Existence of solutions for the dynamic frictional contact problem of isotropic viscoelastic bodies

Nonlinear Analysis 53 (2003) 157 – 181

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Existence of solutions for the dynamic frictional contact problem of isotropic viscoelastic bodies C. Ecka; ∗ , J. Jaru)sekb a Institute

of Applied Mathematics 1, University of Erlangen-Nurnberg, Martensstrasse 3, 91058 Erlangen, Germany b Mathematical Institute, Academy of Science of the Czech Republic, Zitn+ ) a 25, 115 67 Praha 1, Czech Republic Received 20 January 2001; accepted 23 August 2001

Keywords: Unilateral dynamic contact problem; elasticity; Existence of solutions; Penalty method

Parabolic equation;

Coulomb law of friction; Visco-

1. Introduction Contact problems with friction have many applications, for example in machine dynamics, when di2erent parts of a machine touch each other, or in crack problems, when the two faces of a crack come into contact. Other important applications concern e.g. the behaviour of buildings during earthquakes or the plate tectonics of the Earth. Despite this importance there are still few mathematical results concerning existence of solutions available, especially in the dynamic case. The reason for this lack of results is the structure of the functional describing the friction, which is non-monotone and non-compact. The 6rst attempts to solve the dynamic contact problem with friction have been carried out in [3], where the contact problem of a viscoelastic body with given timeindependent friction force and contact condition formulated in velocities has been studied. This result was extended in [5] to a given friction force depending on time. A similar result for the contact condition formulated in displacements was derived in [6]. Other results are available for the normal compliance model of contact description, where the exact Signorini condition, based on the non-penetrability of mass, is relaxed

∗

Corresponding author. Fax: +49-91-3127670. E-mail addresses: [email protected] (C. Eck), [email protected] (J. Jaru)sek).

0362-546X/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 9 1 1 - 7

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and a problem of much simpler mathematical structure is obtained. The main up-to-now published results for the problem with Signorini condition and unknown friction force were derived in [7,8]. There the contact of a viscoelastic body with a rigid foundation is investigated; the Signorini condition is formulated for the velocity 6eld. The existence of solutions is proved, if the coeGcient of friction is small enough. However, in [7] no information about the magnitude of the coeGcient of friction is speci6ed, and its upper bound CF formulated in [8] for generally anisotropic material and a general dimension N ∈ N is rather restrictive since it is below 0.3 (a magnitude the coeGcient of friction is close to for many relevant materials) even in the optimal cases. The Signorini condition is also considered in [2], where a beam satisfying essentially di2erent constitutive relationships in the normal and tangential direction is studied. In the present paper we investigate the frictional contact of a linear viscoelastic body having short memory with a rigid foundation. In order to compute a better upper bound for the magnitude of the coeGcient of friction, we restrict ourselves to the case of material whose viscosity is homogeneous and isotropic and to two space dimensions. This enables us to solve the di2erential equations on a half-space explicitly and to derive certain optimal trace estimates. Although it was not possible to write an explicit formula for CF , the numerical results show that for moderate values of the modulus of viscosity (a counterpart to the Young modulus of elasticity) and the viscous variant of the Poisson ratio not too close to 0.5 this bound can be between 0.4 and 0.5. On the other hand, this approach does not contribute to the solution of the quasistatic problems. Their elliptic nature needs a suitable adaptation of the shift technique combined with a time discretization. Those problems were recently solved (again for bounded coeGcients of friction) by Andersson, see [1]. 2. Description of the problem Let  ⊂ R2 be a bounded domain with a Lipschitz boundary  composed of the three parts U ; F and C which are measurable and mutually disjoint. Let IT := [0; T] be the time interval of the problem, let QT := IT ×  denote the time-space domain and ST := IT ×  be its lateral boundary consisting of the parts SX; T := IT × X for X = U; F; C. The deformation of the body is described by a displacement 6eld u = {ui } which is a solution of the equations of motion uK i − ij; j (u) = fi ;

i = 1; 2;

(1)

with the stress tensor {ij } and the volume force {fi }. Here and in the sequel, the summation convention is employed. The subscript i denotes the derivative of a function with respect to the space variable xi and the dot describes the time derivative. The strain–stress relation is given by the linear viscoelastic law (1) ˙ ij (u) = a(0) ijk‘ ek‘ (u) + aijk‘ ek‘ (u);

i; j = 1; 2

with the linearized strain tensor e(u) ≡ {eij (u)} := { 12 (ui; j +uj; i )}. The tensor {a(0) ijk‘ } of (0) (0) the elastic part is assumed to be symmetric, bounded and positive, i.e. a(0) ijk‘ =ajik‘ =ak‘ij

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159

and (0) 0 6 a(0) ijk‘ ij k‘ 6 A0 ij ij

(2)

for all symmetric tensors {ij } ∈ R2; 2 with a positive real constant A(0) 0 . The viscous part of the strain–stress relation shall be isotropic, E E ij k‘ + a(1) (3) (ik j‘ + i‘ jk ) ijk‘ = 2 + 2 (1 + )(1 − 2) with the modulus of viscosity E, the viscous variant of the Poisson ratio  and the Kronecker symbol ij . The di2erential equations are supplemented by initial conditions, u(0; x) = u0 (x)

and

u(0; ˙ x) = u1 (x)

for x ∈ 

(4)

and by boundary conditions including the contact condition and the Coulomb law of friction, u=U

on SU; T ;

(5)

(n) (u) = b on SF; T ; u˙ n 6 0;

n 6 0;

n u˙ n = 0;

u˙ t = 0 ⇒ |t | 6 F(0)|n |; u˙ t u˙ t = 0 ⇒ t = −F(u)| ˙ n| |u˙ t |

(6)         

on SC; T :

(7)

Here, n denotes the outer normal vector of the boundary, i(n) ≡ ij nj the components of the boundary traction; the subscripts n and t describe the normal and tangential components of the corresponding vectors. In particular, we have n = ij ni nj and t = (n) − n n. The Signorini condition given here is formulated in velocities; this can be interpreted as a 6rst-order approximation with respect to time of the original contact condition in displacements. The coeGcient of friction F may depend on the space variable x and on the velocity of the solution, F = F(x; u). ˙ Before stating the weak formulation of the problem, let us 6x the notation of the employed function spaces. For a domain M ⊂ RN ; N = 1; 2; 3, or a suGciently smooth manifold M of dimension N − 1 and for k ¿ 0, let H k (M ); H k (M )∗ be the isotropic Sobolev–Slobodetskii space of Hilbert type with the maximal order of di2erentiability k and its dual, respectively. If k = (k1 ; k2 ) has two components, then the 6rst one de6nes the smoothness with respect to the time variable and the second one the regularity with respect to the space variables. By HO k (M ) and H −k (M ) the corresponding spaces with zero traces and their duals are denoted, respectively, for k ¿ 12 . Spaces with range in R2 are indicated by bold letters, e.g. H k (M ; R2 ) ≡ H k (M ); k ∈ N; Lp (M ; R2 ) ≡ Lp (M ); p ¿ 1. With u; v M we denote the (possibly generalized) L2 (M )-scalar product of two functions u; v de6ned on M . The contact problem, consisting of the system of di2erential equations (1), the initial conditions (4) and the boundary conditions (5) – (7), has a weak formulation in terms

