (29)

and

〈〈A1zn, zn〉〉V∗×V → 〈〈A1z, z〉〉V∗×V or equivalently 〈〈A1zn, zn − z〉〉V∗×V → 0.

(30)

By the weak continuity of B1, the convergences (25) and (29), we conclude that

ηn = A1zn + B1zn + ξn → A1z + B1z + ξ = η weakly in V∗.

This together with ξ ∈ N1z gives η ∈ Fz. Next, from (24), (27) and (30), we get

lim sup 〈〈B1zn, zn − z〉〉V∗×V ≤ lim sup 〈〈ηn, zn − z〉〉V∗×V−

− lim 〈〈A1zn, zn − z〉〉V∗×V − lim 〈〈ξn, zn − z〉〉V∗×V ≤ 0.

This and (28) implies lim 〈〈B1zn, zn − z〉〉V∗×V = 0, which also yields

〈〈B1zn, zn〉〉V∗×V → 〈〈B1z, z〉〉V∗×V . (31)

Passing to the limit in the equation

〈〈ηn, zn〉〉V∗×V = 〈〈A1zn, zn〉〉V∗×V + 〈〈B1zn, zn〉〉V∗×V + 〈〈ξn, zn〉〉Z∗×Z ,

from (27), (30) and (31), we obtain lim 〈〈ηn, zn〉〉V∗×V → 〈〈η, z〉〉V∗×V with η ∈ Fz.This proves the L-pseudomonotonicity of F .

Since V is a strictly convex Banach space (this follows from the fact that in everyreflexive Banach space there exists an equivalent norm such that this space is strictlyconvex, see [24], p.256), from Claims 1, 2, 3, by Proposition 1, we deduce that theproblem (20) has a solution z ∈ D(L), so z + u1 solves (15), and u = K(z + u1) is asolution of (Ig) in the case when u1 ∈ V.

Let us assume now that u1 ∈ H. Since V ⊂ H is dense, there is a sequence {u1n} ⊂V such that u1n → u1 in H, as n → ∞. Consider a solution un of the problem (Ig),when u1 is replaced by u1n, i.e. a solution of the following problem

⎧⎨

⎩

find un ∈ V such that u′n ∈ W and

u′′n(t) + A(t, u′

n(t)) + Bun(t) + ∂ J(t, g(t), u′n(t)) � f (t) a.e. t ∈ (0, T)

un(0) = u0, u′n(0) = u1n.

Recall that in this problem g ∈ L2(0, T;H) is supposed to be fixed. The existence ofun, for n ∈ N, follows from the first part of the proof. We have

u′′n(t) + A(t, u′

n(t)) + Bun(t) + ξn(t) = f (t) for a.e. t ∈ (0, T) (32)

with

ξn(t) ∈ ∂ J(t, g(t), u′n(t)) for a.e. t ∈ (0, T) (33)


and un(0) = u0, u′n(0) = u1n. From Lemma 17, we have

‖un‖C(0,T;V) + ‖u′n‖W ≤ C (1 + ‖u0‖ + |u1n|H + ‖ f‖V∗) with C > 0.

Since {u1n} is bounded in H, we obtain that {un} and {u′n} are bounded uniformly

with respect to n in V and W , respectively. Hence, by passing to a subsequence ifnecessary, we assume

un → u weakly in V,

u′n → u′ weakly in V,

u′′n → u′′ weakly in V∗.

Our goal is now to show that u is a solution to the problem (Ig).Because un → u, u′

n → u′ both weakly in W and W ⊂ C(0, T; H) continuously,we get un(t) → u(t) and u′

n(t) → u′(t) both weakly in H for all t ∈ [0, T]. Hence u0 =un(0) → u(0) weakly in H, which gives u(0) = u0. Similarily, from the convergencesu1n → u1 in H and u1n = u′

n(0) → u′(0) weakly in H, we get u′(0) = u1.As in Lemma 21(b) and (b’), using H(J) or H(J)1, we obtain from (33) that {ξn}

remains in a bounded subset of Z∗, and so for a subsequence we may assume

ξn → ξ weakly in Z∗. (34)

