Hassan Boualem
email: hassan.boualem@univ-montp2.fr

Robert Brouzet
email: robert.brouzet@univ-perp.fr

To Be (a Circle) or Not to Be?

Montpellier in 1992, where he studied group actions and foliations theory and continues to teach. He is also

Is the curve Ŵ given by √ x + √ y = 1for(x, y) ∈ [0, 1] 2 the arc of a circle? Plotting the curve, Figure 1, shows why this is a natural question. This exercise was given as an open question in a French high school. Despite the visual impression, this curve is not the arc of a circle. In a tour of properties and characterizations of the circle, we detail or outline 14 proofs! After such a definitive resolution to the original question, we establish that the curve is indeed the arc of another conic section. Generalizing, our curve is part of a family Ŵ r of curves whose Cartesian equations have the form x r + y r = 1. For which r is Ŵ r an arc of a conic? Although general conic sections have fewer nice properties than circles, we answer the problem in two ways, using curvature and algebra.

The Ŵ r are part of a family of curves studied by Lamé known as superellipses. We conclude by reviewing some of his results on these curves and looking at our questions in that context.

Many proofs for "not to be"

We begin with an abundance of proofs for the following claim. The 14 arguments, some of which are given as exercises, are divided among four methods: elementary geometry, analytic geometry, parametric curves, and calculus. Proposition 1. The curve Ŵ given by √ x + √ y = 1 for (x, y) ∈ [0, 1] 2 is not an arc of a circle.

As an initial step, write √ y = 1 -√ x and square to see that Ŵ is the graph of the function f

(x) = x -2 √ x + 1forx ∈ [0, 1].
Elementary geometry. Our first proof is based on the fact that three noncollinear points determine a circle. The subsequent exercise leads to a proof using the result that all perpendicular bisectors of chords on a circle must pass through the center.

Proof 1. It is easy to identify three noncollinear points of Ŵ, e.g., (0, 1), (1, 0), (1/4, 1/4). The general equation of a circle is x 2 + y 2 + αx + βy + γ = 0. Substituting the coordinates of our three points shows that α, β, γ satisfy the system

β + γ =-1,α+ γ =-1,α+ β + 4γ =-1 2 .
An easy calculation gives α = β =-7/4 and γ = 3/4. Thus, the only circle that can contain the curve Ŵ has equation

x 2 + y 2 -7
4 x -7 4 y + 3 4 = 0. However, the point (1/9, 4/9) on Ŵ does not satisfy this equation: Verify that its coordinates give -1/81 rather than 0. This gives a contradiction.

Exercise 2. Show that segments with endpoints in Ŵ do not have perpendicular bisectors that pass through the center of the circle.

Hint: Use the points in Proof 1.

Analytic geometry.

Here, we give a very natural and elementary proof using the definition of a circle as the set of points in the plane whose distance to a fixed point (the center) is constant. Then we outline four related proofs whose details are left as exercises.

Proof 3. We proceed by contradiction. Assume that Ŵ is an arc of a circle with center (a, b). In this case, the function defined on [0, 1] by

ϕ(x) = (x -a) 2 + (x -2 √ x + 1 -b) 2
is constant and so has a vanishing derivative on (0, 1]. Therefore, for all x ∈ (0, 1],

ϕ ′ (x) = 2(x -a) + 2(x -2 √ x + 1 -b) 1 - 1 √ x = 0
and so (with

X = √ x) the polynomial 2X 3 -3X 2 + (3 -a -b)X + b -1 has an infinite set of roots, a contradiction.
One can determine the only possible values for the center coordinates a and b in several ways. Here is a method using the fact that the tangent to a circle at a given point is perpendicular to the radius at that point.

The function f has a derivative on (0, 1] given by f ′ (x) = 1 -1/ √ x. Because f ′ (1) = 0, the tangent to Ŵ is horizontal at the point (1, 0). Moreover, the curve is symmetric relative to the line y = x, so the tangent at the point (0, 1) is vertical.

Because the tangent at a point of a circle is perpendicular to the corresponding radius, the circle C potentially containing Ŵ has center (1, 1) and radius 1. Figure 2 shows Ŵ and the relevant quarter of C. The following exercises lead to other proofs of Claim 1 based on the necessary parameters of C. Exercise 4. Show that the only value of x satisfying (x -1) 2 + ( f (x) -1) 2 = 1is x = 1.

Exercise 5. Calculate the distance between the point (1/4, 1/4) on Ŵ and (1, 1). Exercise 6. Show that each point of Ŵ must satisfy x 2 + y 2 -2xy -2x -2y + 1 = 0. Compare this equation with that of the "necessary circle" C. Exercise 7. If Ŵ were an arc of a circle, then its tangent lines would be perpendicular to the radius at each point, and so its normal lines would pass through a fixed point (the center of the circle). Show that it is not the case.

Hint: Proceed by contradiction; give an equation of the normal line at (a, f (a)) and show that all normals must pass through (1, 1). Use algebra to conclude that this will only be true for a = 1 and a = 1/4.

Parametric curves.

Here are proofs based on two parametrizations of the circle.

Proof 8. We again assume that Ŵ is an arc of a circle with center (a, b) and radius R so that Ŵ can be parametrized as

x(θ) = a + R cos θ, y(θ) = b + R sin θ for θ in some interval I ⊂ [0, 2π].
We can specify I because we have already seen that, if Ŵ is an arc of a circle, then it is a quarter of the circle; thus, I = [π, 3π/2]. We also know from other proofs that a = b = R = 1. So, according to the expression of f , for all θ ∈ I we have sin θ = cos θ -2 √ 1 + cos θ + 1. Differentiating with respect to θ gives cos θ =-sin θ + sin θ/ √ 1 + cos θ. Taking θ = 5π/4 ∈ I , we obtain the equality

- √ 2 2 = √ 2 2 - √ 2 2 1 - √ 2 2
, which simplifies to 2 1 -√ 2/2 = 1. But this is impossible because, after squaring, we obtain √ 2 = 3/2 (and, after squaring again, 8 = 9).

Exercise 9. A circle is a unicursal curve, i.e., it admits a rational parametrization, namely the one obtained from the previous trigonometric parametrization by using the parameter t = tan(θ/2). Show that if Ŵ was the quarter of a circle, then 1t = 2(1 + t 2 ) for all t ∈ (-∞, -1] leading, for t =-2, to the impossibility 9 = 10.

Calculus. We finish our proofs with ones using concepts from calculus: area, arc length, curvature, and smoothness.

Exercise 10. If Ŵ were the quarter of circle we have discussed, then the area delimited by it would be π/4. But computing the area between y = 1 and f leads to the impossibility π/4 = 1 -1 0 f (x) dx = 5/6. Proof 11. Instead of the using the area of the quarter of circle, we can consider the arc length, which would be π/2. Since Ŵ is concave up over the region, we know that its arc length is greater than the combined length of the line segments from (0, 1) to (1/4, 1/4) and from (1/4, 1/4) to (1, 0). But this combined length is √ 10/2, which is greater than π/2 since 10 >π 2 .

Although the lower bound in Proof 11 suffices, we can compute the exact arc length of Ŵ as outlined in the next exercise.

Exercise 12. The length of Ŵ is given by the formula [START_REF] Berger | Geometry I and II[END_REF][START_REF] Docarmo | Differential Geometry of Curves and Surfaces[END_REF]

ℓ(Ŵ) = 1 0 1 + f ′ (x) 2 dx. 1. Prove that ℓ(Ŵ) = √ 2 1 0 √ u 2 + 1 du.

Use the hyperbolic functions to show ℓ(

Ŵ) = 1 + √ 2ln 1 + √ 2 /2.
3. Using just the rough inequalities π<3.2 and e < 2.72, prove that

π 2 < 1 + √ 2 2 ln 1 + √ 2 .
Recall that curvature measures the intensity of the force needed to keep the particle on the track of the curve and also measures how quickly the curve changes direction. More precisely, when the curve is parametrized with respect to arc length, independent of the parametrization, the curvature is defined in terms of the acceleration vector. For example, the curvature of the straight line is identically zero so that the circle of radius R has constant curvature 1/R.

Exercise 13. The curvature c(x) of an arc with Cartesian equation y = f (x) at the point (x, f (x)) is given by the formula [1, 2, 3]

c 2 (x) = ( f ′′ (x)) 2 (1 + ( f ′ (x)) 2 ) 3 . (1)
Calculate the curvature of Ŵ at multiple values, say 1/2 and 1, to show that it is not constant, as required for an arc of a circle.

Finally, recall that a smooth (C ∞ ) real function of one real variable has, at all defined points, derivatives of all orders.

Proof 14. Let s denote reflection across the vertical line x = 1. If Ŵ were an arc of a circle, because it would have a horizontal tangent at (1, 0) (as already noted in others proofs), then Ŵ ∪ s(Ŵ) would be an arc of this same circle (actually a half circle, although we will not need that specificity). Let g represent the symmetric arc; for all x ∈ [1, 2] we must have g(x) = f (2x). Thus, a Cartesian equation of the set

Ŵ ∪ s(Ŵ) is y = f (x) = x -2 √ x + 1i f x ∈ [0, 1], g(x) =-x -2 √ 2 -x + 3i f x ∈ [1, 2].
The junction at x = 1 would be smooth if Ŵ ∪ s(Ŵ) were an arc of a circle. However, (3) (1) =-3/4 and g (3) (1) = 3/4. All is well for the two first derivatives but not for the third, so Ŵ ∪ s(Ŵ) is not smooth, and once again, Ŵ is not an arc of a circle. ) that each point of Ŵ satisfies x 2 + y 2 -2xy -2x -2y + 1 = 0. We recognize in this the equation of a conic [START_REF] Audin | Geometry[END_REF][START_REF] Berger | Geometry I and II[END_REF] whose nature is determined by the quadratic form q = x 2 + y 2 -2xy. This one is degenerate because q = (xy) 2 , and so this conic is a parabola; see Figure 3. Precisely, in the new coordinates X = xy and Y = x + y (i.e., use the two bisectors as new axes), we get the equation X 2 -2Y + 1 = 0, and a translation of the origin brings us to Y ′ = X 2 .SoŴ is an arc of a parabola, not an arc of circle.

f ′ (x) = 1 - 1 √ x , f ′′ (x) = 1 2x 3/2 , f (3) (x) = -3 4x 5/2 , g ′ (x) =-1 + 1 √ 2 -x , g ′′ (x) = 1 2(2 -x) 3/2 , g (3) (x) = 3 4(2 -x) 5/2 , so, at x = 1, we have f (1) = g(1) = 0, f ′ (1) = g ′ (1) = 0, f ′′ (1) = g ′′ (1) = 1/2, but f
To explain that Ŵ is an arc of parabola, we can also give the following elementary argument (suggested by one of the anonymous referees). By rotating the axes 45 • clockwise, the equation becomes

X √ 2 + Y √ 2 + X - √ 2 + Y √ 2 = 1, so √ X + Y = 2 1/4 - √ Y -X . Squaring leads to √ Y -X = 2 -3/4 -2 -1/4
X and, squaring again, Y = 2 -3/2 + 2 -1/2 X 2 , clearly the equation of a parabola.

The equal exponent family of curves

After the proofs that the curve √ x + √ y = 1f o r(x, y) ∈ [0, 1] 2 is not an arc of a circle, we showed that it is an arc of a parabola. The curve Ŵ is the r = 1/2 case of a family Ŵ r with Cartesian equation x r + y r = 1forr ∈ (0, +∞) and (x, y)

∈ [0, 1] 2 .
Of course, in the case r = 2, we have an arc of a circle and, for r = 1, an arc of a straight line, i.e., a segment. It is natural to ask for which r the family of curves contains arcs of conics [START_REF] Audin | Geometry[END_REF][START_REF] Berger | Geometry I and II[END_REF]. We show next that we have found all such.

Proposition 2. The curve Ŵ r given by x r + y r = 1 for r ∈ (0, +∞) and (x, y) ∈ [0, 1] 2 is an arc of a conic if and only if r ∈{1/2, 1, 2}.

Just the "only if" direction of the claim requires proof because we already know that in the cases r = 1/2, 1, 2 the curves are conics. We want to prove this assertion using two methods, first using curvature, second by algebra.

Proof. We proceed in two steps:

(i) According to the value of r ∈ (0, +∞) \{1/2, 1, 2}, we show that the curvature approaches either 0 or infinity in a neighborhood of x = 1. (ii) We verify that conics do not have this property.

Choosing x for the parameter, the curvature formula (1) gives c 2 = y ′′2 /(1 + y ′2 ) 3 . In our case, we have y = (1x r ) 1/r for x ∈ (0, 1). Easy calculations give

y ′ =-x r-1 (1 -x r ) 1 r -1 , y ′′ = (1 -r )x r-2 (1 -x r ) 1 r -2
so that we can write c as

c(x) = |1 -r |x r-2 (1 -x r ) 1-2r r 1 + x 2(r-1) (1 -x r ) 2(1-r ) r 3 2 . If r < 1, then c(x) ∼|1 -r |(1 -x r
) (1-2r )/r as x → 1 with limit 0 if r ∈ (0, 1/2) and limit +∞ if r ∈ (1/2, 1).I fr > 1, then c(x) ∼|1r |(1x r ) (r-2)/r as x → 1 with limit +∞ if r < 2 and limit 0 if r > 2. These calculations show that for r ∈ (0, +∞) \{1/2, 1, 2}, the curvature can be as small or as large as we want, according to the values of r , in a neighborhood of the point (1, 0). We now proceed to our second step.

Again, curvature does not depend on the regular parametrization of the curve. So we can calculate curvature of the three kinds of conics using standard parametrizations, given below with their curvatures calculated from (1).

• The ellipse x(t) = a cos t, y(t) = b sin t (we assume a ≥ b > 0) has curvature 3 .

c 2 E (t) = a 2 b 2 (a 2 sin 2 t + b 2 cos 2 t)
• The hyperbola 3 .

x(t) = a cosh t, y(t) = b sinh t (again a ≥ b > 0) has curvature c 2 H (t) = a 2 b 2 (a 2 sinh 2 t + b 2 cosh 2 t)
• The parabola y = x 2 /2 p has curvature c 2 P (x) = p 4 /( p 2 + x 2 ) 3 .

For an ellipse, because

a 2 sin 2 t + b 2 cos 2 t = b 2 + (a 2 -b 2 ) sin 2 t ≥ b 2 , since a ≥ b > 0, we see that, for all real t, a 2 b 2 (a 2 + b 2 ) 3 ≤ c 2 E (t) ≤ a 2 b 4 .
Since that curvature cannot approach zero or infinity, our arc cannot be an arc of an ellipse.

The two others curvatures have upper bounds, namely

c 2 H ≤ a 2 b 4 , c 2 P ≤ 1 p 2 .
These can approach zero at infinity, but that is not a problem since our arc is compact. Thus, our arc cannot be an arc of a hyperbola or a parabola either.

Proof. The general equation of a conic is ax 2 + 2bxy + cy 2 + dx + ey + f = 0 with not all of a, b, c, d, e being zero [START_REF] Audin | Geometry[END_REF].

If our arc is an arc of a conic, then there exists some real numbers a, b, c, d, e, f such that we have, for all x ∈ [0, 1],

ax 2 + 2bx(1 -x r ) 1 r + c(1 -x r ) 2 r + dx + e(1 -x r ) 1 r + f = 0.
The idea now is to look at the limited expansion near 0 of the function appearing in the left-hand side. In this expansion, we can find constant terms and x, x 2 , x r , x 2r , x 3r , ...; x r+1 , x 2r+1 , x 3r+1 , .... Of course, these terms can be the same for certain values of r . We have two cases.

• r > 1 and r = 2. In this case, the only x 2 term is ax 2 Among negative values for the common exponent, r =-1 gives an arc of hyperbola: 1/x + 1/y = 1f o r(x, y) ∈ (0, 1) 2 is equivalent to (x -1)(y -1) = 1, which we recognize a hyperbola with center (1, 1). We remark without proof that the arcs are not arcs of conics for other negative values of r .

Lam é's curves

The Ŵ r curves belong to the family of curves studied by G. Lamé[ 4] in the 19th century and now called Lamé's curves, or superellipses. Namely, such a Lamé's curve has Cartesian equation

x α a α + y α b α = 1,
where a and b are positive real numbers and α is a real number. For some values of α (e.g., α = 2), the Cartesian coordinates x and y can be negative, but for some other values (e.g., our previous case α = 1/2), they must be positive. In order to avoid supplementary difficulties and to stay in the previous framework, we only work with positive coordinates and so define the Lamé's curve L(α, a, b) as

L(α, a, b) = (x, y) ∈ [0, +∞) 2 x α a α + y α b α = 1 .
In order to consider the following proof by Lamé, we introduce the envelope of a family of curves [START_REF] Audin | Geometry[END_REF]. This is the curve that is tangent to each member of the family at some point. Thus, the envelope of a family of curves (C λ ) λ , each of them defined by a Cartesian equation F(x, y,λ) = 0, is obtained by solving the system of equations

1 γ = 1 α + 1 β .
Proof. The parameter here is a pair λ = (m, n) constrained by the condition

m β a β + n β b β = 1
and so we can write the family of curves with only the only parameter m as and therefore to the desired equation,

F(x, y, m) = x α m α + y α b α 1 -m β
x γ a γ + y γ b γ = 1.
Lamé went on to use the this result for some geometric constructions [START_REF] Lamé | Examen des différentes méthodes employées pour résoudre les problèmes de Géométrie[END_REF]. Here we give examples illustrating this theorem in the cases when Lamé's curves are conics, which we know from Proposition 2 to be only those for which the exponent r ∈{ 1/2, 1, 2}. To realize the condition 1/γ = 1/α + 1/β using only these three values, the only possibilities are α = β = 2,γ = 1 and α = β = 1,γ = 1/2.

• From the first case, the envelope of the family of ellipses (given by their equations)

x 2 m 2 + y 2 n 2 = 1 (m,n)∈E
, where E is the ellipse

x 2 a 2 + y 2 b 2 = 1,
is the line x/a + y/b = 1.

• From the second case, the envelope of the family of lines (given by their equations)

x m + y n = 1 (m,n)∈L ,
where L is the line x/a + y/b = 1, is the parabola

x a 1/2 + y b 1/2 = 1,
a slightly generalized form of our motivating problem.

1 1Figure 1 .

 11 Figure 1. The curve Ŵ

1 1Figure 2 .

 12 Figure 2. The curve Ŵ (full line) and the arc of circle C (dashed line)

Figure 3 .

 3 Figure 3. The circle and the parabola

1 -x r ) 1 r 1 e

 111 ,soa = 0. The only x r+1 term is -(2b/r )x r+1 ,sob = 0. Then only dx remains as the only x term, so d = 0. Now x = 1gives f = 0. Simplifying the remaining express gives c (+ e = 0. Again x = 1 implies e = 0 and finally c = 0. So we obtain a = b = c = d = e = 0, a contradiction. • 0 < r < 1 and r = 1/2: The powers of x in the six terms are 2, kr + 1, lr,1,lr,0, successively, for positive integers k, l. The terms x r and x 2r appear only in the c and e factors (because r is not 1/2), thus -= 0, so 2c + e = 0 and (2r )c + (1r )e = 0. It follows that c = e = 0 and then also the constant term f = 0. Now there is only one x r+1 term, so b = 0. We conclude a = d = 0 and so get the same contradiction as in the previous case.

FTheorem 3 (

 3 (x, y,λ) = 0, ∂ F ∂λ (x, y,λ) = 0. Lam é). The envelope of the family of curves L(α, m, n) when the parameter (m, n) describes the curve L(β, a, b) is the curve L(γ, a, b) with

- 1 = a m β a β 1 α + 1 β.= b 1 - m β a β 1 α + 1 β.

 11111 = 0.The equation ∂ F ∂m (x, y,λ) = 0 leads to the relationx α m α+1 + m β-1 a βm β 1 -x α m α = 0from which we can obtain x α = m α+β /a β or x = m(m/a) β/α that we prefer finally to write as x It follows from this calculation and the fact that the point (x, y) belongs to the curve L(α, m, n) that y If γ is the real number defined by 1/γ = 1/α + 1/β, then the previous values of x and y lead to

Acknowledgments. We thank Fanny Ayache for informing us of the motivating problem for this article, our colleague Sridharan Sriraman for his proofreading, the anonymous referees for helpful advice, and the editor for ameliorating the text.

Summary. We begin with a high school exercise, determining whether a particular equation describes an arc of a circle. We give several proofs that it does not, exploring many properties of circles along the way. Yet the curve is the arc of a conic. We then explore which equations of a generalized family are arcs of conics and give some properties of the resulting Lamécurves.


Mircea Sofonea
email: sofonea@univ-perp.fr

Weimin Han
email: weimin-han@uiowa.edu

Mikaël Barboteu
email: barboteu@univ-perp.fr

A Variational-Hemivariational Inequality in Contact Mechanics

This chapter deals with a new mathematical model for the frictional contact between an elastic body and a rigid foundation covered by a deformable layer made of soft material. We study the model in the form of a variational-hemivariational inequality for the displacement field. We review a unique solvability result of the problem under certain assumptions on the data. Then we turn to the numerical solution of the problem, based on the finite element method. We derive an optimal order error estimate for the linear finite element solution. Finally, we present numerical simulation results in the study of a two-dimentional academic example. The theoretically predicted optimal convergence order is observed numerically. Moreover, we provide mechanical interpretations of the numerical results for our contact model.

Introduction

Phenomena of contact involving deformable bodies abound in industry and daily life. Due to their inherent complexity, they lead to mathematical models expressed in terms of nonlinear boundary value problems which, in variational formulation, give rise to challenging inequality problems. Analysis of these problems is based on arguments of nonlinear functional analysis through the theory of variational and hemivariational inequalities.

 and the references therein. It was built on arguments of monotonicity and convexity, including properties

Ŵ 2 and Γ 3 such that meas (Γ 1 )>0. The body is in equilibrium under the action of a body force of density f 0 in Ω and a surface traction of density f 2 on Γ 2 ,isfixedonΓ 1 , and is in frictional contact on Γ 3 with a foundation. We use S d for the space of second order symmetric tensors on R d .Also,"•" and " • " will represent the canonical inner product and the Euclidean norm on the spaces R d and S d . We denote by u : Ω → R d and σ : Ω → S d the displacement field and the stress field, respectively. In addition, we use ε(u) to denote the linearized strain tensor. Let ν be the unit outward normal vector, defined a.e. on Γ . For a vector field v,weusev ν := v • ν and v τ := vv ν ν for the normal and tangential components of v on Γ . Similarly, for the stress field σ , its normal and tangential components on the boundary are defined as σ ν := (σ ν) • ν and σ τ := σ νσ ν ν, respectively.

With the above notation, the contact model to be studied is the following. Problem P. Find a displacement field u : Ω → R d , a stress field σ : Ω → S d and an interface force ξ ν :

Γ 3 → R such that σ = F (ε(u)) in Ω, (1)
Div σ + f 0 = 0 in Ω, (2)
u = 0 on Γ 1 , (3)
σ ν = f 2 on Γ 2 , (4)
u ν ≤ g,σ ν + ξ ν ≤ 0,(u ν -g)(σ ν + ξ ν ) = 0,ξ ν ∈ ∂ j ν (u ν ) on Γ 3 , (5)
σ τ ≤F b (u ν ), -σ τ = F b (u ν ) u τ u τ if u τ = 0 on Γ 3 . (6)
In ( 1)-( 6) and sometimes below, we do not indicate explicitly the dependence of various functions on the spatial variable x ∈ Ω ∪ Γ . We now present a short description of the equations and conditions in Problem P and we refer the reader to the books [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems[END_REF][START_REF] Sofonea | Mathematical Models in Contact Mechanics[END_REF] for more details on the modelling of contact problems. First, Eq. ( 1) is the constitutive law for elastic materials in which F represents the elasticity operator, allowed to be nonlinear. Equation ( 2) is the equilibrium equation and is used here since the process is assumed to be static. Condition (3) represents the displacement condition and condition (4) is the traction condition. Relations ( 5) and ( 6) represent the contact condition and the friction law, respectively. Here g ≥ 0, ∂ j ν denotes the Clarke subdifferential of the given function j ν , and F b denotes a positive function, the friction bound.

Note that condition (5) models the contact with a foundation made of a rigid body covered by a layer of soft material, say asperities. It is obtained through the following considerations:

(a) The penetration is restricted by the rigid body, i.e.

u ν ≤ g, ( 7
)
where g ≥ 0 represents the thickness of the soft layer. We consider the nonhomogeneous case, i.e., g is allowed to be a function of the spatial variable x ∈ Γ 3 . (b) The normal stress has an additive decomposition of the form

σ ν = σ D ν + σ R ν , (8)
where the term σ D ν describes the reaction of the soft layer and σ R ν describes the reaction of the rigid body. (c) The component σ D ν satisfies a multivalued normal compliance condition of the form

-σ D ν ∈ ∂ j ν (u ν ). (9)
Examples of contact conditions of the form (9) can be found in [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems[END_REF], for instance. (d) The component σ R ν satisfies the Signorini unilateral condition in a form with the gap g, i.e.

σ R ν ≤ 0,σ R ν (u ν -g) = 0. ( 10
)
Comments and mechanical interpretation on the contact condition (10) can be found in [START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] and the references therein.

Denote -σ D ν = ξ ν . Then, it is easy to see that the contact condition ( 5) is a direct consequence of relations ( 7)- [START_REF] Han | A class of variational-hemivariational inequalities with applications to frictional contact problems[END_REF].

The friction law (6) was used in [START_REF] Sofonea | Numerical analysis of history-dependent variational inequalities with applications to contact problems[END_REF], associated with a multivalued normal compliance contact condition without unilateral constraint. Here the friction bound F b may depend on the normal displacement u ν , which is reasonable from the physical point of view, as explained in [START_REF] Sofonea | Numerical analysis of history-dependent variational inequalities with applications to contact problems[END_REF].

Note that, due to the strong nolinearities involved, in general Problem P does not have classical solution. Therefore, as usual in Contact Mechanics, its study is made by using a weak formulation, the so-called variational formulation. The formulation will allow one to prove the unique solvability of the problem and to construct numerical schemes for the approximation of the weak solution.

Variational Analysis

In the study of Problem P we use standard notation for Lebesgue and Sobolev spaces. For the stress and strain fields, we use the space Q = L 2 (Ω; S d ), which is a Hilbert space with the canonical inner product

(σ , τ ) Q := Ω σ ij (x)τ ij (x) dx, σ , τ ∈ Q
and the associated norm • Q . The displacement fields will be sought in a subset of the space

V = v = (v i ) ∈ H 1 (Ω; R d ) | v = 0 on Γ 1 .
Since meas (Γ 1 )>0, it is known that V is a Hilbert space with the inner product

(u, v) V := Ω ε(u) • ε(v) dx, u, v ∈ V
and the associated norm • V . We denote by V * the topological dual of V , and by •, • V * ×V the duality pairing of V and V * . When no confusion may arise, we simply write

•, • instead of •, • V * ×V .Forv ∈ H 1 (Ω; R d )
we use the same symbol v for the trace of v on Γ . By the Sobolev trace theorem we have

v L 2 (Γ 3 ;R d ) ≤ Ŵ v V ∀ v ∈ V, (11)
γ being the norm of the trace operator γ :

V → L 2 (Γ 3 ; R d ).
We now turn to the assumptions on the data. First, the elasticity operator F : Ω × S d → S d and the potential function j ν : Γ 3 × R → R, are assumed to have the following properties:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (a) there exists L F > 0 such that for all ε 1 , ε 2 ∈ S d , a.e. x ∈ Ω, F (x, ε 1 ) -F (x, ε 2 ) ≤L F ε 1 -ε 2 ; (b) there exists m F > 0 such that for all ε 1 , ε 2 ∈ S d , a.e. x ∈ Ω, (F (x, ε 1 ) -F (x, ε 2 )) • (ε 1 -ε 2 ) ≥ m F ε 1 -ε 2 2 ; (c) F (•, ε) is measurable on Ω for all ε ∈ S d ; (d) F (x, 0) = 0 for a.e. x ∈ Ω. (12) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (a) j ν (•, r ) is measurable on Γ 3 for all r ∈ R and there exists ē ∈ L 2 (Γ 3 ) such that j ν (•, ē(•)) ∈ L 1 (Γ 3 ); (b) j ν (x, •) is locally Lipschitz on R for a.e. x ∈ Γ 3 ; (c) |∂ j ν (x, r )|≤ c0 +c 1 |r | for a.e. x ∈ Γ 3 , for all r ∈ R with c0 , c1 ≥ 0; (d) j 0 ν (x, r 1 ; r 2 -r 1 ) + j 0 ν (x, r 2 ; r 1 -r 2 ) ≤ α j ν |r 1 -r 2 | 2 for a.e. x ∈ Γ 3 , all r 1 , r 2 ∈ R with α j ν ≥ 0. ( 13
)
On the penetration bound g : Γ 3 → R and the friction bound

F b : Γ 3 × R → R + ,we assume g ∈ L 2 (Γ 3 ), g(x) ≥ 0a .e. on Γ 3 , (14)
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (a) there exists L F b > 0 such that |F b (x, r 1 ) -F b (x, r 2 )|≤L F b |r 1 -r 2 |∀ r 1 , r 2 ∈ R, a.e. x ∈ Γ 3 ; (b) F b (•, r ) is measurable on Γ 3 , for all r ∈ R; (c) F b (x, r ) = 0forr ≤ 0, F b (x, r ) ≥ 0forr ≥ 0, a.e. x ∈ Γ 3 . (15)
Finally, on the densities of the body force and the surface traction, we assume

f 0 ∈ L 2 (Ω; R d ), f 2 ∈ L 2 (Γ 2 ; R d ). ( 16
) Define f ∈ V * by f, v V * ×V = (f 0 , v) L 2 (Ω;R d ) + (f 2 , v) L 2 (Γ 2 ;R d ) ∀ v ∈ V. ( 17
)
Corresponding to the constraint u ν ≤ g on Γ 3 in (5), we introduce the following subset of the space V :

U := {v ∈ V | v ν ≤ g on Γ 3 } . ( 18
)
Also, we use the notation j 0 ν (u, v) for the generalized directional derivative of j ν at u ∈ R in the direction v ∈ R, defined by

j 0 ν (u; v) := lim sup y→u,λ↓0 j ν (y + λv) -j ν (y) λ .
Then, from the definition of Clarke subdifferential the following implication holds:

ξ ν ∈ ∂ j ν (u ν ) a.e. on Ŵ 3 =⇒ j 0 ν (u ν ; v ν ) ≥ ξ ν v ν a.e. on Ŵ 3 , ∀ v ∈ V. ( 19
)
By a standard approach, based on integration by parts and the inequality [START_REF] Migórski | A Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces[END_REF], the following weak formulation of the contact problem P can be derived.

Problem P V . Find a displacement field u ∈ U such that (F (ε(u)), ε(v -u)) Q + Γ 3 F b (u ν ) ( v τ -u τ ) dΓ + Γ 3 j 0 ν (u ν ; v ν -u ν ) dΓ ≥ f, v -u V * ×V ∀ v ∈ U. ( 20
)
Note that the inequality (20) has both a convex and nonconvex structure. Its convex structure is given by the subset of the admisible displacement fields U , which is convex, and the function

v → Γ 3 F b (u ν ) v τ dΓ,
which is a convex function on V . The nonconvex structure of the inequality (20) follows from the term

Γ 3 j 0 ν (u ν ; v ν -u ν ) dΓ
which involves a possibly nonconvex locally Lipschitz functions j ν . We conclude from here that the inequality (20) represents a variational-hemivariational inequality.

The analysis of inequalities of the form (20) has been carried out in [START_REF] Han | Numerical Analysis of Stationary Variational-Hemivariational Inequalities[END_REF][START_REF] Migórski | A Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces[END_REF], in an abstract functional framework. There, a general existence and uniqueness result for inequalities with pseudomonotone operators was provided, under a smallness assumption on the data. The use of this abstract result in the study of (20) is straighforward and, therefore, we skip it. The main point is the use of smallness assumption, that we describe in what follows.

Let λ 1,V > 0 be the smallest eigenvalue of the eigenvalue problem

u ∈ V, Ω ε(u)•ε(v) dx = λ Γ 3 u•v dΓ ∀ v ∈ V,
and let λ 1ν,V > 0 be the smallest eigenvalue of the eigenvalue problem

u ∈ V, Ω ε(u)•ε(v) dx = λ Γ 3 u ν v ν dΓ ∀ v ∈ V.
Assume also that

L F b λ -1 1,V + α j ν λ -1 1ν,V < m F , (21)
Then, using the abstract result in [START_REF] Han | Numerical Analysis of Stationary Variational-Hemivariational Inequalities[END_REF] it follows that, under the assumptions ( 12), ( 12)-( 16) and ( 21), Problem P V has a unique solution u ∈ U . Let u ∈ U be the solution of Problem P V and denote by σ ∈ Q the function given by σ = F ε(ν). The couple (u, σ ) is called a weak solution to the contact problem P. We conclude from the above discussion that the latter has a unique weak solution.

Numerical Analysis

We now consider the finite element method of solving Problem P V . For simplicity, assume Ω is a polygonal/polyhedral domain and express the three parts of the boundary, Γ k ,1≤ k ≤ 3, as unions of closed flat components with disjoint interiors:

Γ k =∪ i k i=1 Γ k,i , 1 ≤ k ≤ 3.
Let {T h } be a regular family of partitions of Ω into triangles/tetrahedrons that are compatible with the partition of the boundary

∂Ω into Γ k,i ,1≤ i ≤ i k ,1≤ k ≤ 3,
in the sense that if the intersection of one side/face of an element with one set Γ k,i has a positive measure with respect to Γ k,i , then the side/face lies entirely in Γ k,i . Construct the linear element space corresponding to T h :

V h = v h ∈ C(Ω) d | v h | T ∈ P 1 (T ) d , T ∈ T h , v h = 0 on Γ 1 ,
and the related finite element subset

U h = V h ∩ U. Assume g is a concave function. Then U h = v h ∈ V h | v h ν ≤ g at node points on Γ 3 .
Note that 0 ∈ U h . Define the following numerical method for Problem P V .

Problem P h V . Find a displacement field u h ∈ U h such that (F (ε(u h )), ε(v h -u h )) Q + Γ 3 F b (u h ν ) v h τ -u h τ dΓ + Γ 3 j 0 ν (u h ν ; v h ν -u h ν ) dΓ ≥ f, v h -u h V * ×V ∀ v h ∈ U h . (22)
For an error analysis, we assume

u ∈ H 2 (Ω) d , σ ν ∈ L 2 (Γ 3 ) d . (23)
Note that for many application problems, σ ν ∈ L 2 (Γ 3 ) d follows from u ∈ H 2 (Ω) d ; e.g., this is the case where the material is linearized elastic with suitably smooth coefficients, or where the elasticity operator F depends on x smoothly. The starting point for obtaining error estimates is the inequality

u -u h 2 V ≤ c u -v h 2 V + u -v h L 2 (Γ 3 ) d + R(v h ) ∀ v h ∈ U h . (24)
This inequality is based on the properties of the operators F , the function F b ,t h e potential j ν and the trace inequality [START_REF] Han | Numerical Analysis of Stationary Variational-Hemivariational Inequalities[END_REF]. Its proof follows from an abstract error estimation result in the study of elliptic variational-hemivariational inequalities which can be found in [START_REF] Han | Numerical Analysis of Stationary Variational-Hemivariational Inequalities[END_REF]. In [START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] and below, c represents a positive constant which does not depend on h and whose value may change from line to line and R(v h ) is a residual term defined by

R(v h ) = (F (ε(u)), ε(v h -u)) Q + Γ 3 F b (u ν ) v h τ -u τ dΓ + Γ 3 j 0 ν (u ν ; v h ν -u ν ) dΓ -f, v h -u V * ×V .
We now derive a bound for this residual term and, to this end, we follow the procedure found in [START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF]. Take v = u ± w with w in the subset Ũ of U defined by

Ũ := w ∈ C ∞ (Ω) d | w = 0 on Γ 1 ∪ Ŵ 3 ,
and derive from (20) that

(F (ε(u)), ε(w)) Q = f, w V * ×V ∀ w ∈ Ũ . Therefore, DivF (ε(u)) + f 0 = 0 in Ω, (25)
σ ν = f 2 on Γ 2 . (26)
Then multiply [START_REF] Sofonea | Numerical analysis of history-dependent variational inequalities with applications to contact problems[END_REF]byvu with v ∈ U , integrate over Ω, and integrate by parts,

∂Ω σ ν•(v -u) dΓ - Ω F (ε(u))•ε(v -u) dx + Ω f 0 •(v -u) dx = 0, i.e., Ω F (ε(u))•ε(v -u) dx = f, v -u V * ×V + Γ 3 σ ν•(v -u) dΓ. ( 27
)
Thus,

R(v h ) = Γ 3 σ ν•(v h -u) + F b (u ν ) v h τ -u τ + j 0 ν (u ν ; v h ν -u ν ) dΓ,
and then,

R(v h ) ≤ c u -v h L 2 (Γ 3 ) d . (28)
Finally, from [START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF], we derive the inequality

u -u h 2 V ≤ c u -v h 2 V + u -v h L 2 (Γ 3 ) d ∀ v h ∈ U h . ( 29
)
Under additional solution regularity assumption

u| Γ 3,i ∈ H 2 (Γ 3,i ; R d ), 1 ≤ i ≤ i 3 , (30)
we have the optimal order error bound

u -u h V ≤ ch. ( 31
)
We comment that similar results hold for the frictionless version of the model, i.e., where the friction condition ( 6) is replaced by

σ τ = 0 on Γ 3 .
Then the problem is to solve the inequality (20) without the term

Γ 3 F b (u ν ) ( v τ -u τ ) dΓ.
The condition [START_REF] Panagiotopoulos | Nonconvex problems of semipermeable media and related topics[END_REF]

reduces to α j ν λ 1 - ν, 1
V < m F . The inequality (29) and the error bound (31) still hold for the linear finite element solution.

Numerical Simulations

This section is devoted to some numerical simulation results in order to illustrate the solution of the frictional contact Problem P h V and to provide a numerical evidence of the theoretical error bound obtained in Sect. 4. We comment that the solution of Problem P h V is based on numerical methods presented in detail in [START_REF] Barboteu | An analytical and numerical approach to a bilateral contact problem with nonmonotone friction[END_REF][START_REF] Barboteu | Analysis of a contact problem with normal compliance, finite penetration and nonmonotone slip dependent friction[END_REF]. Numerous standard numerical methods for contact mechanics can be found for instance in [START_REF] Laursen | Computational Contact and Impact Mechanics[END_REF][START_REF] Wriggers | Computational Contact Mechanics[END_REF].

Numerical example. The physical setting of the numerical example related to Problem P h V is depicted in Fig. 1. There, the unit square body Ω = (0, 1) × (0, 1) ⊂ R 2 is considered and

Γ 1 =[0, 1]×{1},Γ 2 = ({0}×(0, 1)) ∪ ({1}×(0, 1)), Γ 3 =[0, 1]×{0}.
The domain Ω represents the cross section of a three-dimensional linearly elastic body subjected to the action of tractions in such a way that a plane stress hypothesis is assumed. On the part Γ 1 the body is clamped and, therefore, the displacement field vanishes there. Horizontal compressions act on the part ({0}×[0.5, 1)) ∪ ({1}× [0.5, 1)) of the boundary Γ 2 and the part ({0}×(0, 0.5)) ∪ ({1}×(0, 0.5)) is traction free. Constant vertical body forces are assumed to act on the elastic body. We consider that the deformable body is in frictional contact with an obstacle on the subset Γ 3 =[0, 1]×{0} of its boundary. 3 where the dependence of the relative error u refu h E / u ref E with respect to h is plotted for the Problem P h V . Note that these results provide a numerical evidence of the theoretically predicted optimal order estimate obtained in Sect. 4 and highlight the linear asymptotic convergence of the numerical solutions.

Fig. 1 Fig. 3 h = 1 /

 131 Fig. 1 Reference configuration of the two-dimensional example

Acknowledgements

The work of W.H. was partially supported by NSF under grant DMS-1521684.

Let 0 < r 1 ν < r 2 ν be given, and let p ν : R → R, j ν : R → R be the functions defined by

In the numerical example, we consder the frictional contact conditions ( 5) and ( 6)in which the function j ν is given by ( 32), (33) and

where μ ≥ 0 represents a given coefficient of friction. Note that the fuction p ν is continuous but is not monotone and, therefore, j ν is a locally Lipschitz nonconvex function. With this choice, the frictional contact condition we use on Γ 3 takes the following form:

The compressible material response, considered here, is governed by a linear elastic constitutive law defined by the elasticity tensor F given by

where E and κ are Young's modulus and Poisson's ratio of the material and αβ denotes the Kronecker symbol.

For the numerical simulations, the following data are used: In Fig. 2, we plotted the deformed mesh and the interface forces on Γ 3 . We observe that the contact nodes on the extremities of the boundary Γ 3 are in multivalued normal compliance with either backward slip (slip-) or forward slip (slip+); there, the normal displacement u ν does not reach the penetration that is u ν < g.All the remaining nodes of Γ 3 are in unilateral contact; there, the penetration bound is reached, that is u ν = g. Most of these nodes are in the slip status, except the node in the center of the boundary Γ 3 which is in stick status.

Numerical convergence orders. The aim of this part is to illustrate the convergence of the discrete solutions and to provide numerical evidence of the optimal error estimate obtained in Sect. 4. To this end, we computed a sequence of numerical solutions by using uniform discretization of the Problem P h V according to the spatial discretization parameter h. For instance, for h = 1/64, we obtained the deformed configurations and the interface forces plotted in Fig. 2.

The numerical errors uu h E are computed by using the energy norm • E for several discretization parameters of h. The energy norm • E is equivalent to the canonical norm • V . Since it is not possible to calculate the exact solution u in an analytical way, we consider a "reference" solution u ref corresponding to a fine discretization of Ω, instead of the exact solution. Here, each line segment component of the boundary Γ of Ω is divided into 1/ h equal parts. We start with h = 1/4 which is successively halved. The numerical solution u ref corresponding to


Krzysztof Bartosz

Mircea Sofonea
email: sofonea@univ-perp.fr

Subdifferential inclusions for stress formulations of unilateral contact problems

Keywords: Clarke subdifferential, convex subdifferential, pseudomonotone operator, history-dependent operator, inclusion, unilateral contact condition, frictionless contact, weak solution

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Subdifferential inclusions play an important and challenging role in the study of nonlinear boundary value problems, which arise in mechanics, physics and engineering sciences. In particular, they represent a powerful instrument, which enables new and interesting results to be obtained in the study of various classes of variational and hemivariational inequalities. This is because variational inequalities are closely related to inclusions involving the subgradient of convex functions and, in turn, hemivariational inequalities are related to inclusions involving the Clarke subgradient of locally Lipschitz functions. Variational and hemivariational inequalities have been widely used in the study of mathematical models that describe the contact process of deformable bodies [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF][START_REF] Eck | Unilateral contact problems: variational methods and existence theorems[END_REF][START_REF]Advances in variational and hemivariational inequalities: theory, numerical analysis and applications[END_REF][START_REF] Han | Quasistatic contact problems in viscoelasticity and viscoplasticity[END_REF][START_REF] Haslinger | Numerical methods for unilateral problems in solid mechanics[END_REF][START_REF] Hlaváček | Solution of variational inequalities in mechanics[END_REF][START_REF] Haslinger | PD Finite element method for hemivariational inequalities: theory, methods and applications[END_REF][START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF][START_REF] Naniewicz | Mathematical theory of hemivariational inequalities and applications[END_REF][START_REF] Panagiotopoulos | Inequality problems in mechanics and applications[END_REF][START_REF] Panagiotopoulos | Hemivariational inequalities: applications in mechanics and engineering[END_REF][START_REF] Shillor | JJ Models and analysis of quasistatic contact[END_REF][START_REF] Sofonea | Mathematical models in contact mechanics[END_REF].

Various classes of stationary and history-dependent subdifferential inclusions have been studied [START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF][START_REF] Han | A class of variational-hemivariational inequalities with applications to frictional contact problems[END_REF][START_REF] Migórski | History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics[END_REF][START_REF] Migórski | History-dependent variational-hemivariational inequalities in contact mechanics[END_REF][START_REF] Sofonea | Numerical analysis of history-dependent variational inequalities with applications to contact problems[END_REF]. There, existence and uniqueness results were proved, through arguments of surjectivity for pseudomonotone operators and the Banach fixed point theorem. These results have been used in the study of various classes of variational and hemivariational inequalities, for which continuous dependence of the solution on the data was shown, numerical methods were introduced, and convergence was established rigorously. Finally, results on the well-posedness and error estimation of numerical solutions were applied to inequalities arising in the study of new models of contact.

The aim of this paper is to study a new class of subdifferential inclusion and to apply the corresponding results in the analysis of contact problems with unilateral constraints. Unlike the inequalities considered in the papers mentioned in the previous paragraph, the inclusions considered here involve both the subdifferential of the indicator of a closed convex set and the Clarke subdifferential of a locally Lipschitz function. This represents a first trait of novelty of our paper, which allows us to obtain existence and uniqueness results for new classes of variational inequalities, with multivalued operators and unilateral constraints. We apply these results in the study of frictionless models of unilateral contact, both in the static and time-dependent case. The constitutive law of the material is expressed in terms of the Clarke subdifferential, whose argument is the stress function. Consideration of such models, in which the unilateral constraints are formulated in terms of displacement and the constitutive law is governed by a subdifferential in stress, leads to new and challenging mathematical problems. Their variational analysis represents the second trait of novelty of our paper.

The rest of the manuscript is structured as follows. In Section 2, we present some notation and preliminary material. In Section 3, we consider a class of stationary subdifferential inclusions in reflexive Banach spaces, for which we state and prove an abstract existence and uniqueness result, Theorem 2. We extend this result in Section 4 to a class of history-dependent subdifferential inclusions. There, we state and prove our second abstract existence and uniqueness result, Theorem 3. In Section 5, we consider a static frictionless model of contact with unilateral constraints. We list the assumptions on the data, then we derive a variational formulation of the problem, in terms of stress. Next, we use Theorem 2 to prove the unique solvability of the problem. In Section 6, we use Theorem 3 to extend these results to a history-dependent frictionless model of contact. Finally, in Section 7, we provide examples of one-dimensional constitutive laws for which our results work.

Notation and preliminaries

In this section we briefly present the notation and some preliminary material to be used later in this paper. More details on the material presented below can be found elsewhere [START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF][START_REF] Naniewicz | Mathematical theory of hemivariational inequalities and applications[END_REF][START_REF] Clarke | Optimization and nonsmooth analysis[END_REF][START_REF] Denkowski | An introduction to nonlinear analysis: theory[END_REF][START_REF] Zeidler | /B: nonlinear monotone operators[END_REF].

