Boundary Variational Inequality Approach, B

Rend. Lincei Mat. Appl. 23 (2012), 267–293 DOI 10.4171/RLM/628

Continuum Mechanics — Frictionless contact problems for elastic hemitropic solids: Boundary variational inequality approach,byA. Gachechiladze, R. Gachechiladze and D. Natroshvili, communicated on 22 June 2012.

Dedicated to Professor Gaetano Fichera on the occasion of 90th anniversary of his birthday.

Abstract. — The frictionless contact problems for two interacting hemitropic solids with di¤erent elastic properties is investigated under the condition of natural impenetrability of one medium into the other. We consider two cases, the so-called coercive case (when elastic media are ﬁxed along some parts of their boundaries), and the semicoercive case (the boundaries of the interacting elastic media are not ﬁxed). Using the potential theory we reduce the problems to the boundary variational inequalities and analyse the existence and uniqueness of weak solutions. In the semicoercive case, the necessary and su‰cient conditions of solvability of the corresponding contact problems are written out explicitly.

Key words: Hemitropic solids, Cosserat solids, frictionless contact, boundary variational inequality.

2010 Mathematics Subject Classification: 35J86, 47A50, 74A35, 74M15.

Introduction

In the present work we consider a frictionless contact of two elastic hemitropic media with di¤erent physical properties under the condition of natural impenetrability. Here we consider the model of the theory of elasticity in which, unlike the classical theory, an elementary particle of a body along with displacements undergoes rotation, and hence the condition of mechanical equilibrium of the body is described by means of the three-component displacement vector and three-component micro-rotation vector. The origin of the rational theories of polar continua goes back to brothers E. and F. Cosserat [CC1], [CC2], who gave a development of the mechanics of continuous media in which each material point has the six degrees of freedom defined by 3 displacement components and 3 micro-rotation components (for the history of the problem see [Min1], [Now1], [KGBB1], [Dy1], and the references therein). A micropolar continua which is not isotropic with respect to inversion is called hemitropic, noncentrosymmetric, or chiral. Materials may exhibit chirality on the atomic scale, as in quartz and in biological molecules—DNA, as well as 268 a. gachechiladze, r. gachechiladze and d. natroshvili on a large scale, as in composites with helical or screw–shaped inclusions, certain types of nanotubes, bone, fabricated structures such as foams, chiral sculptured thin films and twisted fibers. For more details and applications see the references [Er1], [HZ1], [Sh1], [Ro1], [La1], [Mu1], [Mu2], [Now1], [Dy1], [YL1], [LB1] [CC1]. Refined mathematical models describing the hemitropic properties of elastic materials have been proposed by Aero and Kuvshinski [AK1], [AK2]. In the mathematical theory of hemitropic elasticity there are introduced the asymmetric force stress tensor and couple stress tensor, which are kinematically related with the asymmetric strain tensor and torsion (curvature) tensor via the constitutive equations. All these quantities are expressed in terms of the components of the displacement and micro-rotation vectors. In turn, the displacement and micro- rotation vectors satisfy a coupled complex system of second order partial di¤erential equations. We note that the governing equations in this model become very involved and generate 6 6 matrix partial di¤erential operator of second order. Evidently, the corresponding 6 6 matrix boundary di¤erential operators describing the force stress and couple stress vectors have also an involved structure in comparison with the classical case. In [NGS1], [NGZ1], [NS1], [NGGS1] the fundamental matrices of the associ- ated systems of partial di¤erential equations of statics and steady state oscilla- tions have been constructed explicitly in terms of elementary functions and the basic boundary value and transmission problems of hemitropic elasticity have been studied by the potential method for smooth and non-smooth Lipschitz domains. Particular problems of the elasticity theory of hemitropic continuum have been considered in [EL1], [La1], [LB1], [LVV1], [LVV2], [Now1], [Now2], [NN1], [We1]. Unilateral boundary value problems for hemitropic elastic solids have been studied in [GGN1], while the contact problems were treated in [GaGaNa1] with the help of spatial variational inequality technique. The main goal of the present paper is the study of frictionless contact problems for hemitropic elastic solids, their mathematical modelling as transmission- boundary value problems with natural impenetrability conditions and their analysis with the help of the boundary variational inequality technique based on properties of the corresponding potential operators. This approach reduces the dimension of the problem by one which is important for numerical realiza- tions. Similar unilateral problems of the classical linear elasticity theory with various modifications have been considered in many monographs and papers (see, e.g., [DuLi1], [Fi1], [Fi2], [GaNa1], [HHNL1], [KiOd1], [Ki1], [Rod1], and the references therein). More general problems, including the unilateral problems of non- linear classical elasticity, are studied in [BBGT]. The work consists of five sections and is organized as follows. First, in Section 1, we collect the basic field equations of statics of the theory of elasticity for hemitropic media in vector and matrix forms, introduce the generalized stress operator and the potential energy quadratic form. Then, in Sections 2 and 3, we formulate the contact problem for two elastic homogeneous hemitropic continua with dif- ferent elastic properties under the condition of natural impenetrability of one frictionless contact problems for elastic hemitropic solids 269 body into the other. We consider the coercive case when the interacting bodies are fixed along some parts of their boundaries. With the help of the potential method the problem is reduced equivalently to the boundary variational inequality. In Section 4, we present a detailed analysis of these inequalities and investigate existence and uniqueness of a weak solution of the original contact problem. Finally, in Section 5, we consider a semicoercive case when the contacting bodies are not fixed along their boundaries. In this case, the corresponding mathematical problem is not solvable, in general. We derive the necessary conditions of solvability and formulate also some su‰cient conditions of solvability in explicit form.

1. Basic field equations

Let W a R3 be a bounded simply connected domain with a piecewise smooth boundary S ¼ qW, W ¼ W A S. We assume that W is occupied by a homogeneous > > hemitropic elastic material. Denote by u ¼ðu1; u2; u3Þ and o ¼ðo1; o2; o3Þ the displacement vector and the micro-rotation vector, respectively; here and in what follows the symbol ðÞ> denotes transposition. In the hemitropic elasticity theory we have the following constitutive equations for the force stress tensor ftpqg and the couple stress tensor fmpqg [AK1], [NGS1]:

ð1:1Þ tpq ¼ tpqðUÞ :¼ðm þ aÞqpuq þðm aÞqqup þ ldpq div u þ ddpq div o

þðK þ nÞqpoq þðK nÞqqop 2aepqkok;

ð1:2Þ mpq ¼ mpqðUÞ :¼ ddpq div u þðK þ nÞ½qpuq epqkokþbdpq div o

þðK nÞ½qqup eqpkokþðg þ eÞqpoq þðg eÞqqop;

> where U ¼ðu; oÞ , dpq is the Kronecker delta, q ¼ðq1; q2; q3Þ with qj ¼ q=qxj, epqk is the permutation (Levi-Civita´) symbol, and a, b, g, d, l, m, n, K, and e are the material constants. Throughout the paper summation over repeated indexes is meant from one to three if not otherwise stated. ðnÞ ðnÞ ðnÞ ðnÞ > The components of the force stress vector t ¼ðt1 ; t2 ; t3 Þ and the ðnÞ ðnÞ ðnÞ ðnÞ > couple stress vector m ¼ðm1 ; m2 ; m3 Þ , acting on a surface element with a normal vector n ¼ðn1; n2; n3Þ, read as

ðnÞ ðnÞ ð1:3Þ tq ¼ tpqnp; mq ¼ mpqnp; q ¼ 1; 2; 3:

Denote by Tðq; nÞ the generalized 6 6 matrix di¤erential stress operator [NGS1] T1ðq; nÞ T2ðq; nÞ ð1:4Þ Tðq; nÞ¼ ; Tj ¼½Tjpq33; j ¼ 1; 4; T3ðq; nÞ T4ðq; nÞ 66 270 a. gachechiladze, r. gachechiladze and d. natroshvili where

T1pqðq; nÞ¼ðm þ aÞdpqqn þðm aÞnqqp þ lnpqq;

T2pqðq; nÞ¼ðK þ nÞdpqqn þðK nÞnqqp þ dnpqq 2aepqknk; ð1:5Þ T3pqðq; nÞ¼ðK þ nÞdpqqn þðK nÞnqqp þ dnpqq;

T4pqðq; nÞ¼ðg þ eÞdpqqn þðg eÞnqqp þ bnpqq 2nepqknk:

Here qn ¼ q=qn denotes the usual normal derivative. From formulas (1.1), (1.2) and (1.3) it can be easily checked that

ðtðnÞ; mðnÞÞ> ¼ Tðq; nÞU:

The equilibrium equations of statics in the theory of hemitropic elasticity read as [NGS1]

qptpqðxÞþ%FqðxÞ¼0; q ¼ 1; 2; 3;

qpmpqðxÞþeqlrtlrðxÞþ%MqðxÞ¼0; q ¼ 1; 2; 3;

> where % is the mass density of the elastic material, and F ¼ðF1; F2; F3Þ and > M ¼ðM1; M2; M3Þ are the body force and body couple vectors. Using the constitutive equations (1.1) and (1.2) we can rewrite the equilibrium equations in terms of the displacement and micro-rotation vectors,

ðm þ aÞDuðxÞþðl þ m aÞ grad div uðxÞþðK þ nÞDoðxÞ þðd þ K nÞ grad div oðxÞþ2a curl oðxÞþ%FðxÞ¼0; ð1:6Þ ðK þ nÞDuðxÞþðd þ K nÞ grad div uðxÞþ2a curl uðxÞ þðg þ eÞDoðxÞþðb þ g eÞ grad div oðxÞþ4n curl oðxÞ 4aoðxÞþ%MðxÞ¼0;

2 2 2 where D ¼ q1 þ q2 þ q3 is the Laplace operator. Let us introduce the matrix di¤erential operator generated by the left hand side expressions of the system (1.6): L ðqÞ L ðqÞ ð1:7Þ LðqÞ :¼ 1 2 ; L3ðqÞ L4ðqÞ 66 where

L1ðqÞ :¼ðm þ aÞDI3 þðl þ m aÞQðqÞ;

ð1:8Þ L2ðqÞ¼L3ðqÞ :¼ðK þ nÞDI3 þðd þ K nÞQðqÞþ2aRðqÞ;

L4ðqÞ :¼½ðg þ eÞD 4aI3 þðb þ g eÞQðqÞþ4nRðqÞ:

Here and in the sequel Ik stands for the k k unit matrix and

QðqÞ :¼½qkqj33; RðqÞ :¼½ekjlql33: frictionless contact problems for elastic hemitropic solids 271

It is easy to see that RðqÞu ¼ curl u and QðqÞu ¼ grad div u: Equations (1.6) can be written in matrix form as

LðqÞUðxÞþGðxÞ¼0 with U :¼ðu; oÞ>; G :¼ð%F;%MÞ>:

Note that the operator LðqÞ is formally self-adjoint, i.e., LðqÞ¼½LðqÞ>:

1.1. Green’s formulas

For real-valued vector functions U ¼ðu; oÞ> and U 0 ¼ðu0; o0Þ> from the class ½C 2ðWÞ6 the following Green formula holds [NGS1] Z Z ð1:9Þ ½LðqÞU U 0 þ EðU; U 0Þ dx ¼ fTðq; nÞUgþ fU 0gþ dS; W S where fgþ denotes the trace operator on S from W, while Eð ; Þ is the bilinear form deﬁned by the equality:

ð1:10Þ EðU; U 0Þ¼EðU 0; UÞ 0 0 0 0 ¼fðm þ aÞupqupq þðm aÞupquqp þðK þ nÞðupqopq þ opqupqÞ 0 0 0 0 þðK nÞðupqoqp þ opquqpÞþðg þ eÞopqopq þðg eÞopqoqp 0 0 0 0 þ dðuppoqq þ oqquppÞþluppuqq þ boppoqqg; where upq and opq are the so called strain and torsion (curvature) tensors for hemitropic bodies,

ð1:11Þ upq ¼ qpuq epqkok; opq ¼ qpoq; p; q ¼ 1; 2; 3:

Here and in what follows the central dot a b denotes the usual scalar product of m two vectors a; b a R : a b ¼ ajbj: From formulas (1.10) and (1.11) we get 3l þ 2m 3d þ 2K 3d þ 2K ð1:12Þ EðU; U 0Þ¼ div u þ div o div u0 þ div o0 3 3l þ 2m 3l þ 2m 1 ð3d þ 2KÞ2 þ 3b þ 2g ðdiv oÞðdiv o0Þ 3 3l þ 2m n2 þ e curl o curl o0 a n n þ a curl u þ curl o 2o curl u0 þ curl o0 2o0 a a 272 a. gachechiladze, r. gachechiladze and d. natroshvili m X3 qu qu K qo qo þ k þ j þ k þ j 2 qx qx m qx qx k; j¼1; kAj j k j k qu0 qu0 K qo0 qo0 k þ j þ k þ j qxj qxk m qxj qxk m X3 qu qu K qo qo þ k j þ k j 3 qx qx m qx qx k; j¼1 k j k j qu0 qu0 K qo0 qo0 k j þ k j qxk qxj m qxk qxj K2 X3 1 qo qo qo0 qo0 þ g k þ j k þ j m 2 qx qx qx qx k; j¼1; kAj j k j k 1 qo qo qo0 qo0 þ k j k j : 3 qxk qxj qxk qxj The potential energy density function EðU; UÞ is a positive deﬁnite quadratic form with respect to the variables upq and opq, i.e., there exists a positive constant c0 > 0 depending only on the material parameters, such that

X3 2 2 ð1:13Þ EðU; UÞ b c0 ½upq þ opq: p; q¼1

The necessary and su‰cient conditions for the quadratic form EðU; UÞ to be positive deﬁnite are the following inequalities (see [AK2], [Dy1], [GGN1])

m > 0; a > 0; g > 0; e > 0; l þ 2m > 0; mg K2 > 0; ae n2 > 0; ðl þ mÞðb þ gÞðd þ KÞ2 > 0; ð3l þ 2mÞð3b þ 2gÞð3d þ 2KÞ2 > 0; m½ðl þ mÞðb þ gÞðd þ KÞ2þðl þ mÞðmg K2Þ > 0; m½ð3l þ 2mÞð3b þ 2gÞð3d þ 2KÞ2þð3l þ 2mÞðmg K2Þ > 0:

Let us note that, if the condition 3l þ 2m > 0 is fulﬁlled, which is very natural in the classical elasticity, then the above conditions are equivalent to the following simultaneous inequalities

m > 0; a > 0; g > 0; e > 0; 3l þ 2m > 0; mg K2 > 0; ð1:14Þ ae n2 > 0; ðm þ aÞðg þ eÞðK þ nÞ2 > 0; ð3l þ 2mÞð3b þ 2gÞð3d þ 2KÞ2 > 0:

The following assertion describes the null space of the energy quadratic form EðU; UÞ (see [NGS1]). frictionless contact problems for elastic hemitropic solids 273

Lemma 1.1. Let U ¼ðu; oÞ> a ½C 1ðWÞ6 and EðU; UÞ¼0 in W. Then

uðxÞ¼½a xþb; oðxÞ¼a; x a W; where a and b are arbitrary three-dimensional constant vectors and symbol ½ denotes the cross product of two vectors.

Vectors of type ð½a xþb; aÞ are called generalized rigid displacement vectors. Note that a generalized rigid displacement vector vanishes identically if it is zero at a single point. s s Throughout the paper LpðWÞ with 1 a p < l and H ðWÞ¼H2 ðWÞ with s a R denote Lebesgue and Bessel potential spaces (see, e.g., [LiMa1], [Tr1]). The corresponding norms we denote by symbols and s . Denote by W kkLpðWÞ kkH ðWÞ Dð Þ the class of ClðWÞ functions with support in the domain W.IfS is an open proper part of the manifold qW, i.e., S H qW, S A qW, then by H sðS Þ we denote the restriction of the space H sðqWÞ onto S ,

s s H ðS Þ :¼frS j : j a H ðqWÞg;

where rS denotes the restriction operator onto the set S . Further, let

H~ sðS Þ :¼fj a H sðqWÞ : supp j H S g:

From the positive deﬁniteness of the energy form Eð ; Þ with respect to the variables (1.11) (see (1.13)) it follows that Z ð1:15Þ BðU; UÞ :¼ EðU; UÞ dx b 0: W

Moreover, there exist positive constants c1 and c2, depending only on the material parameters, such that the inequality Z () X3 X3 2 2 2 2 BðU; UÞ b c1 ½ðqpuqÞ þðqpoqÞ þ ½uq þ oq dx W p; q¼1 q¼1 Z X3 2 2 c2 ½uq þ oq dx W q¼1 holds for an arbitrary real-valued vector function U a ½C 1ðWÞ6. By standard limiting arguments we easily conclude that for any U a ½H 1ðWÞ6 the following Korn’s type inequality holds (cf. [Fi1], Part I, §12)

2 2 ð1:16Þ BðU; UÞ b c1kUk 1 6 c2kUk 6 : ½H ðWÞ ½L2ðWÞ Remark 1.1. If U a ½H 1ðWÞ6 and the trace fUgþ vanishes on some open sub- þ surface S of the boundary qW, i.e., rS fUg ¼ 0, then we have the strict Korn’s 274 a. gachechiladze, r. gachechiladze and d. natroshvili inequality

2 BðU; UÞ b ckUk½H 1ðWÞ6 with some positive constant c > 0 which does not depend on the vector U. The constant c depends on the material parameters and on the geometry of the domain W. This follows from (1.15), (1.16) and the fact that in this case BðU; UÞ > 0 for U A 0 (see, e.g., [Ne1], [Mc1], Ch. 2, Exercise 2.17).

Remark 1.2. By standard limiting arguments Green’s formula (1.9) can be extended to Lipschitz domains and to vector functions U a ½H 1ðWÞ6 with 6 0 1 6 LðqÞU a ½L2ðWÞ and U a ½H ðWÞ (see, [Ne1], [LiMa1]), Z 0 0 þ 0 þ ð1:17Þ ½LðqÞU U þ EðU; U Þ dx ¼ 3fTðq; nÞUg ; fU g 4qW; W

1=2 6 where 3 ; 4qW denotes the duality between the spaces ½H ðqWÞ and 1=2 6 6 ½H ðqWÞ , which generalizes the usual ½L2ðqWÞ inner product. By this relation the generalized trace of the stress operator fTðq; nÞUgþ on the boundary qW is correctly determined and fTðq; nÞUgþ a ½H 1=2ðqWÞ6. Note that for arbitrary 0 6 real valued vector functions V; V a ½L2ðqWÞ we have Z 0 0 3V; V 4qW ¼ V V dS: qW

2. Statement of the problems and uniqueness results

3 Let Wq a R , q ¼ 1; 2, be simply connected bounded Lipschitz domains with piecewise smooth, simply connected boundaries Sq :¼ qWq. Further, let W1 and W2 be ﬁlled with hemitropic materials possessing di¤erent elastic properties. The elastic constants corresponding to the elastic medium occupying the domain Wq are denoted by aðqÞ, bðqÞ, gðqÞ, dðqÞ, lðqÞ, mðqÞ, nðqÞ, KðqÞ, and eðqÞ, q ¼ 1; 2. Analo- ðqÞ ðqÞ ðqÞ ðqÞ > ðqÞ ðqÞ ðqÞ ðqÞ > gously, u ¼ðu1 ; u2 ; u3 Þ and o ¼ðo1 ; o2 ; o3 Þ denote the displace- ðqÞ ðqÞ ðqÞ ment and micro-rotation vectors in the domain Wq, E ðU ; U Þ designates the corresponding potential energy density, LðqÞðqÞ and T ðqÞðq; nðqÞÞ are the corresponding di¤erential operators given by formulas (1.7), (1.8) and (1.4), (1.5) respectively. Let the boundaries Sq :¼ qWq fall into three mutually disjoint open portions D N D A N A D B j 2; a 0 0 Sq , Sq and Sc, such that Sq Sq Sc ¼ Sq, Sq Sc ¼ and Sc a C , a a ðqÞ ð0; 1Þ. Denote by n ðxÞ the unit, outward with respect to Wq, normal at the point x a Sq (see Fig. 1). We assume that the elastic hemitropic solids occupy- D D ing the domains W1 and W2 are ﬁxed along the subsurfaces S1 and S2 , along N N the subsurfaces S1 and S2 there are prescribed some stresses, while the two bodies under consideration are in frictionless contact along the subsurface Sc :¼ S1 B S2. frictionless contact problems for elastic hemitropic solids 275

Figure 1. Geometry of the contacting solids

Below we describe mathematically this contact problem by means of the usual D mixed Dirichlet-Neumann type boundary conditions on the sub-manifolds Sq N and Sq , while on Sc the frictionless contact is modeled with the help of the so called natural non-penetration conditions.

2.1. Formulation of the problem

Consider the equation in the domain Wq, q ¼ 1; 2;

ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ > 6 ð2:1Þ L ðqÞU þ G ¼ 0; G :¼ð% F ;% M Þ a ½L2ðWqÞ ; where U ðqÞ ¼ðuðqÞ; oðqÞÞ> are the unknown vectors, LðqÞðqÞ is the matrix di¤erential operator given by formulas (1.7) and (1.8), %ðqÞ are the mass densities of ðqÞ ðqÞ ðqÞ ðqÞ > ðqÞ the elastic materials under consideration, F ¼ðF1 ; F2 ; F3 Þ and M ¼ ðqÞ ðqÞ ðqÞ > ðM1 ; M2 ; M3 Þ are the corresponding body force and body couple vectors. In the sequel, we will be concerned with weak solutions of the corresponding di¤erential equations. By deﬁnition, the vector function U ðqÞ ¼ðuðqÞ; oðqÞÞ> a 1 6 ½H ðWqÞ is called a weak solution of equation (2.1) in the domain Wq, if for 6 every F a ½DðWqÞ Z BðqÞðU ðqÞ; FÞ¼ GðqÞ F dx; q ¼ 1; 2; Wq where the bilinear form BðqÞðU ðqÞ; FÞ is deﬁned by the formula Z BðqÞðU ðqÞ; FÞ :¼ E ðqÞðU ðqÞ; FÞ dx Wq with E ðqÞðU ðqÞ; FÞ given by (1.12). Below, for the force and couple stress vectors we use the notation

ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ T U ¼ T1 u þ T2 o ; M U ¼ T3 u þ T4 o ; 276 a. gachechiladze, r. gachechiladze and d. natroshvili

ðlÞ where the boundary operators Tk , k ¼ 1; 2; 3; 4, l ¼ 1; 2, are given by formulas (1.5). For the normal and tangential components of the force stress vector we will use, respectively, the following notation

ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðT U ÞnðqÞ :¼ðT U Þn ; ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðT U Þt :¼ T U n ðT U ÞnðqÞ : Let

ðqÞ ðqÞ ðqÞ > ~ 1=2 N 6 C ¼ðC1 ; C2 Þ a ½H ðSq Þ ; ðqÞ ðqÞ ðqÞ ðqÞ > ~ 1=2 N 3 Cl ¼ðCl1 ; Cl2 ; Cl3 Þ a ½H ðSq Þ ; q; l ¼ 1; 2; and consider the following boundary-contact problem.

ðqÞ ðqÞ ðqÞ > 1 6 Problem ðAÞ: Find vector functions U ¼ðu ; o Þ a ½H ðWqÞ , q ¼ 1; 2, which are weak solutions of equations (2.1) and satisfy:

(i) the Dirichlet type conditions

ðqÞ þ D r D U 0onS ; q 1; 2; Sq f g ¼ q ¼

(ii) the Neumann type conditions

ðqÞ ðqÞ ðqÞ þ ðqÞ N r N T q; n U r N C on S ; q 1; 2; Sq f ð Þ g ¼ Sq q ¼

(iii) the frictionless non-penetration conditions on the contact subsurface Sc

ð1Þ ð1Þ ð2Þ ð2Þ þ rSc fu n þ u n g a 0onSc; ð1Þ ð1Þ þ ð2Þ ð2Þ þ rSc fðT U Þnð1Þ g ¼ rSc fðT U Þnð2Þ g a 0onSc; ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ 3rSc fðT U Þnð1Þ g ; rSc fu n þ u n g 4Sc ¼ 0onSc; ðqÞ ðqÞ þ rSc fðT U Þtg ¼ 0onSc; q ¼ 1; 2; ðqÞ ðqÞ þ rSc fM U g ¼ 0onSc; q ¼ 1; 2:

To reduce this problem to a boundary variational inequality we need ﬁrst to reduce the nonhomogeneous equation (2.1) to the homogeneous one. To this purpose consider the following auxiliary mixed boundary value problem: Find ðqÞ ðqÞ ðqÞ > 1 6 a vector-function U0 ¼ðu0 ; o0 Þ a ½H ðWqÞ which is a weak solution of equation (2.1) and satisﬁes the following mixed boundary conditions:

ðqÞ þ ðqÞ ðqÞ ðqÞ þ ð2:2Þ rS D fU g ¼ 0; r D fT ðq; n ÞU g ¼ 0: q 0 SqnSq 0 This problem possesses a unique solution (see [NGS1]). Clearly, if W ðqÞ a 1 6 ðqÞ ðqÞ ðqÞ > 1 6 ½H ðWqÞ is a solution of the Problem ðAÞ and U0 ¼ðu0 ; o0 Þ a ½H ðWqÞ frictionless contact problems for elastic hemitropic solids 277 is a solution of the above auxiliary mixed boundary value problem (2.2), then the ðqÞ ðqÞ ðqÞ di¤erence U ¼ W U0 will solve the following problem.

ðqÞ ðqÞ ðqÞ > 1 6 Problem ðA0Þ: Find vector functions U ¼ðu ; o Þ a ½H ðWqÞ , q ¼ 1; 2, which are weak solutions of the equations

ðqÞ ðqÞ ð2:3Þ L ðqÞU ¼ 0inWq and satisfy the following boundary and contact conditions:

ðqÞ þ D 2:4 r D U 0onS ; q 1; 2; ð Þ Sq f g ¼ q ¼ ðqÞ ðqÞ ðqÞ þ ðqÞ N 2:5 r N T q; n U r N C on S ; q 1; 2; ð Þ Sq f ð Þ g ¼ Sq q ¼ ð1Þ ð1Þ ð2Þ ð2Þ þ ð2:6Þ rSc fu n þ u n g a j0 on Sc; ð1Þ ð1Þ þ ð2Þ ð2Þ þ ð2:7Þ rSc fðT U Þnð1Þ g ¼ rSc fðT U Þnð2Þ g a 0onSc; ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ ð2:8Þ 3rSc fðT U Þnð1Þ g ; rSc fu n þ u n g j04Sc ¼ 0onSc; ðqÞ ðqÞ þ ð2:9Þ rSc fðT U Þtg ¼ 0onSc; q ¼ 1; 2; ðqÞ ðqÞ þ ð2:10Þ rSc fM U g ¼ 0onSc; q ¼ 1; 2; where CðqÞ is the same as in the formulation of Problem ðAÞ and

ð1Þ ð1Þ ð2Þ ð2Þ þ ð2:11Þ j0 ¼rSc fu0 n þ u0 n g :

ð1Þ ð2Þ > Below we will investigate the Problem ðA0Þ. Clearly, if a pair ðU ; U Þ solves ð1Þ ð2Þ > ð1Þ ð1Þ ð2Þ ð2Þ > the Problem ðA0Þ, the sum ðW ; W Þ :¼ðU þ U0 ; U þ U0 Þ solves then the Problem ðAÞ.

2.2. Uniqueness theorem

Here we prove the following uniqueness theorem.

Theorem 2.1. Problem ðA0Þ has at most one solution.

Proof. Let U ¼ðU ð1Þ; U ð2ÞÞ> with U ðqÞ ¼ðuðqÞ; oðqÞÞ> and W ¼ðW ð1Þ; W ð2ÞÞ> ðqÞ ðqÞ ðqÞ > with W ¼ðv ; w Þ be two distinct solutions of Problem ðA0Þ. Then the dif- ference U~ ¼ðU~ ð1Þ; U~ ð2ÞÞ> :¼ U W will satisfy the conditions (2.3), (2.4), (2.9), (2.10), and the condition (2.5) with CðqÞ ¼ 0. From condition (2.7) we have

ð1Þ ~ ð1Þ þ ð2Þ ~ ð2Þ þ rSc fðT U Þnð1Þ g ¼ rSc fðT U Þnð2Þ g :

Using Green’s formula (1.17) and taking into account the above conditions, we have 278 a. gachechiladze, r. gachechiladze and d. natroshvili Z X2 E ðqÞðU~ ðqÞ; U~ ðqÞÞ dx q¼1 Wq

X2 ðqÞ ðqÞ ~ ðqÞ þ ~ ðqÞ þ ¼ 3fT ðq; n ÞU g ; fU g 4Sq q¼1

X2 ðqÞ ~ ðqÞ þ ðqÞ ðqÞ þ ¼ 3rSc fðT U ÞnðqÞ g ; rSc fu~ n g 4Sc q¼1 ð1Þ ~ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ ¼ 3rSc fðT U Þnð1Þ g ; rSc fu~ n þ u~ n g 4Sc ð1Þ ð1Þ þ ð1Þ ð1Þ þ ¼ 3rSc fðT U Þnð1Þ g rSc fðT W Þnð1Þ g ; ð1Þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ þ rSc fu n v n þ u n v n g 4Sc ð1Þ ð1Þ þ ð1Þ ð1Þ þ ¼ 3rSc fðT U Þnð1Þ g rSc fðT W Þnð1Þ g ; ð1Þ ð1Þ ð2Þ ð2Þ þ rSc fu n þ u n g j0 ð1Þ ð1Þ ð2Þ ð2Þ þ rSc fv n þ v n g þ j04Sc ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ ¼3rSc fðT U Þnð1Þ g ; rSc fv n þ v n g j04Sc ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ 3rSc fðT W Þnð1Þ g ; rSc fu n þ u n g j04Sc a 0:

Bearing in mind that the quadratic form E ðqÞðU~ ðqÞ; U~ ðqÞÞ is positive deﬁnite (see (1.13)), we have

E ðqÞðU~ ðqÞ; U~ ðqÞÞ¼0; q ¼ 1; 2:

By Lemma 1.1

U~ ðqÞ ¼ð½aðqÞ xþbðqÞ; aðqÞÞ>; q ¼ 1; 2:

ðqÞ þ ðqÞ ðqÞ ðqÞ Since r D U~ 0 we conclude a b 0. Thus U~ 0, q 1; 2. r Sq f g ¼ ¼ ¼ ¼ ¼

2.3. Reduction of Problem ðA0Þ to the boundary variational inequality

To reduce Problem ðA0Þ to the boundary variational inequality equivalently we ðqÞ ðqÞ ðqÞ > 1 6 recall that a solution vector U ