A uniqueness criterion for the Signorini problem with Coulomb friction

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A uniqueness criterion for the Signorini problem with Coulomb friction

Yves Renard

To cite this version:

Yves Renard. A uniqueness criterion for the Signorini problem with Coulomb friction. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2006, 38 (2), pp.452-467.

�hal-00690595�

A UNIQUENESS CRITERION FOR THE SIGNORINI PROBLEM WITH COULOMB FRICTION

YVES RENARD^†

Abstract. The purpose of this paper is to study the solutions to the Signorini problem with Coulomb friction (the so-called Coulomb problem). Some optimal a priori estimates are given, and a uniqueness criterion is exhibited. Recently, nonuniqueness examples have been presented in the continuous framework. It is proved, here, that if a solution satisfies a certain hypothesis on the tangential displacement and if the friction coefficient is small enough, it is the unique solution to the problem. In particular, this result can be useful for the search of multisolutions to the Coulomb problem because it eliminates a lot of uniqueness situations.

Key words. unilateral contact, Coulomb friction, uniqueness of solution

Introduction. The so-called Signorini problem with Coulomb friction (or simply the Coulomb problem) has been introduced by Duvaut and Lions [4]. It does not exactly represent the equilibrium of a solid which encounters an obstacle because when the equilibrium is reached (or any steady state solution) the friction condition is no longer an irregular law. The aim of this problem is in fact to be very close to a time semidiscretization of an evolutionary problem by an implicit scheme. The fact that several solutions could coexist in an implicit scheme (independently of the size of the time step) may be an indication that the evolutionary problem has a dynamical bifurcation.

The first existence results for this problem were obtained by Neˇ cas, Jaruˇsek, and Haslinger in [15] for a two-dimensional elastic strip, assuming that the coefficient of friction is small enough and using a shifting technique previously introduced by Fichera and later applied to more general domains by Jaruˇ sek [11]. Eck and Jaruˇsek [5] give a different proof using a penalization method. We emphasize that most results on existence for frictional problems involve a condition of smallness for the friction coefficient (and a compact support on Γ

_C

).

Recently, examples of nonunique solutions have been given by Hild in [7] and [8]

for a large friction coefficient. As far as we know, for a fixed geometry, it is still an open question whether or not there is uniqueness of the solution for a sufficiently small friction coefficient. In the finite element approximation framework, the presence of bifurcation has been studied in [9].

The present paper gives the first (partial) result of uniqueness of a solution to the Coulomb problem. The summary is the following. Section 1 introduces strong and weak formulations of the Coulomb problem. Section 2 gives optimal estimates on the solutions. In particular, a comparison is made with the solution to the frictionless contact problem. Section 3 gives an additional estimate for the Tresca problem, i.e., the problem with a given friction threshold. And finally, section 4 gives the partial

†MIP, CNRS UMR 5640, INSAT, Complexe scientifique de Rangueil, 31077 Toulouse, France (Yves.Renard@insa-toulouse.fr).

00000000 00000000 11111111 11111111

0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 1111111111111111 1111111111111111 1111111111111111 1111111111111111 1111111111111111

ª

 

 

 

Fig. 1. Elastic body Ω in frictional contact.

uniqueness result. It is proved in Proposition 5 for bidimensional problems and a friction coefficient less than one that there is no multisolution with one of the solutions having a tangential displacement with a constant sign. The major result is given by Proposition 6 using the notion of a multiplier in a pair of Sobolev spaces.

1. The Signorini problem with Coulomb friction. Let Ω ⊂ R

^d

(d = 2 or 3) be a bounded Lipschitz domain representing the reference configuration of a linearly elastic body.

It is assumed that this body is submitted to a Neumann condition on a part of its boundary Γ

_N

, to a Dirichlet condition on another part Γ

_D

, and to a unilateral contact with static Coulomb friction condition on the rest of the boundary Γ

_C

between the body and a flat rigid foundation (see Figure 1). This latter part Γ

_C

is supposed to be of nonzero interior in the boundary ∂ Ω of Ω. The problem consists in finding the displacement field u(t, x) satisfying

− div σ(u) = f in Ω, (1)

σ(u) = A ε(u) in Ω, (2)

σ(u)n = F on Γ

_N

, (3)

u = 0 on Γ

_D

, (4)

where σ(u) is the stress tensor, ε(u) is the linearized strain tensor, n is the outward unit normal to Ω on ∂Ω, F and f are the given external loads, and A is the elastic coefficient tensor which satisfies classical conditions of symmetry and ellipticity.