Since W ⊂ Z compactly and u′n → u′ weakly in W , we know that u′

n → u′ in Z . For anext subsequence we may suppose u′

n(t) → u′(t) in Z for a.e. t ∈ (0, T). Analogously,as in the proof of Lemma 21(d), we apply the Convergence Theorem (cf. [4], p.60) tothe inclusion (33) and we get

ξ(t) ∈ ∂ J(t, g(t), u′(t)) for a.e. t ∈ (0, T). (35)

In what follows we will show that

Au′n → Au′ weakly in V∗. (36)

Recall that we use the notation A and B for the Nemytski operators correspondingto A and B, respectively (cf. Eq. 19). Since lim 〈〈ξn, u′

n − u′〉〉Z∗×Z = 0 (recall thatξn → ξ weakly in Z∗ and u′

n → u′ in Z) and lim 〈〈 f, u′n − u′〉〉V∗×V = 0, from (32), we

have

lim sup 〈〈Au′n, u′

n − u′〉〉V∗×V ≤≤ lim sup 〈〈u′′

n, u′ − u′n〉〉V∗×V + lim sup 〈〈Bun, u′ − u′

n〉〉V∗×V . (37)

Using the integration by parts (cf. [24]), we obtain

〈〈u′′n − u′′, u′

n − u′〉〉V∗×V = 1

2|u′

n(T) − u′(T)|2H − 1

2|u′

n(0) − u′(0)|2Hwhich implies

lim sup 〈〈u′′n − u′′, u′ − u′

n〉〉V∗×V ≤ 0. (38)

On the other hand, by Lemma 20(f), we have

lim sup 〈〈Bun, u′ − u′n〉〉V∗×V = lim sup

(− 〈〈Bu − Bun, u′ − u′

n〉〉V∗×V++〈〈Bu, u′ − u′

n〉〉V∗×V)

≤ lim sup 〈〈Bu, u′ − u′n〉〉V∗×V = 0. (39)


Hence, by using (38) and (39) in (37), we get lim sup 〈〈Au′n, u′

n − u′〉〉V∗×V ≤ 0. Sinceu′

n → u′ weakly in W , applying Lemma 19(e) we have (36).Finally, the convergences (34), (36) and the weak continuity of the operator B (by

an argument analogous to that of Lemma 20(c)), allow us to pass to the limit in theequation u′′

n + Aun + Bun + ξn = f in V∗ (cf. (32)) and we obtain u′′ + Au′ + Bu +ξ = f in V∗. The latter together with (35) and the conditions u(0) = u0 and u′(0) = u1

implies that u is a solution to the inclusion (Ig). This completes the existence proof.Now, we will prove the uniqueness of solutions to (Ig). Let u1 and u2 be solutions

to (Ig), i.e. ui ∈ V , u′i ∈ W and there are ξ1, ξ2 ∈ Z∗ such that

⎧⎨

⎩

u′′i (t) + A(t, u′

i(t)) + Bui(t) + ξi(t) = f (t) a.e. t ∈ (0, T)

ξi(t) ∈ ∂ J(t, g(t), u′i(t)) a.e. t ∈ (0, T)

ui(0) = u0, u′i(0) = u1

(40)

for i = 1, 2. Subtracting the two equations in (40), multiplying the result by u′1(t) −

u′2(t) and integrating by parts, we obtain for t ∈ [0, T]

1

2|u′

1(t) − u′2(t)|2H +

∫ t

0〈A(s, u′

1(s)) − A(s, u′2(s)), u′

1(s) − u′2(s)〉 ds +

+∫ t

0〈Bu1(s) − Bu2(s), u′

1(s) − u′2(s)〉 ds +

+∫ t

0〈ξ1(s) − ξ2(s), u′

1(s) − u′2(s)〉 ds = 0.

Using H(B), we obtain∫ t

0〈Bu1(s)−Bu2(s), u′

1(s)−u′2(s)〉 ds = 1

2

∫ t

0

dds

〈B(u1(s)−u2(s)), u1(s)−u2(s)〉 ds =

= 1

2〈B(u1(t) − u2(t)), u1(t) − u2(t)〉 ≥ 0 (41)

for t ∈ [0, T]. From this inequality, exploiting the monotonicity of A(t, ·) (cf.H(A)(4)) and of ∂ J(t, g(t), ·) (recall that H(J) or H(J)1 hold), we deduce that

1

2|u′

1(t) − u′2(t)|2H ≤ 0 for t ∈ [0, T].

So u′1 = u′

2 on [0, T] which implies that u1 = u2 on [0, T]. Finally, again from the twoequations in (40) we have ξ1 = ξ2 which completes the proof of the theorem. ��

6 The Second Part of the Proof

The goal of this section is to apply the Banach fixed point theorem to the problem(Ig) and deduce the existence and uniqueness of solutions to (4). The main additionalhypothesis of this section is the strong monotonicity of the operator A(t, ·).

From the first part of the proof, we know that under the hypotheses ofTheorem 18, for every g ∈ L2(0, T;H), there exists the unique u = ug solution to (Ig)such that ug ∈ V and u′

g ∈ W . Equivalently, we have that for every g ∈ L2(0, T;H),there is the unique solution ug of the problem (Pg) with the above mentioned


regularity. We take σg(t) = Cε(u′g(t)) + Gε(ug(t)) for a.e t ∈ (0, T) and consider the

operator � : L2(0, T;H) → L2(0, T;H) defined by

�g = σg for g ∈ L2(0, T;H). (42)

We have the following

Theorem 22 Under the hypotheses H(μ), H( f ), H(R), H(p), H( j), H(G), H(C)1,if ‖μ‖L∞(�C×(0,T)) is sufficiently small, then the operator � has a unique fixed pointg∗ ∈ L2(0, T;H). The same assertion holds provided H(p) and H( j) are replaced byH(p)1 and H( j)1, respectively.

Proof From the hypotheses we know (cf. Section 4) that H(A)1, H(B), (H0) andH(J) (or H(J)1 when H(p)1 and H( j)1 hold) are satisfied. Let g1, g2 ∈ L2(0, T;H)

and put ui = ugi , σi = σgi for i = 1, 2. Since ui ∈ V with u′i ∈ W is the solution to (Pgi)

for i = 1, 2, we have

⎧⎨

⎩

u′′i (t) + A(t, u′

i(t)) + Bui(t) + ξi(t) = f (t) a.e. t ∈ (0, T)

ξi(t) ∈ ∂ J(t, gi(t), u′i(t)) a.e. t ∈ (0, T)

ui(0) = u0, u′i(0) = u1

(43)

for i = 1, 2. Similarly as in the proof of Theorem 18, subtracting the two equations in(43), multiplying the result by u′

1(t) − u′2(t) and using the integration by parts formula,

we get for t ∈ [0, T]

1

2|u′

1(t) − u′2(t)|2H +

∫ t

0〈A(s, u′

1(s)) − A(s, u′2(s)), u′

1(s) − u′2(s)〉 ds +

+∫ t

0〈Bu1(s)−Bu2(s), u′

1(s)−u′2(s)〉 ds +

+∫ t

0〈ξ1(s)−ξ2(s), u′

1(s)−u′2(s)〉Z ∗×Z ds = 0.

Using (41) and the strong monotonicity of A(t, ·) (cf. (H(A)1), we have

1

2|u′

1(t) − u′2(t)|2H + m

∫ t

0‖u′

1(s) − u′2(s)‖2 ds ≤

∫ t

0〈ξ1(s) − ξ2(s), u′

2(s) − u′1(s)〉Z ∗×Z ds

(44)

for t ∈ [0, T]. From the definition of the subdifferential, we obtain

〈ξ1(t), u′2(t) − u′

1(t)〉Z ∗×Z ≤ J(t, g1(t), u′2(t)) − J(t, g1(t), u′

1(t))

〈ξ2(t), u′1(t) − u′

2(t)〉Z ∗×Z ≤ J(t, g2(t), u′1(t)) − J(t, g2(t), u′

2(t))


for t ∈ [0, T]. Using the form of the functional J (cf. (3)), H(p) and H( j) (or H(p)1

and H( j)1), we deduce

I ≡∫ t

0〈ξ1(s) − ξ2(s), u′

2(s) − u′1(s)〉Z ∗×Z ds ≤

≤∫ t

0

(J(s, g1(s), u′

2(s))− J(s, g1(s), u′1(s))+ J(s, g2(s), u′

1(s))− J(s, g2(s), u′2(s))

)ds=

=∫ t

0

∫

�C

μ (p(|Rg1N(s)|) − p(|Rg2N(s)|)) (j(x, s, u′

2T(s)) − j(x, s, u′1T(s))

)d�ds ≤

≤ L jLp‖μ‖∫ t

0

∫

�C

|R(g1N(s) − g2N(s))| ‖u′2T(s) − u′

1T(s)‖Rd d�ds ≤

≤ 2L jLp‖μ‖∫ t

0

∫

�C

|R(g1N(s) − g2N(s))| ‖u′2(s) − u′

1(s)‖Rd d�ds

for t ∈ [0, T]. In order to estimate the integrand of the latter, we apply the Cauchyinequality ab ≤ ε2

2 |a|2 + 12ε2 |b |2, a, b ∈ R with ε > 0 such that

ε2 = m

2L jLp‖μ‖ ‖γ ‖2 .

We obtain

|R(g1N(s) − g2N(s))| ‖u′2(s) − u′

1(s)‖Rd ≤

≤ ε2

2‖u′

2(s) − u′1(s)‖2

Rd + 1

2ε2|R(g1N(s) − g2N(s))|2

for s ∈ (0, t). Integrating over �C × (0, t), and using Remark 8, we have∫ t

0

∫

�C

|R(g1N(s) − g2N(s))| ‖u′2(s) − u′

1(s)‖Rd d�ds ≤

≤ ε2

2

∫ t

0‖u′

2(s) − u′1(s)‖2

L2(�C;Rd)ds + 1

2ε2

∫ t

0|R(g1N(s) − g2N(s))|2 ds ≤

≤ ε2‖γ ‖2

2

∫ t

0‖u′

2(s) − u′1(s)‖2 ds + c2

R

2ε2

∫ t

0‖g1(s) − g2(s)‖2

H ds.

Hence

I ≤ m2

∫ t

0‖u′

2(s) − u′1(s)‖2 ds + 2

m(L jLp‖μ‖‖γ ‖cR)2

∫ t

0‖g1(s) − g2(s)‖2

H ds.

Combining this estimate with (44), we find

1

2|u′

1(t) − u′2(t)|2H + m

2

∫ t

0‖u′

1(s) − u′2(s)‖2 ds ≤

≤ 2

m(L jLp‖μ‖‖γ ‖cR)2

∫ t

0‖g1(s) − g2(s)‖2

H ds

for all t ∈ [0, T]. Omitting the first term on the left hand side, for t = T, we obtain

‖u′1 − u′

2‖V ≤ c‖μ‖ ‖g1 − g2‖L2(0,T;H), (45)


where c = 2L jLp‖γ ‖cR/m > 0. Next, since ui(t) = ui(0) + ∫ t0 u′

i(s) ds for t ∈ [0, T],i = 1, 2, we have

‖u1(t) − u2(t)‖ = ‖∫ t

0(u′

1(s) − u′2(s)) ds‖ ≤ ‖u′

1 − u′2‖L1(0,T;V) ≤ √

T‖u′1 − u′

2‖Vand ‖u1 − u2‖V ≤ T‖u′

1 − u′2‖V . On the other hand, by the Lipschitz continuity of

C(x, t, ·) (cf. H(C)(3)) and the estimate ‖Gε(v)‖L2(0,T;H) ≤ cG‖v‖V with cG > 0, itfollows that

‖σ1 − σ2‖L2(0,T;H) ≤ ‖Cε(u′1) − Cε(u′

2)‖L2(0,T;H) + ‖G(ε(u1 − u2))‖L2(0,T;H) ≤≤ LC‖u′

1 − u′2‖V + cG‖u1 − u2‖V .

Combining this estimate with (45), we obtain

‖σ1 − σ2‖L2(0,T;H) ≤ (LC + T)‖u′1 − u′

2‖V ≤ c(LC + T) ‖μ‖ ‖g1 − g2‖L2(0,T;H).

Let

μ0 = m2(LC + T)L jLp‖γ ‖cR

.

For ‖μ‖ < μ0, we have

‖�g1 − �g2‖L2(0,T;H) < ‖g1 − g2‖L2(0,T;H).

The assertion of the theorem is now a consequence of this estimate and the Banachfixed point theorem. The proof of Theorem 22 is complete. ��

Proof of Theorem 12 From the definition of the operator � given by (42) andTheorem 22, if ‖μ‖ is sufficiently small, we deduce that the solution ug∗ of theproblem (Pg∗) is a solution of the variational inequality (4). The uniqueness of thesolution to (4) is a consequence of the uniqueness of the solution of (Pg∗) andthe uniqueness of the fixed point of �. Moreover, we have following regularityof the solution and the corresponding stress tensor u ∈ H1(0, T; V) ∩ C(0, T; V),u′ ∈ C(0, T; H), u′′ ∈ L2(0, T; V∗) and σ ∈ L2(0, T;H) with div σ ∈ L2(0, T; V∗).

��

7 Proofs of Lemmata

In this section we provide the proofs of Lemmata 19 and 21 dealing with theproperties of the operators A1 and N1 which we used in the previous sections.

Proof of Lemma 19 The properties (a), (d) and (e) are established in Lemma 11 of[18]. The Lipschitz condition (c) follows easily from H(A)(3). We prove (b). Letv ∈ V . First, by using the boundedness of A (cf. Remark 14), we infer

∫ T

0〈A(t, v(t) + u1), u1〉 dt ≤ ‖u1‖

∫ T

0‖A(t, v(t) + u1)‖V∗ dt ≤

≤ LC‖u1‖∫ T

0‖v(t) + u1‖ dt + ‖u1‖

∫ T

0a1(t) dt ≤

≤ LC‖u1‖√

T‖v‖V + LCT ‖u1‖2 + ‖u1‖‖a‖L2(0,T).


Exploiting this estimate, H(A)(5), the inequality |r + s|2 ≥ 1/2|r|2 − |s| for r, s ∈ R

and∫ T

0a(t)‖v(t) + u1‖ dt ≤ ‖a‖L2(0,T)‖v‖V + √

T‖a‖L2(0,T)‖u1‖,

we have

〈〈A1v, v〉〉V∗×V =∫ T

0

(〈A(t, v(t) + u1), v(t) + u1〉 − 〈A(t, v(t) + u1), u1〉

)dt ≥

≥∫ T

0

(α‖v(t)+u1‖2−a(t)‖v(t)+u1‖

)dt−

∫ T

0〈A(t, v(t)+u1), u1〉 dt ≥

≥ α

2‖v‖2

V − β1‖v‖V − β2

with β1, β2 ≥ 0. This completes the proof of Lemma 19. ��

Proof of Lemma 21 The property (a) follows from the properties of the subdifferen-tial (cf. Lemma 13 in [18]). We will show (b) and (b’). Let v ∈ V and z ∈ N1v. Hencez(t) ∈ ∂ J(t, g(t), v(t) + u1) for a.e. t ∈ (0, T). If H(J) holds, then

‖z(t)‖Z ∗ ≤ 2c0‖μ‖ ‖γ ‖ ‖pg‖L2(�C) for a.e. t ∈ (0, T)

which immediately yields (b). If H(J)1 holds, then

‖z(t)‖Z ∗ ≤ c(1 + ‖v(t) + u1‖Z ) for a.e. t ∈ (0, T)

which implies

‖z‖Z∗ =(∫ T

0c 2‖μ‖2(1 + ‖v(t) + u1‖Z )2 dt

)1/2

≤ c‖μ‖ (2(T + β2‖v + u1‖2

V ))1/2 ≤

≤ √2 c ‖μ‖

(√T + β‖v‖V + β

√T‖u1‖

)≤ c2‖μ‖(1 + ‖v‖V )

with c2 = √2 c max{β,

√T(1 + β‖u1‖)}. Hence (b’) follows.

In order to demonstrate the property (c), let v ∈ V and z ∈ N1v. Thusz(t) ∈ ∂ J(t, g(t), v(t) + u1) for a.e. t ∈ (0, T). Under the hypotheses H(J), fromLemma 21(b), we have

−〈〈z, v〉〉V∗×V = −〈〈z, v〉〉Z∗×Z ≤ ‖z‖Z∗‖v‖Z ≤ c1β‖μ‖‖v‖V .