First, we remark that all linear spaces used in this paper are assumed to be real. Unless stated otherwise, in this section we denote by X a normed space and use the notation • X and 0 X for the norm and the zero element of X , respectively. We denote by X * its topological dual, and •, • X * ×X will represent the duality pairing of X and X * . The symbol 2 X * is used to represent the set of all subsets of X * . We start with a definition of the subdifferential in the sense of Clarke. Definition 1. Let ϕ : X → R be a locally Lipschitz function. The Clarke generalized directional derivative of ϕ at the point x ∈ X in the direction v ∈ X , is defined by

ϕ 0 (x; v) = lim sup y→x,λ↓0 ϕ(y + λv) -ϕ(y) λ .
The Clarke subdifferential of ϕ at x is a subset of X * given by

∂ Cl ϕ(x) = ζ ∈ X * : ϕ 0 (x; v) ≥ ζ , v X * ×X for all v ∈ X .
We recall the basic properties of the Clarke subdifferential (see Proposition 3.23 (iv) of Migórski et al. [START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF]).

Proposition 1. Let ϕ : X → R be a locally Lipshitz function. Then

1.

For every x ∈ X , the set ∂ Cl ϕ(x) is nonempty, convex and weakly * compact subset of X * .

2.

The graph of the Clark subdifferential

∂ Cl ϕ is closed in X × (w * -X * ) topology, i.e., if {x n } ⊂ X and {ζ n } ⊂ X * are sequences such that ζ n ∈ ∂ Cl ϕ(x n ) and x n → x in X , ζ n → ζ weakly * in X * , then ζ ∈ ∂ Cl ϕ(x).
For a convex function, we recall the definition of its subdifferential in the sense of convex analysis.

Definition 2. Let X be a Banach space and : X → R ∪ {+∞} be a convex function. The subdifferential of at x ∈ X is then defined by

∂ (x) = η ∈ X * | (v) -(x) ≥ η, v -x X * ×X for all v ∈ X .
Moreover, the effective domain of is the set defined by

dom = { x ∈ X | (x) < +∞} .
We now recall the definition of pseudomonotonicity, for both single-valued and multivalued operators.

Definition 3. A single-valued operator A : X → X * is called pseudomonotone if, for any sequence {v n } ∞ n=1 ⊂ X , v n → v weakly in X and lim sup n→∞ Av n , v n -v X * ×X ≤ 0 imply that Av, v -y X * ×X ≤ lim inf n→∞ Av n , v n -y X * ×X ,
for all y ∈ X .

Definition 4.

A multivalued operator A : X → 2 X * is called pseudomonotone if the following conditions hold:

1. A has values that are nonempty, weakly compact and convex. 2. A is upper semicontinuous from every finite dimensional subspace of X into X * endowed with the weak topology.

For any sequence {v

n } ∞ n=1 ⊂ X and any v * n ∈ Av n , v n → v weakly in X and lim sup n→∞ v * n , v n -v X * ×X ≤ 0 imply that for any y ∈ X there exists u(y) ∈ Av, such that u(y), v -y X * ×X ≤ lim inf n→∞ v * n , v n -y X * ×X .
The next proposition corresponds to Proposition 3.58 in Migórski et al. [START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF].

Proposition 2. Let X be a real reflexive Banach space, and assume that A : X → 2

v * n , v n -v ≤ 0, then v * ∈ Av and v * n , v n → v * , v . Then the operator A is pseudomonotone.
The following proposition deals with a multivalued operator that is a perturbation of the Clarke subdifferential of a locally Lipschitz functional. We acknowledge that the idea of the proof of Proposition 3 comes from Migórski et al. [START_REF] Migórski | A class of variational-hemivariational inequalities in reflexive Banach spaces[END_REF]. However, for the convenience of the reader, we provide a detailed proof. Proposition 3. Let X be a real reflexive Banach space and let the operator A : X → X * and the functional J : X → R be such that:

1. A is demicontinuous, i.e., for every sequence {u n } ∞ n=1 if u n → u in X then Au n → Au weakly in X * . 2. A is strongly monotone, i.e., there exists a constant α > 0 such that, for all u 1 , u 2 ∈ X we have Au

1 -Au 2 , u 1 -u 2 X * ×X ≥ α u 1 -u 2 2 X . 3. J is locally Lipschitz. 4. ∂ Cl J is bounded, i.e., if U ⊂ X is a bounded set in X then the set { x * ∈ X * | ∃u ∈ U, such that x * ∈ ∂ Cl J (u) } is bounded in X * .

5.

∂ Cl J is relaxed monotone, i.e., there exists a constant m > 0 such that, for all u 1 ,

u 2 ∈ X , η 1 , η 2 ∈ X * , if η i ∈ ∂ Cl J (u i ), i = 1, 2, then η 1 -η 2 , u 1 -u 2 X * ×X ≥ -m u 1 -u 2 2 X .

6.

α > m.

Then the operator T : X → 2 X * defined by Tu = Au + ∂ Cl J (u) for all u ∈ X is pseudomonotone.

Proof. To prove that operator T is pseudomonotone, we shall apply Proposition 2. To this end, we need to show that T satisfies the three conditions of Proposition 2. First, it follows from condition 3 of Proposition 3 and Proposition 1 that, for all u ∈ X , the set Tu is nonempty, convex and closed and, therefore, condition 1 of Proposition 2 holds. Moreover, by condition 4 of Proposition 3, it follows that T is bounded, which shows that condition 2 of Proposition 2 holds, too. It remains to check assumption 3 of Proposition 2. To this end, let

{u n } ∞ n=1 ⊂ X and {u * n } ∞ n=1 ⊂ X * be sequences such that u n → u weakly in X , u * n → u * weakly * in X * , u * n ∈ Tu n for n ∈ N and lim sup n→∞ u * n , u n -u X * ×X ≤ 0. ( 1
)
Our goal is to show that u * ∈ Tu and u * n , u n X * ×X → u * , u X * ×X as n → ∞. Since u * n ∈ Tu n , it follows that, for all n ∈ N, there exists

η n ∈ X * such that u * n = Au n + η n (2)
and

η n ∈ ∂ Cl J (u n ). (3)
Let us choose arbitrary ξ ∈ ∂ Cl J (u). Using conditions 2 and 5 of Proposition 3, we get

(α -m) u n -u 2 X ≤ Au n + η n -Au -ξ , u n -u X * ×X (4)
= u * n , u n -u X * ×X -Au + ξ , u n -u X * ×X .
Combining equations ( 4) with (1), using assumption 6 of Proposition 3 and the fact that u n → u weakly in X , we conclude that u n → u in X .

Since the sequence {u n } ∞ n=1 converges in X , it follows that it is bounded in X . Conversely, using assumption 4 of Proposition 3 and equation (3), we see that the sequence {η n } is bounded in X * . Since the space X * is reflexive, there exists η ∈ X * such that for a subsequence, still denoted η n , we have

η n → η weakly in X * . ( 6
)
Using equations (3), ( 5) and ( 6) and applying condition 2 of Proposition 1, we see that

η ∈ ∂ Cl J (u). (7)
Conversely, using equation ( 5) and assumption 1 of Proposition 3, we obtain that

Au n → Au weakly in X * . ( 8
)
Next, combining equations ( 2), ( 6) and ( 8), we get u * n → Au + η weakly in X * . Thus, by the uniqueness of the weak limit, it follows that u * = Au + η. Combining this equality with equation ( 7) we obtain that u * ∈ Tu. Finally, from equation ( 5) and the fact that u * n → u * weakly in X * , it follows that u * n , u n X * ×X → u * , u X * ×X , which completes the proof of the proposition.

The next proposition deals with an existence result for an abstract elliptic inclusion and corresponds to Theorem 2.2. in Lea [START_REF] Lea | Range and existence theorem for pseudomonotone perturbations of maximal monotone operators[END_REF].

Proposition 4. Let X be a real reflexive Banach space, F

: D(F) ⊂ X → 2 X * a maximal monotone operator, G : D(G) = X → 2 X * a multivalued pseudomonotone operator and L ∈ X * . Assume that there exist v 0 ∈ X and R ≥ v 0 X , such that D(F) ∩ B R (0 X ) = ∅ and ξ + η -L, v -v 0 X * ×X > 0, ( 9
)
for all v ∈ D(F) with v X = R and all ξ ∈ F(v), η ∈ G(v).
Then there exists at least an element u ∈ D(F), such that

F(u) + G(u) ∋ L. ( 10
)
Note that in the statement of Proposition 4 we denote by D(F) and D(G) the effective domains of the operators F and G, respectively, and that B R (0 X ) represents the sphere of radius R and centre 0 X .

We end this section with some preliminaries useful in the study of history-dependent inclusions. Thus, for T > 0, we use the usual notation L 2 (0, T; X ) for the Bochner-Lebesgue space and we recall the following definition.

Definition 5. An operator

S : L 2 (0, T; X ) → L 2 (0, T; X ) is called a history-dependent operator if the following condition holds:        There exists L S > 0 such that (Su 1 )(t) -(Su 2 )(t) X ≤ L S t 0 u 1 (s) -u 2 (s) X ds ∀ u 1 , u 2 ∈ L 2 (0, T; X ), a.e. t ∈ (0, T). ( 11
)
We also recall the following fixed point result.

Theorem 1. Assume that X is a Banach space and S : L 2 (0, T; X ) → L 2 (0, T; X ) is a history-dependent operator. Then S has a unique fixed point, i.e., there exists η * ∈ L 2 (0, T; X ) such that Sη * = η * .

A proof of Theorem 1 could be found, for instance, in Migórski et al. [START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF] or Sofonea and Matei [START_REF] Sofonea | Mathematical models in contact mechanics[END_REF].

Stationary subdifferential inclusions

Everywhere in this section, we assume that Q is a real reflexive Banach space. We denote by • Q its associated norm and by •, • Q * ×Q the duality pairing between Q and its dual Q * . Let be a subset of Q, A : Q → Q * a given operator, J : Q → R a locally Lipschitz function and f ∈ Q * . We denote by ∂ Cl J the subdifferential of the function J in the sense of Clarke, by ψ : Q → R ∪ {+∞} the indicator function of the set and by ∂ψ its subdifferential in the sense of convex analysis. Recall that

ψ (σ ) = 0 for σ ∈ , +∞ for σ ∈ Q \ (12)
and, in addition,

∂ψ (σ ) = ξ ∈ Q * | ξ , τ -σ Q * ×Q ≤ 0 for all τ ∈ if σ ∈ , ∅ if σ ∈ Q\ . ( 13
)
With these data, we consider the following subdifferential inclusion.

Problem P. Find an element σ ∈ Q, such that Aσ + ∂ Cl J (σ ) + ∂ψ (σ ) ∋ f . ( 14
)
In the study of Problem P we consider the following hypotheses:

H( ) The set is a convex, nonempty, closed subset of Q.

H(A)

The operator A : Q → Q * is Lipschitz continuous, strongly monotone, i.e.:

1. There exists a constant L A > 0, such that

Aσ 1 -Aσ 2 Q * ≤ L A σ 1 -σ 2 Q for all σ 1 , σ 2 ∈ Q.
2. There exists a constant α A > 0, such that

Aσ 1 -Aσ 2 , σ 1 -σ 2 Q * ×Q ≥ α A σ 1 -σ 2 2 Q for all σ 1 , σ 2 ∈ Q. H(J ) The function J : Q → R is such that: 1. J is locally Lipschitz. 2.
There exists a constant c J > 0, such that

ξ Q * ≤ c J (1 + σ Q ) for all σ ∈ Q, all ξ ∈ ∂ Cl J (σ ).
3. There exists a constant m J > 0, such that

ξ 1 -ξ 2 , σ 1 -σ 2 Q * ×Q ≥ -m J σ 1 -σ 2 2 Q for all σ 1 , σ 2 ∈ Q, ξ 1 , ξ 2 ∈ Q * , with ξ i ∈ ∂ Cl J (σ i ), i = 1, 2. H(f ) f ∈ Q * .
Finally, we consider the smallness assumption

α A > max {m J , c J } (15)
where, recall, α A , m J and c J are the positive constants that appear in assumptions H(A) and H(J ).

Our main existence and uniqueness result in this section is the following.

Theorem 2. Assume that H( ), H(A), H(J ), H(f ) and (15) hold. Then Problem P has a unique solution σ ∈ .

Proof. We consider three multivalued operators T 1 , T 2 and T : Q → 2 Q * , defined by

T 1 σ = ∂ψ (σ ), T 2 σ = Aσ + ∂ Cl J (σ ), Tσ = T 1 σ + T 2 σ , for all σ ∈ Q.
We show that operator T is surjective, i.e., for all F ∈ Q * there exists σ ∈ Q, such that Tσ ∋ F.

To this end, we apply Proposition 4. First, we note that assumption H( ) implies that ψ is a convex, proper, lower semicontinuous function. Then, it follows that operator T 1 is maximal monotone as a subdifferential of a convex, proper and lower semicontinuous function and, moreover, D(T 1 ) = . Conversely, by Proposition 3, operator T 2 is pseudomonotone and D(T 2 ) = Q. Let τ 0 ∈ be fixed, F ∈ Q * and, for simplicity, denote by 0, the zero element of the space Q, i.e., 0 = 0 Q . We define the constants

C = 5 (α A -c J ) 2 F 2 Q * + c 2 J + A0 2 Q * + (L 2 A + c 2 J ) τ 0 2 Q + 2 α A -c J ( A0 Q * + c J + F Q * ) τ 0 Q , R 1 = C.
It follows from equation ( 15) that C > 0 and, therefore, R 1 is well defined. Let τ ∈ , ξ ∈ T 1 τ and η ∈ T 2 τ . The last inclusion shows that there exists

ζ ∈ ∂ Cl J (τ ) such that η = Aτ + ζ . Suppose that τ Q ≥ R 1 .
Then, using equation ( 15), we have

(α A -c J -5ε) τ 2 Q ≥ 1 4ε F 2 Q * + c 2 J + A0 2 Q * + (L 2 A + c 2 J ) τ 0 2 Q + ( A0 Q * + c J + F Q * ) τ 0 Q , ( 16
)
where ε := (α Ac J )/10. Conversely, using the Hölder inequality and H(J )(2), we have

F -ζ , τ Q * ×Q + Aτ + ζ -F, τ 0 Q * ×Q ≤ (c J + 4ε) τ 2 Q + 1 4ε F 2 Q * + c 2 J + (L 2 A + c 2 J ) τ 0 2 Q + ( A0 Q * + c J + F Q * ) τ 0 Q . (17)
Moreover, from H(A), we find that

Aτ , τ Q * ×Q = Aτ -A0, τ -0 Q * ×Q + A0, τ Q * ×Q (18)
≥ α A τ 2 Q + A0, τ Q * ×Q ≥ (α A -ε) τ 2 Q - 1 4ε A0 2 Q * .
Combining equations ( 16) to ( 18), we get

Aτ , τ Q * ×Q ≥ F -ζ , τ Q * ×Q + Aτ + ζ -F, τ 0 Q * ×Q
In addition, it follows from equation ( 13) that

ξ , τ 0 -τ Q * ×Q ≤ 0.
Summing up the last two inequalities we obtain

ξ + η -F, τ -τ 0 Q * ×Q ≥ 0. ( 19
) Since D(T 1 ) = = ∅, there exists R 2 > 0, such that D(T 1 ) ∩ B R 2 (0) = ∅. Define R = max {R 1 , R 2 , τ 0 Q }. Then, R ≥ τ 0 Q and D(T 1 ) ∩ B R (0) = ∅. Moreover, for all τ ∈ , such that τ Q = R, and for all ξ ∈ T 1 τ , η ∈ T 2 τ , inequality (19) holds.
Thus, we are in a position to apply Proposition 4 to conclude that T is surjective, i.e., there exists σ ∈ Q such that equation ( 14) holds. Hence, σ is a solution of Problem P. We now show that σ ∈ . It follows from equation ( 14) that there exists ξ ∈ ∂ Cl J (σ ), such that

-Aσ -ξ + f ∈ ∂ψ (σ ).
The last inclusion implies that ∂ψ (σ ) = ∅ and, using equation [START_REF] Sofonea | Mathematical models in contact mechanics[END_REF], it follows that σ ∈ , which concludes the existence part of the theorem.

To prove the uniqueness part, we suppose that σ 1 , σ 2 ∈ are two solutions of Problem P. Then, there exist

ξ 1 , ξ 2 ∈ Q, such that Aσ i + ξ i -f , τ -σ i Q * ×Q ≥ 0 for all τ ∈ , with ξ i ∈ ∂ Cl J (σ i ), i = 1, 2. ( 20
)
We add equation [START_REF] Zeidler | /B: nonlinear monotone operators[END_REF] for i = 1, 2, taking τ = σ 2 for i = 1 and τ = σ 1 for i = 2. Therefore, we obtain

Aσ 1 + ξ 1 -Aσ 2 -ξ 2 , σ 1 -σ 2 Q * ×Q ≤ 0. Now, from H(A)(2) and H(J )(3), we get (α A -m J ) σ 1 -σ 2 2 Q ≤ 0.
Using the smallness assumption, equation [START_REF] Migórski | History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics[END_REF], we conclude that σ 1 = σ 2 , which completes the proof.

Remark 1.

Using the definition of the subdifferential ∂ψ , it is easy to see that Problem P can be formulated, equivalently, as follows.

Problem P ′ . Find an element σ ∈ with the property that there exists ξ ∈ Q * such that

Aσ , τ -σ Q * ×Q + ξ , τ -σ Q * ×Q ≥ f , τ -σ Q * ×Q for all τ ∈ , ( 21
) ξ ∈ ∂ Cl J (σ ). ( 22
)
Note that Theorem 2 provides an existence and uniqueness result for Problem P ′ . Nevertheless, if σ is a solution of this problem, the element ξ that satisfies equation ( 22) cannot be uniquely determined.

History-dependent subdifferential inclusions

We now introduce a history-dependent version of Problem P. To this end, we consider a time interval (0, T), with T > 0, and we allow the set , the function J and the element f to depend on the time variable. More precisely, we assume in what follows that H(A) holds and we replace the assumptions H( ), H(J ) and H(f ) with the following assumptions.

H hd ( ) The set (t) is a convex, nonempty, closed subset of Q, for a.e. t ∈ (0, T).

H hd (J ) The function J : (0, T) × Q → R satisfies:

1. J (•, σ ) is measurable on (0, T) for all σ ∈ Q.

2. J (t, •) is locally Lipschitz for a.e. t ∈ (0, T).

3. There exists a function c J : (0, T) → R, such that c J (t) > 0 a.e. t ∈ (0, T) and

ξ Q * ≤ c J (t)(1 + σ Q ) for all σ ∈ Q, all ξ ∈ ∂ Cl J (t, σ ), a.e. t ∈ (0, T).
4. There exists a function m J : (0, T) → R, such that m J (t) > 0 a.e. t ∈ (0, T) and

ξ 1 -ξ 2 , σ 1 -σ 2 Q * ×Q ≥ -m J (t) σ 1 -σ 2 2 Q for all σ 1 , σ 2 ∈ Q, ξ 1 , ξ 2 ∈ Q, with ξ i ∈ ∂ Cl J (t, σ i ), i = 1, 2, a.e. t ∈ (0, T). H hd (f ) f ∈ L 2 (0, T; Q * ).
We also consider the following smallness assumption.

There exists δ > 0 s.t.

α A > max {m J (t), c J (t)} + δ a.e. t ∈ (0, T). ( 23
)
Finally, let S be an operator such that H hd (S) S : L 2 (0, T; Q) → L 2 (0, T; Q * ) is a history-dependent operator.

We now consider the following subdifferential inclusion.

Problem P hd . Find a function σ : (0, T) → Q, such that

Aσ (t) + (Sσ )(t) + ∂ Cl J (t, σ (t)) + ∂ψ (t) (σ (t)) ∋ f (t), a.e. t ∈ (0, T). ( 24
)
Our main existence and uniqueness result in this section is the following. The proof of this theorem will be carried out in several steps, which we present next. In the first step, we consider a given element η ∈ L 2 (0, T, Q * ) together with the following intermediate problem.

Problem P η hd . Find a function σ η : (0, T) → Q, such that Aσ η (t) + η(t) + ∂ Cl J (t, σ η (t)) + ∂ψ (t) (σ η (t)) ∋ f (t) a.e. t ∈ (0, T). ( 25
)
We have the following existence and uniqueness result.

Lemma 1.

Assume that H hd ( ), H(A), H hd (J ), H hd (f ) and equation [START_REF] Kalita | A class of subdifferential inclusions for elastic unilateral contact problems[END_REF] hold. Then Problem P η hd has a unique solution with regularity σ η ∈ L 2 (0, T; ).

Proof. The following equalities and inequalities hold for a.e. t ∈ (0, T), supposed to be fixed. We note that for such t the set (t) satisfies assumption H( ). Moreover, the functions J (t, •) and f (t) satisfy assumptions H(J ) and H(f ), respectively, and, in addition, equation [START_REF] Migórski | History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics[END_REF] follows from equation [START_REF] Kalita | A class of subdifferential inclusions for elastic unilateral contact problems[END_REF]. Therefore, using Theorem 2 we deduce that there exists a unique element σ η (t) ∈ (t), which solves equation (25) at time t.

We now prove that the function (0, T) ∋ t → σ η (t) ∈ Q belongs to the space L 2 (0, T; Q). To this end, let η i ∈ L 2 (0, T; Q * ), f i ∈ L 2 (0, T; Q * ) be given and, for simplicity, denote σ η i = σ i , i = 1, 2. Let t ∈ (0, T) be fixed. It follows from equation (25) that there exists ξ i (t) ∈ ∂ Cl J (t, σ i ) such that

-Aσ i (t) -η i (t) -ξ i (t) + f i (t) ∈ ∂ψ (t) (σ i (t))
and, therefore,

Aσ i (t) + η i (t) + ξ i (t) -f i (t), τ -σ i (t) Q * ×Q ≥ 0 for all τ ∈ (t), i = 1, 2. ( 26
)
We add equation ( 26) for i = 1, 2, taking τ = σ 2 (t) for i = 1 and τ = σ 1 (t) for i = 2. As a result, we obtain

Aσ 1 (t) -Aσ 2 (t) + ξ 1 (t) -ξ 2 (t), σ 1 (t) -σ 2 (t) Q * ×Q + η 1 (t) + f 2 (t) -η 2 (t) -f 1 (t), σ 1 (t) -σ 2 (t) Q * ×Q ≤ 0.
Now, using assumptions H(A)( 2) and H hd (J )(4), we find that

(α -m J (t)) σ 1 (t) -σ 2 (t) Q ≤ η 1 (t) -η 2 (t) Q * + f 1 (t) -f 2 (t) Q *
and, therefore, equation ( 23) implies that

σ 1 (t) -σ 2 (t) Q ≤ 1 δ η 1 (t) -η 2 (t) Q * + f 1 (t) -f 2 (t) Q * ). (27)
Inequality ( 27) provides the continuity of the function

Q * × Q * ∋ (η, f ) → σ ∈ Q
, where σ = σ η is the unique solution of equation ( 25) corresponding to (η, f ). Since, clearly, the functions η and f are measurable, we deduce that the function (0, T) ∋ t → σ η (t) ∈ Q is measurable as a composition of continuous and measurable functions. In addition, equation (27) shows that σ η satisfies an inequality of the form

σ η (t) Q ≤ c (1 + η(t) Q * ), (28)
where c denotes a positive constant that does not depend on t. Now, since η ∈ L 2 (0, T; Q * ) we deduce from equation (28) that σ η ∈ L 2 (0, T; Q). Conversely, recall that σ η (t) ∈ a.e. t ∈ (0, T). This concludes the existence part of the lemma. The uniqueness follows from the uniqueness of the solution of equation (25) for a.e. t ∈ (0, T), guaranteed by Theorem 2.

We now use Lemma 1 to define the operator :

L 2 (0, T; Q * ) → L 2 (0, T; Q * ) by equality η = Sσ η for all η ∈ L 2 (0, T; Q * ). ( 29
)
We have the following fixed point result.

Lemma 2.

Assume that H hd ( ), H(A), H hd (S), H hd (J ), H hd (f ) and equation [START_REF] Kalita | A class of subdifferential inclusions for elastic unilateral contact problems[END_REF] hold. Then the operator has a unique fixed point η * ∈ L 2 (0, T; Q * ).

Proof. Let η 1 , η 2 be two elements in the space L 2 (0, T; Q * ) . Then using inequality (27) we deduce that

σ η 1 (t) -σ η 2 (t) Q ≤ 1 δ η 1 (t) -η 2 (t) Q * a.e. t ∈ (0, T). ( 30
)
We now combine inequality (30) with assumption H hd (S) to see that is a history-dependent operator. Then, we use Theorem 1 to conclude the proof.

We are now in a position to present the proof of Theorem 3.

Proof. We use Lemma 2. The solution of the auxiliary Problem P η * hd represents the unique solution to Problem P hd .

Remark 2. Using the definition of the subdifferential ∂ψ , it is easy to see that Problem P hd can be formulated, equivalently, as follows.

Problem P ′ hd . Find a function σ : (0, T) → Q with the property that σ (t) ∈ (t) a.e. t ∈ (0, T) and there exists ξ : (0, T) → Q * , such that

Aσ (t), τ -σ (t) Q * ×Q + (Sσ )(t), τ -σ (t) Q * ×Q + ξ (t), τ -σ (t)) Q * ×Q ≥ f (t), τ -σ (t) Q * ×Q for all τ ∈ (t), ξ (t) ∈ ∂ Cl J (t, σ (t)) (31)
a.e. t ∈ (0, T).

As in the previous section, we note that Theorem 3 provides an existence and uniqueness result for Problem P ′ hd . Nevertheless, if σ is a solution of this problem, we have no information on the uniqueness and the regularity of the function ξ that satisfies equation (31).

A static model of contact

Theorem 2 is useful in the study of various models of contact with deformable bodies. To provide an example, in this section we consider a frictionless contact problem for elastic bodies. Let ⊂ R d (d = 1, 2, 3) be the reference configuration of an elastic body, Ŵ the boundary of and Ŵ 1 , Ŵ 2 , Ŵ 3 a partition of Ŵ such that meas (Ŵ 1 ) > 0. Here, and in the following, meas (Ŵ 1 ) denotes the d -1 dimensional Lebesgue measure of the set Ŵ 1 . We denote by S d the space of second-order symmetric tensors on R d or, equivalently, the space of symmetric matrices of order d = 1, 2, 3. The inner product and norm on R d and S d are defined by

u • v = u i v i , v = (v • v) 1 2 for all u, v ∈ R d , σ • τ = σ ij τ ij , τ = (τ • τ ) 1 2
for all σ , τ ∈ S d .

Here, and in the following, the indices i and j run between 1 and d and, unless stated otherwise, the summation convention over repeated indices is used. We use the notation x = (x i ) for a typical point in ∪ Ŵ. For a vector field v : → R d , we use the notation v = (v i ), and a tensor field σ : → S d will be denoted σ = (σ ij ). An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable x, e.g., u i,j = ∂u i /∂x j . We use ε and Div for the deformation and divergence operators, respectively, i.e.,

ε(v) = (ε ij (v)), ε ij (v) = 1 2 (v i,j + v j,i ), Div σ = (σ ij,j ).
Let ν = (ν i ) be the outward unit normal at Ŵ. Given a vector field v : Ŵ → R d , we define its normal and tangential components by equalities v ν = v • ν and v τ = vv ν ν, respectively. Similarly, for a tensor field σ : Ŵ → S d , we define its normal and tangential components by σ ν = (σ ν) • ν and σ τ = σ νσ ν ν, respectively. We use the standard notation for Sobolev and Lebesgue spaces associated with and Ŵ. In addition, we consider the space

Q = σ = (σ ij ) : σ ij = σ ji ∈ L 2 ( ) ,
which is a real Hilbert spaces endowed with the canonical inner product given by

(σ , τ ) Q = σ ij τ ij dx.
The associated norm is denoted by • Q . For an element v ∈ H 1 ( ) d , we still write v for the trace of v. Recall also that for a regular stress function σ the following Green's formula holds:

σ • ε(v) dx + Div σ • v dx = Ŵ σ ν • v da for all v ∈ H 1 ( ) d . ( 32
)
We consider the space

V = v ∈ H 1 ( ) d : v = 0 on Ŵ 1 . α A > max { m j , c j , c j meas ( ) }, (43)
where meas ( ) represents the d -dimensional measure of . Finally, we assume that there exists an element g ∈ V such that g ν = g on Ŵ 3 , (

and we refer the reader to Kalita et al. [START_REF] Kalita | A class of subdifferential inclusions for elastic unilateral contact problems[END_REF] and Sofonea et al. [START_REF] Sofonea | Primal and dual variational formulation of a frictional contact problem[END_REF] for examples and details of this condition. We use Riesz's representation theorem to define the element f ∈ V by equality

( f , v) V = f 0 • v dx + Ŵ 2 f 2 • v da for all v ∈ V . (45)
Then, we introduce the set of admissible displacements U and the set of admissible stress fields defined by

U = { v ∈ V : v ν ≤ g a.e. on Ŵ 3 }, (46)
= τ ∈ Q : (τ , ε(v) -ε ( g)) Q ≥ f , v -g V for all v ∈ U . ( 47
)
We now turn to the variational formulation of the contact Problem P and, to this end, we assume that u, σ are regular functions that satisfy equations (34) to (39). Then, multiplying equation ( 35) by (vu), where v ∈ U and, using Green's formula, we have

σ • (ε(v) -ε(u)) dx + Div σ • (v -u) dx = Ŵ σ ν • (v -u) da.
We now split the surface integral on Ŵ 1 , Ŵ 2 and Ŵ 3 ; then we use the equilibrium equation ( 35), the boundary conditions (36), (37) and the definition (45) to deduce that

σ • (ε(v) -ε(u)) dx = f , v -u V + Ŵ 3 σ ν • (v -u) da. (48)
Next, using equations ( 38) and (39), it is easy to see that

σ ν • (v -u) = σ ν (v ν -u ν ) ≥ 0 a.e. on Ŵ 3
and, therefore,

Ŵ 3 σ ν • (v -u) da ≥ 0. (49)
We now combine equations ( 48) and (49) to see that

σ • (ε(v) -ε(u)) dx ≥ f , v -u V . (50)
Note that assumption (44) implies that 2ug ∈ U and g ∈ U. This allows us to test equation (50

) with v = 2u -g and v = g, to deduce that σ • (ε(u) -ε( g)) dx = f , u -g V . (51)
Next, we add inequality (50) and equation (51) and use definition (47) to deduce that

σ ∈ . ( 52
)
Consider now an arbitrary element τ ∈ . Then using equations ( 47) and (51) it is easy to see that

(τ -σ ) • (ε(u) -ε(g)) dx ≥ 0. ( 53
)
for which we apply Theorem 2 with Q = Q and f = ε * ( g). To this end, we note that the set given by equation ( 47) is nonempty since, for instance, it contains the element ε( f ). Conversely, it is easy to check that is a closed convex subset of Q and, therefore, it satisfies assumption H( ). The assumption of equation ( 40) on the elasticity operator A implies that operator A defined by equation ( 57) is Lipschitz continuous and strongly monotone, i.e., it satisfies assumption H(A) with L A = L A and α A = α A . We also note that, using Lemma 3, it follows that condition H(J ) holds, too. Finally, the smallness assumption, equation (43), combined with equation (60), implies equation [START_REF] Migórski | History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics[END_REF]. Therefore, since the element f = ε * ( g) obviously satisfies assumption H(f ), it follows from Theorem 2 that there exists a unique solution σ ∈ to equation (62). Using Remark 1, we obtain the existence of a unique element σ ∈ with the property that there exists ξ in Q * such that

Aσ , τ -σ Q * ×Q + ξ , τ -σ Q * ×Q ≥ ε * ( g), τ -σ Q * ×Q for all τ ∈ , (63)
ξ ∈ ∂ Cl J (σ ). ( 64
) Let η = ι * ξ . Then, it follows that ξ , τ Q * ×Q = η • τ dx ∀ τ ∈ Q. ( 65
)
We now use equations ( 57), ( 58) and (65) in equation ( 63) to see that inequality (55) holds. Conversely, (64) and (61) imply that equation (56) holds, too. We conclude from here that σ ∈ is a solution to Problem P V , which concludes the existence part of the proof.

The uniqueness part could be proved directly, by using arguments similar to those used in the proof of Theorem 2. For this reason, we skip the details. We restrict ourselves to note that it follows by using equations (55), (56), assumptions (40)(c), (41)(d) and the smallness assumption, equation (43).

A history-dependent model of contact

We now present a history-dependent version of Problem P. The classical formulation of the problem is the following.

Problem P hd . Find a displacement field u : × (0, T) → R d and a stress field σ : × (0, T) → S d , such that

ε(u(t)) ∈ Aσ (t) + ∂ Cl j(t, σ (t)) + t 0 B(t -s)σ (s) ds in , ( 66
) Div σ (t) + f 0 (t) = 0 in , (67) u(t) = 0 on Ŵ 1 , (68) σ (t)ν = f 2 (t) on Ŵ 2 , (69) u ν (t) ≤ g, σ ν (t) ≤ 0, σ ν (t)(u ν (t) -g) = 0 on Ŵ 3 , (70) σ τ (t) = 0 on Ŵ 3 , (71)
for all t ∈ (0, T).

These equations and boundary conditions are similar to those in Problem P. The difference arises in the fact that now the constitutive equation ( 34) is replaced with the history-dependent constitutive equation (66), in which B represents the relaxation tensor. Note that now the function j is assumed to depend on time, which makes the problem more general from a mathematical point of view. From a physical point of view, this dependence could model the dependence of j with respect to the temperature, which is considered as given. We assume that

B ∈ L 2 (0, T; Q ∞ ), (72)
where Q ∞ represents the space of fourth-order tensor fields, given by

Q ∞ = E = (E ijkl ) : E ijkl = E jikl = E klij ∈ L ∞ ( ), 1 ≤ i, j, k, l ≤ d .
We recall that Q ∞ is a real Banach space with the norm

E Q ∞ = max 1≤i,j,k,l≤d E ijkl L ∞ ( ) .
Moreover, a simple calculation shows that

Eτ Q ≤ d E Q ∞ τ Q for all E ∈ Q ∞ , τ ∈ Q. ( 73
)
In the study of Problem P hd we keep assumptions (40) and (44) and use the notation of equation ( 46) for the set of admissible displacement fields. Nevertheless, we replace assumptions (41) to (43) with the following. (74)

                           (a) j : (0, T) × S d → R. (b) j(•, σ ) is measurable, for all σ ∈ Q. (c) j(t,
f 0 ∈ L 2 (0, T; L 2 ( ) d ), f 2 ∈ L 2 (0, T; L 2 (Ŵ 2 ) d ). (75)
There exists δ > 0 s.t.

α A > max { m j (t), c j (t), c j (t) meas ( )} + δ, (76)
a.e. t ∈ (0, T).

Next, we define the function f : (0, T) → V by

f (t), v V = f 0 (t) • v dx + Ŵ 2 f 2 (t) • v da for all v ∈ V , a.e. t ∈ (0, T) (77)
and, for a.e. t ∈ (0, T), we define the set

(t) = τ ∈ Q : (τ , ε(v) -ε( g)) Q ≥ f (t), v -g V for all v ∈ U . (78)
Then, the variational formulation of the contact Problem P hd is obtained by arguments similar to those used in the previous section and is as follows.

Problem P V

hd . Find a stress field σ : (0, T) → Q with the property that σ (t) ∈ (t) a.e. t ∈ (0, T) and there exists η : (0, T) → Q, such that

Aσ (t) • (τ -σ (t)) dx + t 0 B(t -s)σ (s) ds • (τ -σ (t)) dx (79) + η(t) • (τ -σ (t)) dx ≥ ε( g) • (τ -σ ) dx ∀ τ ∈ (t), η(t) ∈ ∂ Cl j(t, σ (t)) a.e. in , (80)
for a.e. t ∈ (0, T).

Our main result in this section is the following.

Theorem 5. Assume that equations (40), ( 44), ( 72), (74), ( 75) and (76) hold. Then, Problem P V hd has a unique solution, which satisfies σ ∈ L 2 (0, T; Q).

To provide the proof of Theorem 5, we consider the operator S : L 2 (0, T; Q) → L 2 (0, T; Q * ) and the functional J :

(0, T) × Q → R, defined by (Sσ )(t), τ Q * ×Q = t 0 B(t -s)σ (s) ds • τ dx for all τ , σ ∈ L 2 (0, T; Q), a.e. t ∈ (0, T). (81) J (t, τ ) = j(t, τ ) dx for all τ ∈ Q, a.e. t ∈ (0, T). ( 82
)
The next lemma deals with the properties of the function J .

Lemma 4. Assume that equation (74) holds. Then, the functional J is well defined and satisfies assumption H hd (J ) on the space Q = Q, with the functions

c J (t) = max c j (t), c j (t) meas ( ) , m J (t) = m j (t). ( 83
)
In addition, the following implication holds:

ξ ∈ ∂ Cl J (t, τ ) =⇒ ι * ξ ∈ ∂ Cl j(t, τ ) a.e. in , for a.e. t ∈ (0, T). ( 84
)
Proof. We use Theorem 3.47 of Migórski et al. [START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF] to see that the function J satisfies conditions 1 to 3 of hypothesis H hd (J ) and, moreover, equation (84) holds. The value of functions c J , that appears in equation ( 83) follows from a direct calculation based on equation (84). In addition, the validity of condition 4 of hypothesis H hd (J ), with m J (t) = m j (t) follows straightforwardly from equations ( 84) and ( 74)(e).

We now pass to the proof of Theorem 5.

Proof. We consider the subdifferential inclusion

Aσ (t) + (Sσ )(t) + ∂ Cl J (t, σ (t)) + ∂ψ (t) (σ (t)) ∋ ε * ( g) a.e. t ∈ (0, T), (85)
for which we apply Theorem 3 with Q = Q and f = ε * ( g). To this end, we note that for a.e. t ∈ (0, T) the set (t) given by equation ( 78) is a nonempty closed convex subset of Q and, therefore, satisfies assumption H hd ( ). The assumption (40) on the elasticity operator A implies that operator A is Lipschitz continuous and strongly monotone and, therefore, satisfies assumption H(A) with L A = L A and α A = α A . We also note that Lemma 4 and condition (74) on the function j shows that the assumption H hd (J ) holds. Next, the smallness assumption of equation (76) combined with equation (83) implies equation [START_REF] Kalita | A class of subdifferential inclusions for elastic unilateral contact problems[END_REF]. We now use inequality (73) to see that the operator S satisfies condition [START_REF] Panagiotopoulos | Hemivariational inequalities: applications in mechanics and engineering[END_REF]. We conclude from here that S is a history-dependent operator, i.e., it satisfies condition H hd (S). Therefore, since the element f = ε * ( g) obviously satisfies the assumption H hd (f ), it follows now from Theorem 3 that there exists a unique function σ ∈ L 2 (0, T; Q), such that σ (t) ∈ (t) a.e. t ∈ (0, T). Moreover, equation (85) holds, too. Using Remark 2, we obtain the existence of a unique function σ ∈ L 2 (0, T; Q) with the property that σ (t) ∈ (t) a.e. t ∈ (0, T) and, moreover, there exists ξ : (0, T) → Q * such that

Aσ (t), τ -σ (t) Q * ×Q + (Sσ )(t), τ -σ (t) Q * ×Q (86) + ξ (t), τ -σ (t) Q * ×Q ≥ ε * ( g), τ -σ (t) Q * ×Q for all τ ∈ , ξ (t) ∈ ∂ Cl J (t, σ (t)) (87)
a.e. t ∈ (0, T). Let η(t) = ι * ξ (t). Then, using arguments similar to those used in the proof of Theorem 4, it is easy to see that σ is a solution to Problem P V hd , which concludes the existence part of the proof. The uniqueness part could be proved directly, by using equations (79) and (80). It is based on the historydependence of the operator (S), assumptions (40)(c), (74)(e) and the smallness assumption of equation (76).

One-dimensional examples

In this section we present some examples of potential functions j for which our results apply. For simplicity, we restrict ourselves to the one-dimensional time-independent case, i.e., we assume in what follows that d = 1 and j does not depend explicitly on time. Then, assumptions (41) and ( 74 (89)

Then, it is easy to see that j is a C 1 function and, therefore, condition (88)(b) is satisfied. Moreover,

∂ Cl j(σ ) =    -ασ 0 if σ < -σ 0 , ασ if |σ | ≤ σ 0 , ασ 0 if σ > σ 0 .
This imply that |∂ Cl j(σ )| ≤ ασ 0 for all σ ∈ R and, hence, condition (88)(c) holds with c j = ασ 0 . In addition, since ∂ Cl j is a monotone function, we deduce that condition (88)(d) holds with any m j > 0. It is easy to see that j satisfies condition (88)(b). Moreover, using the definition of the Clarke subdifferential it follows that

∂ Cl j(σ ) =    (α -1)e σ -α if σ < 0, [-1, 1] if σ = 0, (1 -α)e -σ + α if σ > 0.
In addition, it is easy to check that |ξ | ≤ 1 for all ξ ∈ ∂ Cl j(σ ) and σ ∈ R and, therefore, condition (88)(c) holds with c j = 1. Finally, a simple calculation shows that condition (88)(d) holds with m j = 1α.

Example 3. Let α ≥ 0 and let j : R → R be the function defined by j(σ ) = 0 if σ < 0, -e -σ + ασ + 1 if σ ≥ 0.

(91)

It is easy to see that j satisfies condition (88)(b). Moreover, using elementary computation it follows that

∂ Cl j(σ ) =    0 if σ < 0, [0, 1 + α] if σ = 0, e -σ + α if σ > 0.
In addition, it is easy to check that |ξ | ≤ 1 + α for all ξ ∈ ∂ Cl j(σ ) and σ ∈ R and, therefore, condition (88)(c) holds with c j = 1 + α. Finally, a simple calculation shows that condition (88)(d) holds with m j = 1.

We conclude from Examples 1 to 3 that Theorem 2 could be applied in the study of contact problems with elastic constitutive laws of the form ε ∈ aσ + ∂ Cl j(σ ),

where a > 0 is a given compliance coefficient and j represents one of the functions of equations ( 89) to (91). Note that in the case of equation ( 89) the constitutive law of equation ( 92) is single-valued and is given by

ε =    aσ -ασ 0 if σ < -σ 0 , (a + α)σ if |σ | ≤ σ 0 , aσ + ασ 0 if σ > σ 0 . (93)
In contrast, in the case of equations ( 90) and (91), the constitutive law (92) is multivalued. In addition, Theorem 3 could be applied in the study of contact problems with viscoelastic constitutive laws of the form ε(t) ∈ aσ (t) + ∂ Cl j(σ (t)) + t 0 c(ts)σ (s) ds.

Here, a > 0 is a given compliance coefficient, j represents one of equations ( 89) to (91) and c ∈ L 2 (0, T) is a relaxation function. Both theorems provide the existence of a unique solution, in terms of stress, to the corresponding frictionless unilateral contact problems.

Example 1 .-

 1 ) are identical and can be formulated as follows.                 (a) j : R → R. (b) j is locally Lipschitz. (c) There exists c j > 0 such that |ξ | ≤ c j (1 + |σ |) for all σ ∈ R, all ξ ∈ ∂ Cl j(σ ). (d) There exists m j > 0 such that (ξ 1ξ 2 ) • (σ 1σ 2 ) ≥ -m j |σ 1σ 2 | 2 for all σ 1 , σ 2 ∈ R, ξ 1 , ξ 2 ∈ R, with ξ i ∈ ∂ Cl j(σ i ), i = 1, 2.Let α > 0, σ > 0 and let j : R → R be the function defined byj(σ ) = ασ 0 (σ + σ 0 ) if σ < -σ 0 , 1 2 ασ 2 if |σ | ≤ σ 0 , 1 2 ασ 2 0 + ασ 0 (σσ 0 ) if σ > σ 0 .

Example 2 .

 2 Let α ∈ [0, 1) and let j : R → R be the function defined byj(σ ) = (α -1)e -|σ | + α|σ | ∀ σ ∈ R.(90)

  •) is locally Lipschitz for a.e. t ∈ (0, T). (d) There exists a function c j : (0, T) → R such that c j (t) > 0 for a.e. t ∈ (0, T) and ξ ≤ c j (t)(1 + σ ) for all σ ∈ S d , all ξ ∈ ∂ Cl j(t, σ ), a.e. t ∈ (0, T). (e) There exists a function m j : (0, T) → R such that m j (t) > 0 for a.e. t ∈ (0, T) and (ξ 1 -ξ 2 ) • (σ 1 -σ 2 ) ≥ -m j (t) σ 1 -σ 2 2 for all σ 1 , σ 2 , ξ 1 , ξ 2 ∈ S d with ξ i ∈ ∂ Cl j(t, σ i ) for a.e. t ∈ (0, T), i = 1, 2.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Centre of Poland under the Maestro Advanced Project (grant number DEC-2012/06/A/ST1/00262).

It is well known that V is a real Hilbert space endowed with the inner product

and the associated norm • V . Completeness of the space (V , • V ) follows from the assumption meas (Ŵ 1 ) > 0, which allows the use of Korn's inequality. We also recall that there exists c 0 > 0, which depends on , Ŵ 1 and

Inequality (33) represents a consequence of the Sobolev trace theorem. With these preliminaries, the classical formulation of the unilateral frictionless contact problem that we study in this section is the following.

Problem P. Find a displacement field u : → R d and a stress field σ : → S d , such that

We recall that equation (34) represents the constitutive law in which A is the compliance operator and j is a nonlinear potential. Examples of such laws will be provided in Section 7. Here we restrict ourselves to remark that equation (34) shows that the strain tensor has an additive decomposition into a single-valued part, Aσ , and a multivalued part, ∂ Cl j(σ ). Equation ( 35) is the equilibrium equation in which f 0 denotes the density of body forces. We use it here, since we assume that the mechanical process is static. Conditions (36) and (37) are the displacement-traction boundary conditions, in which f 2 represents the density of traction on Ŵ 2 . Condition (38) represents the Signorini contact condition in a form with a gap g. Finally, condition (39) represents the frictionless condition, which states that the tangential component of the stress, denoted σ τ , vanishes on the contact surface Ŵ 3 .

The assumptions on the data of Problem P are the following.

(41)

Next, the constitutive law, equation (34), shows that there exists a function η : → S d such that

We now gather relations (52) to (54) to deduce the following variational formulation of Problem P, in terms of stress.

Problem P V . Find a stress field σ ∈ with the property that there exists η : → S d such that

Our main result in this section is the following.

Theorem 4. Assume that equations (40) to (44) hold. Then, Problem P V has a unique solution, which satisfies σ ∈ .

To provide the proof of Theorem 4, we consider the dual of the space Q, denoted Q * and let ι : Q → Q * be the isometry provided by the Riesz representation theorem defined by

We also denote by ι * : Q * → Q, the inverse of ι and define the operators

Then, we have the following result.

Lemma 3.Assume that equation (41) holds. Then, the function J is well defined and satisfies assumption H(J ) on the space Q = Q with the constants

In addition, for all τ ∈ Q, the following implication holds:

Proof. It follows from Theorem 3.47 of Migórski et al. [START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems[END_REF] that the function J satisfies conditions 1 and 2 of hypothesis H(J ). Moreover, the same theorem guarantees that equation (61) holds. Note that the value of the constant c J that appears in equation (60) follows from a direct calculations, based on equation (61). Moreover, the validity of condition 3 of hypothesis H(J ) with the constant m J = m j follows straightforwardly from equations (61) and (41)(d), which concludes the proof.

We are now in a position to provide the proof of Theorem 4.

Proof. We start by considering the subdifferential inclusion

Aσ + ∂ Cl J (σ ) + ∂ψ (σ ) ∋ ε * ( g), (62)