On Γ

_C

, it is usual to decompose the displacement and the stress vector in normal and tangential components as follows:

u

_N

= u · n, u

_T

= u − u

_N

n,

σ

_N

(u) = (σ(u)n) · n, σ

_T

(u) = σ(u)n − σ

_N

(u)n.

To give a clear sense to this decomposition, we assume Γ

_C

to have the C

¹

regular- ity. The unilateral contact condition is expressed by the following complementary condition:

u

_N

≤ g, σ

_N

(u) ≤ 0, (u

_N

− g)σ

_N

(u) = 0, (5)

where g is the normal gap between the elastic solid and the rigid foundation in refer-

ence configuration (see Figure 2).

0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 1111111111111111 1111111111111111 1111111111111111 1111111111111111 1111111111111111

Rigid foundation

Ω g

n

Fig. 2. Normal gap between the elastic solid Ω and the rigid foundation.

Denoting by F ≥ 0 the friction coefficient, the static Coulomb friction condition reads as follows:

if u

_T

= 0, then | σ

_T

(u) | ≤ −F σ

_N

(u), (6)

if u

_T

= 0, then σ

_T

(u) = F σ

_N

(u) u

_T

| u

_T

| . (7)

The friction force satisfies the so-called maximum dissipation principle

− σ

_T

(u) · u

_T

= sup

μT∈Rd−1

|μT|≤−FσN(u)

( − μ

_T

· u

_T

).

(8)

1.1. Classical weak formulation. We present here the classical weak formu- lation proposed by Duvaut [3] and Duvaut and Lions [4]. Let us introduce the Hilbert spaces

V = { v ∈ H

¹

(Ω; R

^d

), v = 0 on Γ

_D

} , X = { v |

ΓC

: v ∈ V } ⊂ H

^1/2

(Γ

_C

; R

^d

), X

_N

= { v

_N

|

ΓC

: v ∈ V } , X

_T

= { v

_T

|

ΓC

: v ∈ V } , and their topological dual spaces V

, X

, X

N

, and X

T

. It is assumed that Γ

_C

is suffi- ciently smooth such that X

_N

⊂ H

^1/2

(Γ

_C

), X

_T

⊂ H

^1/2

(Γ

_C

; R

^d−1

), X

N

⊂ H

^−1/2

(Γ

_C

), and X

T

⊂ H

^−1/2

(Γ

_C

; R

^d−1

).

Classically, H

^1/2

(Γ

_C

) is the space of the restriction on Γ

_C

of traces on ∂ Ω of functions of H

¹

(Ω), and H

^−1/2

(Γ

_C

) is the dual space of H

^1/2

00

(Γ

_C

), which is the space of the restrictions on Γ

_C

of functions of H

^1/2

(∂Ω) vanishing outside Γ

_C

. We refer to [1] and [12] for a detailed presentation of trace operators.

Now, the set of admissible displacements is defined as K = { v ∈ V, v

_N

≤ g a.e. on Γ

_C

} . (9)

The maps

a(u, v) =

Ω

A ε(u) : ε(v)dx, l(v) =

Ω

f · vdx +

Γ_N

F · vdΓ, j( F λ

_N

, v

_T

) = −F λ

_N

, | v

_T

|

X N,XN

represent the virtual work of elastic forces, the external load, and the “virtual work”

of friction forces, respectively. Standard hypotheses are as follows:

a( · , · ) is a bilinear symmetric V-elliptic and continuous form on V × V : (10)

∃ α > 0, ∃ M > 0, a(u, u) ≥ α u

²_V

, a(u, v) ≤ M u

V

v

V

∀ u, v ∈ V, l( · ) is a linear continuous form on V ; i.e., ∃ L > 0, | l(v) | ≤ L v

V

∀ v ∈ V, (11)

g ∈ X

_N

, (12)

F ∈ M X

_N

is a nonnegative multiplier in X

_N

. (13)

The latter condition ensures that j( F λ

_N

, v

_T

) is linear continuous on λ

_N

and convex lower semicontinuous on v

_T

when λ

_N

is a nonpositive element of X

N

(see, for instance, [2]). To satisfy condition (10), it is necessary that Γ

_D

is of nonzero interior in the boundary of Ω and that the elastic coefficient tensor is uniformly elliptic (see [4]).

We refer to Maz’ya and Shaposhnikova [14] for the theory of multipliers. The set M X

_N

denotes the space of multipliers from X

_N

into X

_N

, i.e., the space of function f : Γ

_C

−→ R of finite norm

f

M XN

= sup

vN∈XN vN=0

f v

_N

_X

v

_N

XNN

.

This is the norm of the linear mapping X

_N

v −→ (f v) ∈ X

_N

. Of course, if F is a constant on Γ

_C

, one has F

M X

N

= F . From the fact that Ω is supposed to be a bounded Lipschitz domain and Γ

_C

is supposed to have the C

¹

regularity, it is possible to deduce that for d = 2 the space H

^1/2+ε

(Γ

_C

) is continuously included in M X

_N

for any ε > 0, and for d = 3 the space H

¹

(Γ

_C

) ∩ L

^∞

(Γ

_C

) is included in M X

_N

, continuously for the norm f

_H1(Γ

C)

+ f

_L∞(ΓC)

(see [14]). In particular, the space of Lipschitz continuous functions is continuously included in M X

_N

.

Condition (10) implies in particular that a( · , · ) is a scalar product on V and that the associated norm

v

a

= (a(v, v))

^1/2

is equivalent to the usual norm of V :

√ α v

V

≤ v

a

≤ √

M v

V

∀ v ∈ V.

The continuity constant of l( · ) can also be given with respect to: ·

^a

:

∃ L

_a

> 0, | l(v) | ≤ L

_a

v

^a

∀ v ∈ V.

Constants L and L

_a

can be chosen such that

√ αL

_a

≤ L ≤ √ M L

_a

.

The classical weak formulation of problem (1)–(7) is given by

⎧ ⎨

⎩

Find u ∈ K satisfying

a(u, v − u) + j( F σ

_N

(u), v

_T

) − j( F σ

_N

(u), u

_T

) ≥ l(v − u) ∀ v ∈ K.

(14)

The major difficulty about (14) is due to the coupling between the friction thresh-

old and the contact pressure σ

_N

(u). The consequence is that this problem does not

represent a variational inequality, in the sense that it cannot be derived from an

optimization problem.

1.2. Neumann to Dirichlet operator. In this section, the Neumann to Dirich- let operator on Γ

_C

is introduced together with its basic properties. This will allow to restrict the contact and friction problem to Γ

_C

and obtain useful estimates.

Let λ = (λ

_N

, λ

_T

) ∈ X

; then, under hypotheses (10) and (11), the solution u to

⎧ ⎨

⎩

Find u ∈ V satisfying a(u, v) = l(v) + λ, v

X,X

∀ v ∈ V (15)

is unique (see [4]). So it is possible to define the operator

E ^{: X}

−→ X λ −→ u |

Γ

C

.

This operator is affine and continuous. Moreover, it is invertible and its inverse is continuous. It is possible to express E

⁻¹

as follows: For w ∈ X , let u be the solution to the Dirichlet problem

⎧ ⎨

⎩

Find u ∈ V satisfying u |

ΓC

= w and a(u, v) = l(v) ∀ v ∈ V, v |

ΓC

= 0;

(16)

then E

⁻¹

(w) is equal to λ ∈ X

defined by

λ, v

X,X

= a(u, v) − l(v) ∀ v ∈ V.

In a weak sense, one has the relation E

⁻¹

^{(u) = σ(u)n on Γ}

C

. Now, under hypotheses (10) and (11) one has

E ^(λ

¹

^{) −} E ^(λ

²

⁾

X

≤ C

₁²

α λ

¹

− λ

²

_X

, (17)

where C

₁

is the continuity constant of the trace operator on Γ

_C

and α is the coercivity constant of the bilinear form a( · , · ). One can verify it as follows. Let λ

¹

and λ

²

be given in X

T

and let u

¹

, u

²

be the corresponding solutions to (15); then α u

¹

− u

²

²_V

≤ a(u

¹

− u

²

, u

¹

− u

²

) = λ

¹

− λ

²

, u

¹

− u

²

X,X

≤ C

₁

λ

¹

− λ

²

_X

u

¹

− u

²

V

(18)

and, consequently,

u

¹

− u

²

V

≤ C

₁

α λ

¹

− λ

²

_X

. (19)

Conversely, one has

E

⁻¹

^(u

¹

^{) −} E

⁻¹

^(u

²

⁾

X

≤ M C

₂²

u

¹

− u

²

X

, (20)

where M is the continuity constant of a( · , · ) and C

₂

> 0 is the continuity constant

of the homogeneous Dirichlet problem corresponding to (16) (i.e., with l(v) ≡ 0 and

C

₂

= sup

v∈X v=0

w_V

v_X

, where w |

Γ C

= v and a(w, z) = 0 ∀ z ∈ V ). This latter estimate can be performed as follows:

E

⁻¹

^(u

¹

^{) −} E

⁻¹

^(u

²

⁾

X

= sup

v∈Xv=0

E

⁻¹

^(u

¹

⁾ − E

⁻¹

^(u

²

^{), v}

_X,X

v

X

= sup

v∈Xv=0

⎛

⎝ inf

{w∈V:w

|

ΓC

=v}

a(u

¹

− u

²

, w) v

X

⎞

⎠

≤ M γ u

¹

− u

²

V

, (21)

where γ = sup

v∈X v=0

inf

{w∈V:w

|

ΓC

=v} w_V

v_X

is the continuity constant of the homoge- neous Poisson problem with respect to a Dirichlet condition on Γ

_C

. Using γ ≤ C

₂

, this gives (20).

It is also possible to define the following norms on Γ

_C

relative to a( · , · ):

v

a,Γ_C

= inf

w∈V, w

|

Γ C

=v

w

a

,

λ

−a,Γ_C

= sup

v∈Xv=0

λ, v

X,X

v

a,Γ_C

= sup

v∈Vv=0

λ, v

X,X

v

a

. These are equivalent, respectively, to the norms in X and X

:

√ α

C

₁

v

X

≤ v

^a,Γ_C

≤ √

M γ v

X

,

√ 1

M γ λ

_X

≤ λ

−a,Γ_C

≤ C

₁

√ α λ

_X

.

With these norms, the estimates are straightforward since the following lemma holds.

Lemma 1. Let λ

¹

and λ

²

be two elements of X

and let u

¹

= E ^(λ

¹

^{), u}

²

⁼ E ^(λ

²

^);

then under hypotheses (10) and (11) one has

u

¹

− u

²

^a,Γ_C

= u

¹

− u

²

^a

= λ

¹

− λ

²

−a,Γ_C

. Proof. On the one hand, one has

u

¹

− u

²

²a,Γ_C

= inf

w∈V w

|

Γc

=u1−u2

w

²a

= u

¹

− u

²

²a

,

because u

¹

− u

²

is the minimum of

¹₂

w

²a

under the constraint w |

ΓC

= u

¹

− u

²

. This implies

u

¹

− u

²

²a,Γ_C

= a(u

¹

− u

²

, u

¹

− u

²

) = λ

¹

− λ

²

, u

¹

− u

²

_X,X

≤ λ

¹

− λ

²

−a,Γ_C

u

¹

− u

²

a,Γ_C

, and finally

u

¹

− u

²

^a,Γ_C

≤ λ

¹

− λ

²

−a,Γ_C

.

On the other hand, one has λ

¹

− λ

²

−a,Γ_C

= sup

v∈Xv=0

λ

¹

− λ

²

, v

X,X

v

^a,Γ_C

= sup

v∈Xv=0 w∈V

inf

w

|

ΓC

=v

a(u

¹

− u

²

, w) v

^a,Γ_C

,

≤ sup

v∈Xv=0 w∈V

inf

w

|

ΓC

=v

u

¹

− u

²

a

w

a

v

a,Γ_C

= u

¹

− u

²

a

= u

¹

− u

²

a,Γ_C

, which ends the proof of the lemma.

1.3. Direct weak inclusion formulation. Let

K

_N

= { v

_N

∈ X

_N

: v

_N

≤ 0 a.e. on Γ

_C

}

be the (translated) set of admissible normal displacements on Γ

_C

. The normal cone in X

N

to K

_N

at v

_N

∈ X

_N

is defined as N

_K

N

(v

_N

) = { μ

_N

∈ X

N

: μ

_N

, w

_N

− v

_N

_X

N,X N

≤ 0 ∀ w

_N

∈ K

_N

} . In particular, N

_K

N

(v

_N

) = ∅ if v

_N

∈ / K

_N

. The subgradient of j( F λ

_N

, u

_T

) with respect to the second variable is given by

∂

₂

j( F λ

_N

, u

_T

) = { μ

_T

∈ X

T

: j( F λ

_N

, v

_T

)

≥ j( F λ

_N

, u

_T

) + μ

_T

, v

_T

− u

_T

_X

T,XT

∀ v

_T

∈ X

_T

} . With this notation, problem (14) is equivalent to the following problem:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

Find u ∈ V, λ

_N

∈ X

N

, and λ

_T

∈ X

T

satisfying (u

_N

, u

_T

) = E ^(λ

N

, λ

_T

),

− λ

_N

∈ N

_K

N

(u

_N

− g) in X

N

,

− λ

_T

∈ ∂

₂

j( F λ

_N

, u

_T

) in X

T

. (22)

More details on this equivalence can be found in [13].

Remark 1. Inclusion − λ

_N

∈ N

_K

N

(u

_N

− g) is equivalent to the complementarity relations

u

_N

≤ g, λ

_N

, v

_N

_X

N,XN

≥ 0 ∀ v

_N

∈ K

_N

, λ

_N

, u

_N

− g

_X

N,XN

= 0,

which is the weak formulation of the strong complementarity relations (5) for the contact conditions. Similarly, the second inclusion − λ

_T

∈ ∂

₂

j( F λ

_N

, u

_T

) represents the friction condition.

1.4. Hybrid weak inclusion formulation. We will now consider the sets of admissible stresses. The set of admissible normal stresses on Γ

_C

can be defined as

Λ

_N

= { λ

_N

∈ X

N

: λ

_N

, v

_N

X N,XN

≥ 0 ∀ v

_N

∈ K

_N

} .

This is the opposite of K

^∗

N

, the polar cone to K

_N

. The set of admissible tangential stresses on Γ

_C

can be defined as

Λ

_T

( F λ

_N

) = { λ

_T

∈ X

T

: − λ

_T

, w

_T

_X

T,XT

+ F λ

_N

, | w

_T

|

_X

N,XN

≤ 0 ∀ w

_T

∈ X

_T

} . With this, problem (14) is equivalent to the following problem:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

Find u ∈ V, λ

_N

∈ X

N

, and λ

_T

∈ X

T

satisfying (u

_N

, u

_T

) = E ^(λ

N

, λ

_T

),

− (u

_N

− g) ∈ N

_Λ

N

(λ

_N

) in X

_N

,

− u

_T

∈ N

_Λ

T(Fλ

N)

(λ

_T

) in X

_T

, (23)

where the two inclusions can be replaced by inequalities as follows:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

Find u ∈ V, λ

_N

∈ X

N

, and λ

_T

∈ X

T

satisfying (u

_N

, u

_T

) = E ^(λ

N

, λ

_T

),

λ

_N

∈ Λ

_N

, μ

_N

− λ

_N

, u

_N

− g

X N,XN

≥ 0 ∀ μ

_N

∈ Λ

_N

, λ

_T

∈ Λ

_T

( F λ

_N

), μ

_T

− λ

_T

, u

_T

_X

T,XT

≥ 0 ∀ μ

_T

∈ Λ

_T

( F λ

_N

).

(24)

Remark 2. The inclusion − u

_T

∈ N

_Λ

T(Fλ

N)

(λ

_T

) implies the complementarity relation

λ

_T

, u

_T

X T,XT

= F λ

_N

, | u

_T

|

X N,XN

and the weak maximum dissipation principle

− λ

_T

, u

_T

_X

T,X T

= sup

μT∈Λ_T(Fλ

N)

− μ

_T

, u

_T

_X

T,X T

, which is the weak formulation of (8).

2. Optimal a priori estimates on the solutions to the Coulomb problem.

For the sake of simplicity, a vanishing contact gap (g ≡ 0) will be considered in the following.

Remark 3. In the case of a nonvanishing gap, it is possible to find u

_g

∈ V such that u

_g

|

ΓC

= gn, and then w = u − u

_g

is solution to the problem

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

Find w ∈ V, λ

_N

∈ X

N

, and λ

_T

∈ X

T

satisfying a(w, v) = l(v) − a(u

_g

, v) + λ

_N

, w

_N

_X

N,XN

+ λ

_T

, w

_T

_X

T,XT

,

− w

_N

∈ N

_Λ

N

(λ

_N

) in X

_N

,

− w

_T

∈ N

_Λ

T(Fλ

N)

(λ

_T

) in X

_T

, (25)

i.e., a contact problem without gap but with a modified source term.

Following Remarks 1 and 2, a solution (u, λ) to problem (22) (i.e., a solution u to problem (14)) satisfies the complementarity relations

λ

_N

, u

_N

X N,XN

= 0, λ

_T

, u

_T

_X

T,XT

= F λ

_N

, | u

_T

|

_X

N,XN

. This implies

λ, u

X,X

≤ 0,

which expresses the dissipativity of contact and friction conditions. The first conse- quence of this is that solutions to problem (14) can be bounded independently of the friction coefficient.

Proposition 1. Assuming hypotheses (10), (11), and (13) are satisfied and g ≡ 0, let (u, λ) be a solution to problem (22), which means that u is a solution to problem (14); then

u

a

≤ L

_a

, λ

−a,Γ_C

≤ L

_a

, u

V

≤ L

α , λ

_X

≤ Lγ M

α . Proof. One has

u

²a

= a(u, u) = l(u) + λ, u

X,X

≤ L

_a

u

^a

,

which states the first estimates. The estimate on λ

−a,Γ_C

can be performed using the intermediary solution u

^N

to the following problem with a homogeneous Neumann condition on Γ

_C

:

a(u

^N

, v) = l(v) ∀ v ∈ V.

(26)

Since u

^N

a

≤ L

_a

for the same reason as for u, and using Lemma 1, one has λ − 0

²₋a,Γ_C

= a(u − u

^N

, u − u

^N

) = λ, u − u

^N

X,X

≤ − λ, u

^N

X,X

≤ L

_a

λ

−a,Γ_C

. The two last estimates can be stated thanks to equivalence of norms introduced in section 1.1.

It is possible to compare u

^a

to the corresponding norm of the solution u

^c

to the Signorini problem without friction defined as follows:

⎧ ⎨

⎩

Find u

^c

∈ K satisfying

a(u

^c

, v − u

^c

) ≥ l(v − u

^c

) ∀ v ∈ K.

(27)

It is well known that under hypotheses (10) and (11), this problem has a unique solution (see [12]).

Proposition 2. Assuming hypotheses (10), (11), and (13) are satisfied and g ≡ 0, let u be a solution to problem (14), let u

^c

be the unique solution to problem (27), and let u

^N

be the solution to problem (26); then

u

^a

≤ u

^c

^a

≤ u

^N

^a

.

Proof. One has

a(u

^N

, u

^N

) = l(u

^N

), a(u

^c

, u

^c

) = l(u

^c

), a(u, u) = l(u) + λ

_T

, u

_T

_X

T,XT

.

Since u

^c

is the solution to the Signorini problem without friction, it minimizes over K the energy functional

¹₂

a(v, v) − l(v). The solution u

^N

minimizes this energy functional over V . Thus, since u ∈ K , one has

1

2 a(u

^N

, u

^N

) − l(u

^N

) ≤ 1

2 a(u

^c

, u

^c

) − l(u

^c

) ≤ 1

2 a(u, u) − l(u), and the following relations allow one to conclude that

a(u

^c

, u

^c

) − a(u

^N

, u

^N

) = l(u

^c

− u

^N

), a(u, u) − a(u

^c

, u

^c

) = l(u − u

^c

) + λ

_T

, u

_T

X T,XT

because then

1

2 a(u

^c

, u

^c

) − 1

2 a(u

^N

, u

^N

) ≤ 0, 1

2 a(u, u) ≤ 1

2 a(u

^c

, u

^c

) + F λ

_N

, | u

_T

|

X N,XN

≤ 1

2 a(u

^c

, u

^c

).

It is also possible to estimate how far from u

^c

is a solution u to problem (14). Let us introduce the following norms on Γ

_C

. For v ∈ X let us define

v

_T

^a,Γ_C

= inf

w∈V wT=vT

w

^a

= inf

z∈X zT=vT

z

^a,Γ_C

, v

_N

^a,Γ_C

= inf

w∈V wN=vN

w

^a

= inf

z∈X zN=vN

v

^a,Γ_C

. One has

v

_T

^a,Γ_C

≤ v

^a,Γ_C

, v

_N

^a,Γ_C

≤ v

^a,Γ_C

. Now, for λ ∈ X

, let us define

λ

_T

−a,Γ_C

= sup

vT∈X vT=0T

λ

_T

, v

_T

_X

T,X T

v

_T

^a,Γ_C

= sup

v∈Xv=0

λ

_T

, v

_T

_X

T,X T

v

^a,Γ_C

,

λ

_N

−a,Γ_C

= sup

vN∈XN vN=0

λ

_N

, v

_N

X N,XN

v

_N

a,Γ_C

= sup

v∈Xv=0

λ

_N

, v

_N

X N,XN

v

a,Γ_C

. Then, the following equivalences of norms are immediate:

√ α

C

₁

v

_N

XN

≤ v

_N

^a,Γ_C

≤ γ √

M v

_N

XN

,

√ α

C

₁

v

_T

XT

≤ v

_T

^a,Γ_C

≤ γ √

M v

_T

XT

,