If H(J)1 holds, then by Lemma 21(b’), we get

−〈〈z, v〉〉V∗×V ≤ ‖z‖Z∗‖v‖Z ≤ c2β‖μ‖(1 + ‖v‖V )‖v‖V = c2β‖μ‖‖v‖2V + c2β‖μ‖‖v‖V

which ends the proof of (c’).Finally, we will show (d). Let vn, v ∈ V , vn → v in Z , zn, z ∈ Z∗, zn → z weakly

in Z∗ and zn ∈ N1vn. By passing to a subsequence if necessary, we have vn(t) →v(t) in Z for a.e. t ∈ (0, T). Taking into account that ∂ J(t, g(t), ·) : Z → 2Z ∗

is uppersemicontinuous with closed and convex values (cf. [5], since J(t, g(t), ·) is convex, ∂ Jcoincides with the Clarke subdifferential), we apply the Convergence Theorem (cf.[4], p.60) to the inclusion

zn(t) ∈ ∂ J(t, g(t), vn(t) + u1) for a.e. t ∈ (0, T)


and we get z(t) ∈ ∂ J(t, g(t), v(t) + u1) for a.e. t ∈ (0, T). This implies that z ∈ N1v,proves (d) and completes the proof of the lemma. ��Acknowledgement The authors would like to thank the referees for their comments.

References

1. Amassad, A., Fabre, C.: On the analysis of viscoplastic contact problem with time dependentTresca’s friction law. Electron. J. Math. Phys. Sci. 1(1), 47–71 (2002)

2. Amassad, A., Fabre, C.: Analysis of viscoelastic unilateral contact problem involving Coulombfriction law. J. Optim. Theory Appl. 116(3), 465–483 (2003)

3. Andersson, L.-E.: Existence results for quasistatic contact problems with Coulomb friction. Appl.Math. Optim. 42, 169–202 (2000)

4. Aubin, J.P., Cellina, A.: Differential Inclusions. Set-valued Maps and Viability Theory. Springer,Berlin Heidelberg New York (1984)

5. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)6. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis:

Theory. Kluwer, Boston, MA (2003)7. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Appli-

cations. Kluwer, Boston, MA (2003)8. Duvaut, G.: Loi de frottement non locale. J. Méc. Thé. Appl. Special issue, pp. 73–78 (1982)9. Duvaut, G., Lions, J.-L.: Les Inéquations en Mécanique et en Physique. Dunod, Paris (1972)

10. Eck, C., Jarušek, J.: Existence of solutions for the dynamic frictional contact problem of isotropicviscoelastic bodies. Nonlinear Anal. 53, 157–181 (2003)

11. Han, W., Sofonea, M.: Time-dependent variational inequalities for viscoelastic contact problems.J. Comput. Appl. Math. 136, 369–387 (2001)

12. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity.American Mathematical Society, Providence, RI (2002)

13. Hild, P.: An example of nonuniqueness for the continuous static unilateral contact model withCoulomb friction. C. R. Acad. Sci. Paris Sér. I Math. 337, 685–688 (2003)

14. Jarušek, J., Eck, C.: Dynamic contact problems with small Coulomb friction for viscoelasticbodies. Existence of solutions. Math. Methods Appl. Sci. 9, 11–34 (1999)

15. Kuttler, K.L., Shillor, M.: Dynamic bilateral contact with discontinuous friction coefficient.Nonlinear Anal. 45, 309–327 (2001)

16. Kuttler, K.L., Shillor, M.: Dynamic contact with Signorini’s condition and slip rate dependentfriction. Electron. J. Differential Equations 2004(83), 1–21 (2004)

17. Lions, J.L.: Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod,Paris (1969)

18. Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem withnormal damped response and friction. Appl. Anal. 84, 669–699 (2005)

19. Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J.Elasticity 83, 247–275 (2006)

20. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities andApplications. Marcel Dekker, New York (1995)

21. Rochdi, M., Shillor, M.: Existence and uniqueness for a quasistatic frictional bilateral contactproblem in thermoviscoelasticity. Quart. Appl. Math. LVIII(3), 543–560 (2000)

22. Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer,Berlin Heidelberg New York (2004)

23. Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standardmodel for contact, friction and wear. Internat J. Solids Structures 33, 1817–1836 (1996)

24. Zeidler, E.: Nonlinear Functional Analysis and Applications II A/B. Springer, Berlin HeidelbergNew York (1990)

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