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of a variational inequality. The set of admissible functions is a convex cone, K := {v ∈ H 1=2; 1 (QT ); v = U˙ on SU; T and vn 6 0 on SC; T }; and the contact problem is given by the variational inequality Find a function u ∈ H 1 (QT ) with u(0; ·) = u0 ; u(0; ˙ ·) = u1 ; u˙ ∈ K; uK ∈ H 1=2 (IT ; ∗ L2 ()) such that for all v ∈ K there holds ˙ v − u) ˙ u; K v − u ˙ QT + a(0) (u; v − u) ˙ + a(1) (u; + F(u)| ˙ n (u)|; |vt | − |u˙ t | SC; T ¿ L(v − u): ˙ Here a(%) (u; v) :=

 QT

(8)

(%) aijk‘ ek‘ (u)eij (v) d x d&

(9)

denote the bilinear forms of elastic energy (%=0) and viscous energy dissipation (%=1) and   L : v → f · v d x d& + b · v dsx d& QT

SF;T

is the linear functional containing all given forces. 3. Approximate contact problem The proof of the existence of solutions to this problem is carried out with the method described in [8]. In its 6rst step, the penalty method is employed; this leads to a problem of normal compliance type. Replacing the contact condition in (7) by the nonlinear boundary condition 1 n (u) = − [u˙ n ]+ with [ · ]+ := max{·; 0} and  ¿ 0;  we arrive at the variational inequality ˙ ·) = u1 ; u˙ ∈ U˙ + U and uK ∈ U∗ , Find a function u ∈ H 1 (QT ) with u(0; ·) = u0 ; u(0; U = {v ∈ H 1=2; 1 (QT ); v = 0 on SU; T }; such that for all v ∈ U˙ + U there holds u; K v − u ˙ QT + a(0) (u; v − u) ˙ + a(1) (u; ˙ v − u) ˙ + 



1 [u˙ n ]+ ; vn − u˙ n 

 1 + F(u) ˙ [u˙ n ]+ ; |vt | − |u˙ t | ¿ L(v − u): ˙  SC; T

 SC; T

(10)

Here we assume that U is extended onto the whole domain . In the next step we replace the non-di2erentiable norms |vt | and |u˙ t | by di2erentiable approximations )* (vt ) and )* (u˙ t ). Here * ¿ 0 is an approximation parameter and )* : R2 → R+ is a convex, continuously di2erentiable function having its minimum at 0 and satisfying |∇)* (·)| 6 1 and the approximation property |)* (·) − | ·  6 * (for an

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161

example see [8], formula (2:13)). The resulting variational inequality is equivalent to the variational equation ˙ ·) = u1 ; u˙ ∈ U˙ + U and uK ∈ U∗ such that for Find u ∈ H 1 (QT ) with u(0; ·) = u0 ; u(0; all v ∈ U there holds   1 ˙ v) + u; K v QT + a(0) (u; v) + a(1) (u; [u˙ n ]+ ; vn  SC; T   1 + F(u) ˙ [u˙ n ]+ ∇)* (u˙ t ); vt = L(v): (11)  SC; T Problem (11) has been solved in [8], where the following result is proved: Theorem 1. In addition to the assumptions concerning the domain  and its parts of boundary U ; F and C ; let the coe
(12)

with a constant depending only on the geometry of the domain and on the given data; but not on the approximation parameters  and *. Remark. The assumption f ∈ L2 (IT ; H 1 ()∗ ) used in [8] can be weakened to f ∈ H 1=2; 1 (QT )∗ . The result is proved in this case with the help of a better a priori estimate for the solution of the parabolic di2erential equations (see Propositions 1 and 2 below and their corollary). In order to prove the existence of solutions for the original contact problem we investigate the limits * → 0 and  → 0 of the solutions of the approximate contact problem. However, due to the non-monotone and non-compact character of the friction functional in (8) the a priori estimates established in (12) are not suGcient. It is necessary to study carefully the regularity of the solution on the contact part of the boundary. This is carried out in [8] for anisotropic material. The regularity result obtained there is only valid if the coeGcient of friction is below a certain value. To keep the magnitude of this value suGciently high, it is essential to use the best trace theorems available. In the two-dimensional isotropic case investigated here we can derive better trace estimates than those used in [8]. This is done in the next section. 4. Trace estimates for the isotropic case and a half-space domain In this section we consider the Dirichlet problem for purely viscous isotropic material on a half-space Q = R × R × R+ . This is suGcient, because the localization technique

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used later transforms any domain onto a half-space and the elastic part of the strain– stress relation is of lower order and therefore does not inQuence the regularity proof substantially. Using the abbreviations C1 = (2 + 2)=E; C2 = (1 + )(1 − 2)=(E(1 − )) and the relation E=(2(1 + )(1 − 2)) = 1=C2 − 1=C1 , the di2erential equation (1) is transformed to  1 1 1 1 u˙ 1 = u2; 12 + u1; 11 + − u1; 22 + f1 ; C2 C2 C1 C1  1 1 1 1 u˙ 2 = u1; 12 + u2; 11 + − u2; 22 + f2 : (13) C1 C2 C1 C2 Here and in the rest of this section we omit one of the time derivatives, i.e. we replace u˙ by u and uK by u˙ for simplicity of the notation. The above equation is considered on the domain Q with the boundary condition u(&; x) = w(&; x1 )

for (&; x) = (&; x1 ; x2 ) ∈ S = R × R × {0}:

(14)

Our aim is the derivation of three generalized trace estimates. Since we work on the half-space, we cannot expect the L2 -norms of the functions to be bounded. Denoting the semi-norms by  · H k (M ) ≡ ( · 2H k (M ) −  · 2L2 (M ) )1=2 ; M = Q; S and k = ( 12 ; 1), we use the space 1=2; 1 V 1=2; 1 (Q) := {v ∈ Hloc (Q); vH 1=2; 1 (Q) ¡ + ∞; v|S ∈ H 1=4; 1=2 (S)};

equipped with the norm  · V 1=2; 1 (Q) =  · H 1=2; 1 (Q) . This is a norm due to the condition v|S ∈ H 1=4; 1=2 (S), because the boundary data of any non-trivial constant function do not belong to this space. Let VO 1=2; 1 (Q) := {v ∈ V 1=2; 1 (Q); v|S = 0} and let VO 1=2; 1 (Q)∗ ; V −1=2; −1 (Q) denote the dual spaces to V 1=2; 1 (Q) and VO 1=2; 1 (Q), respectively. The estimates required in the regularity proof are valid for functions belonging to the set L(f) = {v ∈ V 1=2; 1 (Q); v solves (13)} of (energy-) solutions of problem (13) with right-hand side f. Let

   E  a(u; v) = div(u) div(v) + eij (u) eij (v) d x dt; 1 +  R R×R+ 1 − 2 and let  · a = a(·; ·) be the corresponding energy norm. Proposition 1. Let f ∈ V −1=2; −1 (Q). Then there exist constants D0 ; D1 ; D2 and K such that any function u ∈ L(f) and its boundary data w = u|S satisfy the estimates sup

v∈L(0); v1 =0 on S; v|S H 1=4;1=2 (S) 61

[u; ˙ v Q + a(u; v)] 6 D0 ua + KfV −1=2; −1 (Q) ;

w1 H 1=4; 0 (S) 6 D1 ua + KfV −1=2; −1 (Q)

(15)

(16)

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163

and w1 H 0; 1=2 (S) 6 D2 ua + KfV −1=2; −1 (Q) :

(17)

The estimates (15) – (17) shall be proved with constants D0 ; D1 ; D2 as small as possible. The special form of these estimates is related with the shift technique carried out in the next section. Proof. We prove Proposition 1 for f = 0 at 6rst. Employing the Fourier transform with respect to the time variable & and the tangential space variable x1 ;  1 u(&; x1 ; x2 )e−i(&#+x1 ) d& d x1 ; u(#; ˜ ; x2 ) = 23 R2 the partial di2erential equations (13) are transformed into the parameter-dependent set of ordinary di2erential equations   1  2 1 1 u˜ 1 + iu˜2 ; − u˜ 1 = i# + C1 C2 C1 C2   1  1 1 2 u˜ 2 + iu˜1 : − u˜ = i# + C1 C1 C2 C2 2

(18)

Here the prime  denotes the derivative with respect to the variable x2 ≡ y. This system has four independent solutions   a a eay ; u˜ (2) = e−ay ; u˜ (1) = −i i   i i (3) (4) by u˜ = e ; u˜ = e−by b −b with the complex roots a = a1 + ia2 ; b = b1 + ib2 de6ned by



4 + C 2 #2 +  2  4 + C12 #2 − 2 1 a = 2 + iC1 # = + i sign(#) ; 2 2



4 + C 2 #2 +  2  4 + C22 #2 − 2 2 b = 2 + iC2 # = + i sign(#) : 2 2 The solution of the boundary value problem (13); (14) is that linear combination of the above-mentioned solutions which satis6es the boundary conditions  w˜ 1 (#; ) and u(#; ˜ ; y) → 0 for y → +∞: u(#; ˜ ; 0) = w(#; ˜ ) = w˜ 2 (#; )

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This function is uniquely determined by

   a i e−ay + (iw˜ 1 − aw˜ 2 ) e−by u(#; ˜ ; y) = D−1 (bw˜ 1 + iw˜ 2 ) i −b

(19)

with D = ab − 2 . Motivated by the form of this solution; we perform the change of variables bw˜ 1 + iw˜ 2 iw˜ 1 − aw˜ 2 p1 = ; p2 = : (20) D D Observe that this relation can be easily inverted; w˜ 1 = ap1 + ip2 ;

w˜ 2 = ip1 − bp2 :

In the new variables; the solution is written as   a i −ay e e−by : + p2 u(#; ˜ ; y) = p1 i −b

(21)

In order to prove the trace estimates (16) and (17), we write the energy norm in Fourier-transformed variables as a(u; u) = M   × R

R×R+



 ] 2 + (1 − 2)(| ] |div(u)| (u)|2 ) dy d d# u1; 1 |2 + | u2; 2 |2 + 2|e12 (22)

with M = E=((1 + )(1 − 2)). The Fourier-transformed values in this integral are given by @u˜ 1 ] 2e12 (u) = + iu˜ 2 = −(a2 + 2 )p1 e−ay − 2ibp2 e−by ; @y ] = iu˜ 1 + @u˜ 2 = (b2 − 2 )p2 e−by : div(u) @y Employing these formulae in (22) and performing the inner integration with respect to y yield after some calculation the representation    2|a|2 2 + 12 |a2 + 2 |2 a(u; u) = M (1 − 2) |p1 |2 2a1 R R  2 |b − 2 |2 |b|4 + 22 |b|2 + 4 |p2 |2 +  + (1 − 2) 2b1 2b1  2 T − 2 Re((1 − 2)i(ab +  )p1 pT2 ) d d#: From the special form of the complex roots a = a1 + ia2 and b = b1 + ib2 it is easy to derive the formulae 2a1 a2 = C1 #; 2b1 b2 = C2 # and |a2 + 2 |2 = |a2 − 2 |2 + 42 Re(a2 ) = C12 #2 + 42 Re(a2 ); |b|4 + 22 |b|2 + 4 = |b2 − 2 |2 + 22 Re(b2 ) + 22 |b|2 = C22 #2 + 4b21 2 :

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165

With their application the energy norm is transformed into  

1 − 2 a(u; u) = M |p1 |2 (C1 #a2 + 4a1 2 ) 2 R R − 2 Re((1 − 2)i(abT + 2 )p1 pT2 ) + |p2 |2 ((1 − )C2 #b2  + 2(1 − 2)b1 2 ) d d#: The de6nition of the values C1 and C2 yields 2(1 − )C2 = (1 − 2)C1 . Hence the result for the energy norm is    

E C1 #a2 C1 #b2 a(u; u) = |p1 |2 + 2a1 2 + |p2 |2 + 2b1 2 1+ R R 2 2  − 2 Re (i(abT + 2 )p1 pT2 ) d d#: (23) Now it is possible to derive formulae for the constants D1 and D2 . The Sobolev– Slobodetskii semi-norm of the space H 5; 6 (S) with 0 6 5; 6 ¡ 1 is given in the Fourier transformed representation by   2 w1 H 5; 6 (S) = |w˜ 1 (#; )|2 (c1 (5)|#|25 + c1 (6)||26 ) d d# R

R

with the renormation constants  sin2 (t) c1 (5) = 22−25 dt 1+25 R |t|

for 5 ¿ 0 and c1 (0) = 0

(cf. [4]). From this and the representation w˜ 1 = ap1 + ip2 it is seen that the optimal constants D1 ; D2 are the solutions of the optimization problems 1+ D12 = E (|a|2 |p1 |2 + 2 Im(ap1 pT2 ) + 2 |p2 |2 )c1 ( 14 )|#|1=2 ; 2 1 2 T 2 )p1 pT2 )+|p2 |2 ( 1 C1 #b2 + 2b1 2 ) p1 ;p2 ∈C |p1 | ( C1 #a2 +2a1  )+2 Im((ab+

× sup

;#∈R

2

2

(24) D22 =

1+ E

(|a|2 |p1 |2 + 2 Im(ap1 pT2 ) + 2 |p2 |2 )c1 ( 12 )|| : 2 1 2 T 2 )p1 pT2 )+|p2 |2 ( 1 C1 #b2 +2b1 2 ) p1 ;p2 ∈C |p1 | ( C1 #a2 +2a1  )+2 Im((ab+

× sup

;#∈R

2

2

(25)

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Observe, that the functions in both optimization problems di2er only in the second factor of the numerator. The results of the optimization problems do not change if we restrict the optimization to non-negative values of # only. This can be seen as follows: T changing the sign of # in a and b leads to the conjugate complex values aT and b. Hence the values a1 ; b1 ; #a2 and #b2 do not depend on the sign of #. Using the T 1 pT2 ) = Im(c(−pT1 )p2 ) in the imaginary parts arising in the functions to be formula Im(cp optimized it is seen that the ranges of these functions are preserved by the restriction to non-negative values of #. Due to the continuity of the numerator and denominator of these functions it is possible to exclude the parameters  = 0; # = 0 and p2 = 0 from the optimization without changing its result. Hence we are able to use the new variables p1 = zp2 with a complex parameter z and C1 # = 82 2 with a non-negative parameter 8. A close look at the optimization problems and at the de6nition of the complex roots a; b shows that the variable  cancels completely. The old roots a; b are thereby transformed to



4+1 1 + 8 1 + 84 − 1 x = 1 + i82 = +i = x1 + ix2 ; 2 2



2 84 + 1 1 + k 1 + k 2 84 − 1 y = 1 + ik82 = +i = y1 + iy2 2 2 with the parameter k = C2 =C1 . The result of this transform of variables are the optimization problems 1 D12 = sup C1 c1 ( 14 )8G(8; z) 2 z∈C 8¿0

and D22 =

1 sup C1 c1 ( 12 )G(8; z) 2 z∈C 8¿0

in the two variables 8 ¿ 0 and z ∈ C with G(8; z) =

|x|2 |z|2 + 2 Im(xz) + 1 : ( 12 82 x2 + 2x1 )|z|2 + 2 Im((xyT + 1)z) + ( 12 82 y2 + 2y1 )

The optimization with respect to the complex parameter z can be performed analytically. The function G has the representation G(8; z) =

|x|2 |z + a|2 2 T + |c|2 + 2x1 |z| + 2 Re(bz)

1 2 2 8 x2

with the complex parameters i a= ; x

2(xyT + 1) ; bT = −i 2 8 x2 + 4x1

|c|2 =

82 y2 + 4y1 : 82 x2 + 4x1

(26)

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167

The optimization problem sup z∈C

|z + a|2 T + |c|2 |z|2 + 2 Re(bz)

(27)

can be easily reformulated to sup z∈C

(r + |a − b|)2 |z + (a − b)|2 : = sup 2 2 2 2 2 2 |z| + |c| − |b| r¿0 r + |c| − |b|

It is possible to prove with the help of the formulae 82 =2x1 x2 ; x21 =x22 +1 and y21 =y22 +1 that the expression |c|2 − |b|2 equals to 84 x2 y2 + 8x22 y2 (x2 − y2 ) + 8y1 (x1 − y1 ) + 8x22 y1 (x1 − y1 ) (82 x2 + 4x1 )2 which is strictly positive, provided 8 ∈ (0; + ∞). Therefore the optimization problem has a unique solution with the optimal value 1+

|a − b|2 : |c|2 − |b|2

(28)

Plugging in the known values for a; b; c and using the notation 5x =

82 x2 + 2x1 ; 2

5y =

82 y2 + 2y1 ; 2

s = xyT + 1;

(29)

the result of the optimization (27) yields sup G(8; z) = z∈C

T |x|2 5y + 5x − 2 Re(sx) 82 (|x|2 y2 + x2 ) = 2 : 5x 5y − |s|2 84 x2 y2 + 482 Im(xy) − 4|xy − 1|2

This leads to the formulae D12 = sup

4 8¿0 8 x2 y2

q1 83 (|x|2 y2 + x2 ) ; + 482 Im(xy) − 4|xy − 1|2

(30)

q2 82 (|x|2 y2 + x2 ) (31) 4 2 2 8¿0 8 x2 y2 + 48 Im(xy) − 4|xy − 1| √ √ with q1 = 4 C1 c1 ( 14 ) = 4 23C1 and q2 = C1 c1 ( 12 ) = 23C1 . Due to the complicated structure of the complex roots x and y, the authors have not succeeded in calculating the results of these optimization problems analytically. It is, however, possible to prove that the functions to be optimized are continuous for 0 ¡ 8 ¡ + ∞ and that their limits for 8 → 0 and 8 → + ∞ exist and are bounded. Hence the optimization problems are solvable and the results can be calculated numerically with standard optimization techniques. In order to calculate the constant D0 of the trace estimate (15) we 6rst evaluate the expression D22 = sup

u; ˙ v Q + a(u; v)

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for two functions u; v ∈ L(0). The tangential component v1 of v is assumed to vanish on the boundary S. Consequently, the function u is de6ned by (21) in terms of its boundary data w via formula (20) and v is given by the same formula, if p1 and p2 are replaced by q1 =

iv˜2 D

and

q2 = −

av˜2 D

(32)

with the normal component of boundary data v2 . The bilinear form a(u; v) can be easily computed with the help of formula a(u; v) = 12 (a(u + v; u + v) − a(u; u) − a(v; v)): The result is given by   

1 E C1 #a2 a(u; v) = (|p1 + q1 |2 − |p1 |2 − |q1 |2 ) + 2a1 2 1+ R R 2 2  1 C1 #b2 2 2 2 2 + 2b1  + (|p2 + q2 | − |p2 | − |q2 | ) 2 2   2 1 T − 2 Re i(ab +  ) ((p1 + q1 )(pT2 + qT2 ) − p1 pT2 − q1 qT2 ) d d#: 2 This is easily simpli6ed to   

E C1 #a2 2 Re(p1 qT1 ) a(u; v) = + 2a1  2 1+ R R  C1 #b2 2 + Re(p2 qT2 ) + 2b1  2   p1 qT2 + q1 pT2 d d#: − 2 Re i(abT + 2 ) 2 It remains to calculate the acceleration part u; ˙ v Q . This expression is given in Fourier transformed variables by   u; ˙ v Q = Re i# u(#; ˜ ; y)v(#; ˜ ; y) dy d d# : R

R×R+

Using the representations of u˜ and v˜ and performing the inner integration with respect to y yields u; ˙ v Q = Re

  R

 |a|2 + 2 |b|2 + 2 i# p1 qT1 + p2 qT2 + i(p2 qT1 − p1 qT2 ) d d# : 2a1 2b1 R

With the formulae |a|2 + 2 = 2a21 and |b|2 + 2 = 2b21 we obtain   u; ˙ v Q = Re [i# a1 p1 qT1 + i# b1 p2 qT2 + #(p1 qT2 − p2 qT1 )] d d# : R

R

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169

Adding both expressions for a(u; v) and u; ˙ v Q and using the de6nition of C1 = (2 + 2)=E the following result is derived: u; ˙ v Q + a(u; v)   E Re = [(iC1 # aT + 4a1 2 )p1 qT1 + (iC1 # bT + 4b1 2 )p2 qT2 2 + 2 R R

2 2 T T +  ))p2 qT1 ] d d# : − i(iC1 # + 2(ab +  ))p1 qT2 + i(iC1 # + 2(ab

This can be simpli6ed with the formula (32) for the expressions q1 and q2 . After some calculation one obtains   1 u; ˙ v Q + a(u; v) = Re [2i ap1 − (22 + iC1 #)p2 ]v˜T2 d d# : C1 R R Now a standard HKolder inequality gives u; ˙ v Q + a(u; v) 1 6 v2 H 1=4; 1=2 (S) C1

 R2

|2i ap1 − (22 + iC1 #)p2 |2 d d# c1 ( 14 )|#|1=2 + c1 ( 12 )||

1=2 :

The second factor on the right-hand side of the previous formula shall be estimated by D0 ua . Due to the representation (23) of u2a it can be seen that D02 is the solution of the optimization problem 1 D02 = sup (c1 ( 14 )|#|1=2 + c1 ( 12 )||)−1 2C 1 p1 ;p2 ∈C ;#∈R

|p1

|2 (C

1 #a2 =2

+ 2a1

2 )

|2i ap1 − (22 + iC1 #)p2 |2 : + |p2 |2 (C1 #b2 =2 + 2b1 2 ) − 2 Re(i(abT + 2 )p1 pT2 )

As in the optimization problems for the calculation of D1 ; D2 we use now the transform of variables p1 = zp2 with z ∈ C and C1 # = 82 2 . The resulting problem is   1 |z + a|2 4|x|2 √ sup sup T + |c|2 2( C1 c1 ( 14 )8 + C1 c1 ( 12 )) 5x z∈C |z|2 + 2 Re(bz) 8¿0 with 5x from (29) and the parameters  82 i 1+i a= x 2 and b; c de6ned as in (26). The result of the optimization with respect to z is given by formula (28). Bearing in mind the de6nitions of a; b and c, we can derive the remaining one-parametric optimization problem 84 (82 x2 + 4x1 ) − 482 (|x|2 y2 + x2 ) : 4 2 2 8¿0 (q1 8 + q2 )(8 x2 y2 + 48 Im(xy) − 4|xy − 1| )

D02 = sup

(33)

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As in the problems before, the authors are not able to give an analytical solution. If the Poisson ratio  satis6es −1 ¡  ¡ 12 then it can be proved that the function which is optimized is non-negative, continuous and that its limits for 8 → 0 and 8 → + ∞ exist and are bounded. Therefore the problem has a solution. Up to now we have proved the validity of the generalized trace estimates (15) – (17) for solutions u ∈ L(0) of the homogeneous di2erential equations only. The general result can be derived easily from this result. Each solution u ∈ L(f) can be decomposed into a solution u(1) ∈ L(0) of the homogeneous di2erential equations with the given boundary data and a solution u(2) ∈ L(f) of the inhomogeneous equations with homogeneous boundary data u(2) = 0 on S. Since the function u(1) satis6es the trace estimates it remains to prove the a priori estimate u(2) a 6 c)2 fV −1=2; −1 (Q) for the second part. This is done in the next proposition. Then the combination of the estimates for u(1) and u(2) gives the trace estimates (15) – (17) for the general case. Proposition 2. Let u ∈ L(f) be the solution of the inhomogeneous parabolic system on the half-space Q with homogeneous boundary data u = 0 on S and a functional f ∈ V −1=2; −1 (Q). Then there holds ua 6 c)3 uV 1=2; 1 (Q) 6 c)4 fV −1=2; −1 (Q)

(34)

with constants c)3 ; c)4 independent of f. Proof. The solution u satis6es the variational equation u; ˙ v Q + a(u; v) = f; v Q

(35)

for all v ∈ VO 1=2; 1 (Q). Therefore the a priori estimate e(u)2L2 (Q;R4 ) 6 c)5 fV −1=2; −1 (Q) uV 1=2; 1 (Q)

(36)

and the dual estimate u ˙ V −1=2; −1 (Q) 6 c)6 fV −1=2; −1 (Q) + c)7 e(u)L2 (Q;R4 )

(37)

follow. Due to the vanishing boundary data the solution u can be extended from the half-space Q onto the whole space R3 by u(&; x1 ; −x2 ) = −u(&; x1 ; x2 )

for x2 ¡ 0

1=2; 1 (R3 ); vH 1=2; 1 (R3 ) ¡ = ∞}. Then for any such that u ∈ V 1=2; 1 (R3 ) := {w ∈ Hloc v ∈ V 1=2; 1 (R3 ) there holds

u; ˙ v R3 = u; ˙ v R2 ×R+ + u; ˙ v R2 ×R− = u; ˙ v+ − v− Q with v+ (&; x1 ; x2 ) = v(&; x1 ; x2 ) and v− (&; x1 ; x2 ) = v(&; x1 ; −x2 ) for x2 ¿ 0. The di2erence v+ − v− has vanishing boundary data on R2 × {0} and is therefore an admissible test

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171

function for variational equation (35). Moreover; v+ − v− V 1=2; 1 (Q) 6 c)8 vH 1=2; 1 (R3 ) is valid. Consequently there holds u ˙ V −1=2; −1 (R3 ) ≡

sup

v∈V 1=2;1 (R3 ) vH 1=2;1 (R3 ) =1

u; ˙ v R3 6 c)9 fV −1=2; −1 (Q) + c)10 e(u)L2 (Q;R4 ) :

(38) Via the Fourier transform with respect to all variables the following estimate is derived:  u2 |u| ˆ 2 c1 ( 12 )|#| d d# = H 1=2; 0 (R3 ) 6 c1 ( 12 )

R3

1=2  1=2 |#|2 2 2 1 d d# |u| ˆ (c1 ( 2 )|#|+|| ) d d# |u| ˆ c1 ( 12 )|#|+||2 R3 R3



2

˙ V −1=2; −1 (Q) uV 1=2; 1 (Q) : 6 c)11 u Observe that the Fourier transforms of elements from V 1=2; 1 (R3 ) exist in the sense of Fourier transforms of distributions; but they need not be square-integrable. The above-performed calculation yields u2 ˙ V −1=2; −1 (Q) uV 1=2; 1 (Q) : H 1=2; 0 (Q) 6 c)12 u

(39)

Due to the Korn inequality for the half-space; the norms ∇·L2 (Q;R4 ) and e(·)L2 (Q;R4 ) are equivalent. The proof of the Korn inequality for the half-space directly repeats the corresponding procedure made in [10] for in6nite strips with the only change of the Fourier series by the Fourier transform also in the normal variable. Hence from estimates (36) – (39) the second inequality in (34) follows easily. Corollary 1. Let  be a bounded domain with Lipschitz boundary  and let u be the solution of the parabolic system (13) on QT with Dirichlet data u = w ∈ H 1=4; 1=2 (ST ) on ST ; f ∈ H 1=2; 1 (QT )∗ and initial data u(0; ·)=u0 ∈ L2 (). The boundary data have an extension w ∈ H 1=2; 1 (QT ) such that w˙ ∈ H 1=2; 1 (QT )∗ . Then the a priori estimate uH 1=2; 1 (QT ) 6 c)13 (wH 1=2; 1 (QT ) + w ˙ H 1=2; 1 (QT )∗ + fH 1=2; 1 (QT )∗ + u0 L2 () ) (40) is valid with a constant c)13 independent of w; f and u0 . Proof. The function u˜ = u − w satis6es the equation with vanishing boundary data; modi6ed initial data u˜ 0 = u0 − w(& = 0) and modi6ed functional f : v → f; v QT − w; ˙ v QT − a(w; v). This functional still belongs to H 1=2; 1 (QT )∗ and there holds fH 1=2; 1 (QT )∗ 6 fH 1=2; 1 (QT )∗ + c)14 wH 1=2; 1 (QT ) + w ˙ H 1=2; 1 (QT )∗ . Hence it suGces to show the assertion for w = 0. Like in the preceding two Propositions we get the a priori estimate e(u)2L2 (QT ;R4 ) + u(T; ·)2L2 () 6 c)15 (fH 1=2; 1 (QT )∗ uH 1=2; 1 (QT ) + u0 2L2 () )

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C. Eck, J. Jaru)sek / Nonlinear Analysis 53 (2003) 157 – 181

and the dual estimate u ˙ HO 1=2; 1 (Q )∗ 6 c)16 (e(u)L2 (QT ;R4 ) + fH 1=2; 1 (QT )∗ ) T

with HO 1=2; 1 (QT ) := {v ∈ H 1=2; 1 (QT ); v = 0 on ST }. By interpolation we obtain 1=2 uH 1=2; 0 (QT ) 6 c)17 u ˙ 1=2 uH 1=2; 1 (Q ) ∗ ˙ 1=2; 1 T H

(QT )

and from the coerciveness of stresses and the vanishing boundary data there follows uL2 (QT ) 6 c)18 e(u)L2 (Q;R4 ) : Combining these estimates we obtain 2 u2H 1=2; 1 (QT ) 6 c)19 (u2 H 1=2; 0 (QT ) + e(u)L2 (QT ;R4 ) ) 1=2 1=2 6 c)20 (fH 1=2; 1 (QT )∗ + fH 1=2; 1 (Q )∗ uH 1=2; 1 (Q ) T T

+ u0 L2 () )uH 1=2; 1 (QT ) + c)21 u0 2L2 () and the assertion is proved. For a slightly more regular right-hand side f (just f ∈ H 1=2−<; 1 (QT )∗ for any small < ¿ 0 suGces) and boundary data the standard estimate of sup&∈IT u(&; ·)L2 () is easily proved. For such a case the meaning of the initial condition is obvious. Every f ∈ H 1=2; 1 (QT )∗ can be approximated by a sequence of more regular fk in the H 1=2; 1 (QT )∗ norm. For the di2erences uk1 − uk2 of solutions the estimates in the proof remain valid with right-hand side fk1 − fk2 (the problem solved here is linear), this explains the meaning of the initial condition for the limit u. ˙ The validity of Corollary 1 is easily extended to more general boundary conditions including that of Theorem 1, provided mes U ¿ 0. The proof for the case mes U = 0 requires certain changes. Remark. The existence of uniform estimates for uH 1=2 (IT ;L2 ()) and u(T; ·)L2 () together with the initial condition yield the uniform estimate for u˙ in the space H 1=2 (IT ; L2 ())∗ . This can be veri6ed after application of the standard extension technique via the Fourier transform in time and with the help of the inequality |#|2 (1 + c1 ( 12 )|#|)−1 6 (c1 ( 12 ))−2 (1 + c1 ( 12 )|#|). In particular; this shows that the requirement to uK in (8) is ful6lled.

5. Regularity of the solution of the contact problem The regularity of the solution of the contact problem is studied with the method outlined in [8] for the anisotropic case. The only di2erence in the present study are the better trace estimates for the isotropic case presented in the preceding section. The result of the regularity investigation is the following: Proposition 3. In addition to the assumptions of Theorem 1 let C ∈ C 6 for an arbitrary 6 ¿ 2; f ∈ H 1=4; 1=2 (QT )∗ and u1 ∈ H 3=2 (). The coe
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173



form a(0) shall belong to C 6 () with an exponent 6 ¿ 12 . The support of F is contained in R2 × F with F having a positive distance to  \ C and there holds −1   FL∞ (C ×R2 ) ¡ CF = D0 D12 + D22 with the constants from Proposition 1. Then the solution of the approximate contact problem (11) satis=es the estimate u˙ ; * H 1=2; 1 (SF ) +  1 [(u˙ ; * )n ]+ L2 (SF ) 6 c)22

(41)

with a constant c)22 independent of the approximation parameters ; * and with SF = IT × F . Proof. The main steps of the proof are the same as in [8]; proof of Proposition 4. For convenience of the proof we apply the shift technique to the smoothed problem (11) here; whereas in [8] it was applied to the penalized but non-smoothed problem. We consider some small neighbourhood Ox0 of a point x0 ∈ F and a cut-o2 function >x0 ∈ C 2 (R2 ; [0; 1]) with supp(>x0 ) ⊂ Ox0 and >x0 ≡ 1 near x0 . In addition to this cut-o2 function for the space variables we use a parameter-dependent cut-o2 function ’@ (&) = ’0 (& − @) for the time variable as described in [8]. The function ’0 ∈ C 2 (R; [0; 1]) is supposed to be non-increasing and to satisfy ’0 (&) = 1 for & ¡ − 1 and supp(’0 ) = (− ∞; 0]. The product of the cut-o2 functions for the time and the space variables is %(&; x) := >x0 (x)’@ (&). We use a local transformation R of Ox0 onto a subset of the half-space R × R+ , de6ned by R = B ◦ O, where O is a combination of a rotation and a translation transforming the tangential plane of  at x0 onto R × {0} and B : (x1 ; x2 ) → (x1 ; x2 − (x1 )). Here, is the function by which the boundary  is locally represented as graph over its tangential plane. Due to the required regularity of the boundary there holds ∈ C 6 and  C 1 6 c)23 < with < denoting the diameter of the support of the cut-o2 function >. We localize problem (11) by using the test function %v and transform the localized version onto a half-space Q = R × R × R+ with boundary S by employing the transform of variables xnew = R(xold ). Thus we obtain the variational equation   1 ˙ v) + a(1) (u; V v) + u; VO v Q + a(0) (u; [uV n ]+ J; vn  S   1 (42) + F [uV n ]+ ∇)* (u˙ t )J; vt = F; v Q :  S The functions uV and uO are suitable extensions of %(&; x)(u˙; * (&; x) − u1 (x)) and %(&; x) (u; * (&; x) − u0 (x)) from R(Ox0 ) onto the half-space. The bilinear forms a(0) , a(1) are given by the original de6nition in (9), if the domain of integration QT there is replaced by Q. For the de6nition of the elastic bilinear form a(0) the coeGcients a(0) ijk‘ must be extended in a suitable way. The term J represents the density of surface measure arising from the straightening of the boundary. The linear functional F depends in a technically rather complicated way on the old functional f, on the bilinear forms a(0) , a(1) , on the initial data u0 ; u1 , on lower-order derivatives or small terms of the solution u;* , on the cut-o2 function % and on the transformation function employed for the

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local straightening of the boundary. Eq. (42) is the basis for the regularity proof. For a function g : Q → R and parameters h ∈ R, q ∈ R let g−h (&; x) := g(&; x1 + h; x2 ) g−q (&; x) := g(& + q; x)

and

and

Gh g := g−h − g;

Gq g := g−q − g

denote the shifted functions and corresponding di2erence operators. Observe that the space shift is performed with respect to the tangential variable x1 only. We put the test function v := uV −h − uV into variational equation (42). Then we shift the whole equation (42) into the direction of h and put the test function v−h := uV − uV −h into the shifted equation. We add both equations, multiply the result by |h|−2 and integrate the product with respect to h ∈ R1 . The same is done for the time shift q and the factor |q|−3=2 instead of |h|−2 . We add the results of both the space shift and the time shift procedures and estimate most of the terms by techniques described in [8]. For the estimates of terms containing the appropriate di2erences of the coeGcients of the bilinear forms their indicated HKolder continuity is suGcient. In the sequel we use the abbreviations S = 1 [uV n ]+ J , G = F(u)∇) ˙ * (u˙ t )S,   −2 (1) A1 = |h| a (Gh u; V Gh u) V dh; A2 = |q|−3=2 a(1) (Gq u; V Gq u) V dq; R

 A3 =  J=

R

R3

R

|h|−2 a(1) (Gh uO ; Gh uO ) dh; |h|−2 Gh GGh (uV t ) dsx d& dh +

 R3

|q|−3=2 Gq GGq (uV t ) dsx d& dq:

The estimate of the penalty and friction terms is delicate, because the normal vector n (of the original boundary) depends on the space variable. From the space shifts of the penalty functional we get the terms S−h (uV −h · n−h − uV · n−h ) + S(uV · n − uV −h · n): Using the alternative representation of this term by Gh SGh (uV · n) − Gh SuG V h n + SGh uG V h n; the property n ∈ C 6−1 and J ∈ C 6−1 with 6 ¿ 2 valid due to the regularity of the boundary, the monotonicity of the penalty functional Gh [uV n ]+ Gh uV n ¿ 0, and the estimate uvH 5 (M ) 6 c)24 uH 5 (M ) vL∞ (M ) + c)25 (<)uL2 (M ) vC 5+< (M ) valid for < ¿ 0, the space shifts of the penalty term are estimated by  − |h|−2 (S−h (uV −h · n−h − uV · n−h ) + S(uV · n − uV −h · n)) dsx d& dh R3

6 c)26 SL2 (S) u V H 1=4; 1=2 (S) + c)27 u V H 1=2; 1 (S) SH −1=4; −1=2 (S) : From the friction functional there arises G−h T−h (uV −h − u) V + GT(uV − uV −h )

(43)

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175

with Tv := vt . Due to the regularity of the boundary, the function T satis6es TC 6−1 6 c)28 . Using a representation of (43) by V V − Gh GGh T uV + GGh TGh u; Gh GGh (Tu) we derive the estimate  V + GT(uV − uV −h )) dsx d& dh |h|−2 (G−h T−h (uV −h − u) R3

 6

R3

|h|−2 Gh GGh (uV t ) dsx d& dh + c)29 GL2 (S) u V H 0; 1=2 (S) :

The estimates of all other terms are described in [8]. Thus we arrive at the important inequality V H 1=2; 1 (S) + c)33 : A1 + A2 6 (1 + <)|J| + c)30 A3 + c)31 SL2 (S) + c)32 u

(44)

The constants c)% , % = 30; 31; : : : ; here and in the sequel may depend on the input data, on lower-order derivatives of the solutions and on the small parameter <, but are independent of the approximation parameters , *. The term < in the above formula can be made arbitrarily small by choosing the support of the cut-o2 function % in the localization procedure suGciently small. The crucial term in this estimate is the friction part J. Using the Fourier transform it can be estimated by      1  1  1=2   ˆ ˆ c1 2 || + c1 4 |#| Re(G uV t ) d d# |J| =  2 R

6 FL∞ (C ×R2 ) SL2 (S) H1=2 with

 H≡



R2

c1

1 2

|| + c1

1 4

(45)

2 |#|1=2 |uVˆ t |2 d d#:

Here it is important that |∇)* | 6 1 and therefore |G| 6 |FS| holds. Application of the HKolder inequality with parameter p ¿ 1, of the inequality uV t H 0; 5 (S) 6 u V H 0; 5 (S) + c1 u V L2 (S) and of the renormation technique as described in [8] yields   V H2 0; 1=2 (S) dh H 6 (1 + p) |h|−2 Gh u 

R

1 + 1+ p

 R



|q|−3=2 Gq u V H2 1=4; 0 (S) dq + c)34 :

Here the special trace estimates for the isotropic problem derived in the preceding section are employed, this results in the inequality  

1 2 2 D1 A2 + c)35 : H 6 (1 + <) (1 + p)D2 A1 + 1 + p This relation is valid for all parameters < ¿ 0, p ¿ 1. Optimization with respect to p leads to  2 H 6 (1 + <) D2 A1 + D1 A2 + c)36 : (46)

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Now the remaining task is the estimate of the term SL2 (S) . This expression may be represented by  2 2 SL2 (S) = SH −1=4; −1=2 (S) + |h|−2 Gh S2H −1=4; −1=2 (S) dh  +

R

R

|q|−3=2 Gq S2H −1=4; −1=2 (S) dq:

(47)

The Green formula, derived from variational equation (42), yields − S; w S = u; V˙ w Q + a(1) (u; V w) + a(0) (u; ˙ w) − F; w Q ;

(48)

valid for functions w having only a normal component on the boundary. Remember that n is the normal with respect to the old (unstraightened) boundary. For a given value h ∈ R we use the special test function w(h) de6ned as the solution of the homogeneous purely viscous problem (13) with boundary data wt(h) = 0 and  1  1=2   −1  wˆ n(h) (#; ) = G[ + c1 12 || h S(#; ) 1 + c1 4 |#| on S. This function satis6es the relations wn(h) 2H 1=4; 1=2 (S) = Gh S2H −1=4; −1=2 (S) = Gh S; wn(h) S : Due to the employed recti6cation of the boundary and the required smoothness of the boundary there holds n − e2 C 0 (R2 ) 6 c)37 < with < arbitrarily small, if the support of the employed cut-o2 function % is chosen small enough. Here, e2 = (0; 1) is the normal to the boundary of the half-space domain. As a consequence there holds   w − wn · e2 H 5; 25 (S) 6 c)38 (<wH 5; 25 (S) + wL2 (s) ); 5 ∈ 0; 12 : This relation in particular enables us to use the special inverse trace estimate (15) with constant D0 computed for functions whose component w1 (not wt as in our case) vanishes. We put w(h) into the Green formula (48), then we shift (48) into the direction of h and put −w(h) into the result. We add both equations, multiply the result with |h|−2 and integrate with respect to h. Due to the construction of w(h) we obtain after some estimation including the special trace estimate (15) the inequality   −2 2 |h| Gh SH −1=4; −1=2 (S) dh 6 (1 + <) |h|−2 D0 Gh u V a Gh SH −1=4; −1=2 (S) dh R

+ (c)39



 A3 + c)40 )

R

R

1=2

|h|−2 Gh S2H −1=4; −1=2 (S) dh

+ c)41

with < ¿ 0 arbitrarily small. From this we obtain  |h|−2 Gh S2H −1=4; −1=2 (S) dh 6 (1 + <)D02 A1 + c)42 A3 + c)43 : R

(49)

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177

Repeating the same procedure for the time shifts with the special test function de6ned 1=2 + c1 ( 12 )||)−1 , the analogous relation by wˆ n(q) = G[ q S(#; )(1 + c1 (1=4)|#|  |q|−3=2 Gq S2H −1=4; −1=2 (S) dq 6 (1 + <)D02 A2 + c)44 (50) R

is established. Hence S2L2 (S) is bounded by S2L2 (S) 6 (1 + <)D02 (A1 + A2 ) + c)45 A3 + c)46 :

(51)

As a consequence of the inequalities (46) and (51) the inequality (44) can be modi6ed to A1 + A2 6 (1 + <)FL∞ (C ×R2 ) D0 A1 + A2 (D2 A1 + D1 A2 ) + c)47 A3 + c)48 :

√

√

(52)

−1=2 Using (D2 A1 + D1 A2 ) the HKolder inequality, we estimate the term (A1 + A2 ) by D12 + D22 . Hence the result of all our estimates is the equation −1 A1 + A2 6 (1 + <)FL∞ (C ×R2 ) CF (A1 + A2 ) + c)49 A3 + c)50

(53)

with the constant CF de6ned in the proposition. In order to 6nish the proof it is necessary to 6nd an estimate for the term A3 . For this estimate, the special form of the cut-o2 function ’@ with respect to the time variable is essential. The expressions A1 , A2 and A3 implicitly depend on the parameter @. Observe that none of the constants c)i , i = 49; 50, in (53) depends on @, since the C 2 -norm of ’@ is independent of @. Using the formula @’0 (& − @)=@@ = −@’0 (& − @)=@& it is easy to prove with some partial integration the inequality d A3 (@) 6 2 A3 (@)( A1 (@) + c)51 ): d@ From the special properties of ’0 it is seen that A3 is a non-decreasing function of @ with the initial value A3 (0) = 0. Consequently, the integration of the previous formula with respect to @ yields  @ A3 (@) 6 c)52 A1 (H) dH + c)53 (54) 0

for all @ ¡ T with c)52 , c)53 depending only on T . Hence, if the crucial estimate FL∞ (C ×R2 ) ¡ CF for the coeGcient of friction is satis6ed, then from (53) and (54) there follows with the Gronwall lemma the estimate A1 + A2 6 c)54 : The constant c)54 depends on the H 1=4; 1=2 (QT )∗ -norm of the given volume data f, on the coeGcients of the bilinear forms a(0) and a(1) , on the C 2 -norms of the cut-o2 functions >x0 and ’@ , on the C 6 -norms of the local maps from the straightening of the boundary and on the H 1=2; 1 (QT )-norm of the solution u; * . However, due to the assumptions of the proposition and due to the a priori estimates of Theorem 1 for the

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solution u; * all these terms are bounded, hence c)54 can be chosen independently of the solution u; * and of the local observation point x0 . The localized H 1=2; 1 (SF )-norm of u; * (for a suitably chosen covering of SF ) is equivalent to u V H 1=2; 1 (S) up to some 1 constant and the localized L2 (SF )-norm of  [u˙ n ]+ is related with SL2 (S) . Hence √ both expressions are bounded in terms of A1 + A2 and the proposition is proved. For details see [8]. 6. Existence of solutions The a priori estimates of Theorem 1 and Proposition 3 ensure that there exists a sequence {uk ;*k } of solutions of the approximate contact problem with k → 0 and *k → 0 such that uk ;*k → u and u˙ k ;*k → u˙ weakly in H 1=2; 1 (QT ) and strongly in Lp (SF ) for all p ¡ 6, n (uk ;*k ) = −(1=k )[(u˙ k ;*k )n ]+ → n (u) weakly in L2 (SF ), u˙ k ;*k (T ; ·) → u(T ˙ ; ·) weakly in L2 () and F(u˙ k ;*k ) → F(u) ˙ in Lp (SF ) for all p ¡ + ∞. With these convergence properties it is possible to prove that the limit function u is a solution of the original contact problem with friction (8). Let us sum up the assumptions: Assumption 1. Let  be a bounded domain with C 0; 1 -boundary . The contact part of the boundary C is an element of C 6 for some 6 ¿ 2. The bilinear form a(0) is symmetric and bounded in the sense of (2) with coeGcients being HKolder continuous with an exponent 6 ¿ 12 . For the viscosity the relation (3) is assumed. The given data satisfy f ∈ H 1=4; 1=2 (QT )∗ ; b ∈ L2 (IT ; H 1=2 (F )∗ ); U ∈ H 2 (QT ); U ≡ 0 on SC; T ; U (0; ·) = u0 ; U˙ (0; ·) = u1 ; u1 ∈ H 3=2 (). The coeGcient of friction F = F(x; u) ˙ is non-negative; its support is contained in F × R2 with dist(F ;  \ C ) ¿ 0; satis6es the CarathXeodory property and is strictly smaller than the constant CF . Then the following result is valid: Theorem 2. If Assumption 1 holds; then the contact problem with Coulomb friction has at least one solution. The upper bound CF for the coeGcient of friction depends on the modulus of viscosity E and the Poisson ratio  of the viscous part of the strain–stress relation. It is given in terms of the values D0 , D1 and D2 which are solutions of the one-parametric optimization problems (33), (30) and (31). Unfortunately, the authors have not succeeded in solving these problems and they are therefore not able to present a formula for the admissible magnitude of the coeGcient of friction. However, the mentioned optimization problems are well suited for standard numerical optimization procedures. We have employed a simple optimization algorithm based on curve 6tting with a three-point pattern as described e.g. in [9, Section 7:3]. The results of this optimization are shown in the Figs. 1–3. In Fig. 1 the admissible coeGcient of friction is shown for the modulus of viscosity E = 1 in dependence on the viscous Poisson ratio. In Figs. 2 and 3 the coeGcient of friction is depicted in dependence on E and  from

C. Eck, J. Jaru)sek / Nonlinear Analysis 53 (2003) 157 – 181

179

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.1

0.2

0.3

0.4

0.5

–1

–0.8 –0.6 –0.4 –0.2

0

0.2

0.4

Fig. 1. Upper bound for the coeGcient of friction for E = 1.

di2erent viewpoints for positive and negative values of the Poisson ratio. The admissible coeGcient of friction converges to 0 for both very small and very large values of E. Hence our existence result does not give any information about both limit problems, the quasistatic contact problem and the purely elastic contact problem without any viscous damping. However, for moderate values of E and  the magnitude of the admissible coeGcient of friction varies between 0.4 and 0.5. This is suGcient for many cases of practical relevance where the coeGcient of friction has values around 0.3. The case of Poisson ratio  = 0:5 is not included in our investigation, since the representation (18) of the di2erential equation used in the proof of the special trace estimates (15) – (17) becomes singular in this case. In the limit  → 0:5 the admissible coeGcient of friction we have obtained here also tends to zero. The requirement of continuous dependence of the coeGcient of friction F on the solution u˙ from Assumption 1 can be easily weakened to the following special requirement: for almost all x ∈ C and u˙ n = 0, F(·; 0; u˙ t ) is continuous outside u˙ t = 0, limu˙ t →0 F(·; 0; u˙ t ) exists and F(·; 0; 0) ¿ limu˙ t →0 F(·; 0; u˙ t ). Indeed, the value of F at u˙ n ¡ 0 is unimportant for the original problem since then n (u) is 0. Moreover, if we de6ne F0 (·; 0; u˙ t ) = F(·; 0; u˙ t ) for u˙t = 0, F0 (·; 0; 0) = limu˙ t →0 F(·; 0; u˙ t ) and F0 (·; u˙ n ; ·) ≡ F0 (·; 0; ·) for u˙ n ¡ 0, then F0 satis6es Assumption 1. Theorem 2 ensures the existence of solutions for the problem with the coeGcient of friction F0 . A look at the inequality (8) shows that such solutions solve the problem for the original coeGcient of friction F, too. On the other hand, the solutions of the problem with F need not be, in general, solutions of the problem with F0 . However, this shows the solvability of problems with di2erent coeGcient of friction of slip and stick and probably will result in a general non-uniqueness of solutions of frictional contact problems.

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C. Eck, J. Jaru)sek / Nonlinear Analysis 53 (2003) 157 – 181

0.4

0

0.4 –10

0.2

0.2

0.1 0.2

5

0.3 0.4

0.2

0 10

log 1

0

0E

–5 0 0

0.1

0.3 5

10

0.5

0

log E 10

0.4 –5

–10

0.5

0.4 0.3 0.4 0.2 0.3 0.1 0.2 0

0.1 0.2

0.5 0.4

0 –10

0.4 10

–5

0

5

–10

0.3 –5

0.2 0

log E 10

log 10E

0.1

5

10 0

Fig. 2. Upper bound for the coeGcient of friction in dependence on the modulus of viscosity (E) and the viscous Poisson ratio () for 0 6  ¡ 0:5.

0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 –10

–5

0 log10 E

5

0 –0.1 –0.2 –0.3 –0.4 –0.5 10

0

0 –0.1 –0.2 –0.3 –0.4

10 5 –5 –10

0 E log 10

Fig. 3. Upper bound for the coeGcient of friction in dependence on the modulus of viscosity (E) and the viscous Poisson ratio () for −1 ¡  ¡ 0.

C. Eck, J. Jaru)sek / Nonlinear Analysis 53 (2003) 157 – 181

181

Acknowledgements This work was partially supported by the Czech Academy of Sciences by the Grants A 1075707 and A 1075005. References [1] L.-E. Andersson, Existence result for quasistatic contact problems with Coulomb friction, Appl. Math. Optim. 42 (2000) 169–202. [2] K.T. Andrews, M. Shillor, S. Wright, Dynamic evolution of an elastic beam in frictional contact with an obstacle, in: M. Raous, E. Pratt (Eds.), Contact Mechanics, Proceedings of the Second Cont. Mech. International Symposium Carry-le-Rouet, Plenum Press, New York, 1995, pp. 49–56. [3] G. Duvaut, J.L. Lions, Les inXequations en mXecanique et en physique, Dunod, Paris, 1972. [4] J. Jaru)sek, Contact problems with bounded friction. Coercive case, Czechosl. Math. J. 33 (1983) 237–261. [5] J. Jaru)sek, Contact problems with given time-dependent friction force in linear viscoelasticity, Commun. Math. Univ. Carolinae 31 (1990) 257–262. [6] J. Jaru)sek, Dynamic contact problems with given friction for viscoelastic bodies, Czechosl. Math. J. 46 (1996) 475–487. [7] J. Jaru)sek, C. Eck, Dynamic contact problems with friction in linear viscoelasticity, C. R. Acad. Sci. Paris SXer. I 322 (1996) 497–502. [8] J. Jaru)sek, C. Eck, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions, Math. Models Methods Appl. Sci. 9 (1) (1999) 11–34. [9] D.G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, MA, 1973. [10] J. Necas, ) J. Jaru)sek, J. Haslinger, On the solution of the variational inequality to the Signorini problem with small friction, Bollettino U.M.I. 5 (17-B) (1980) 796–811.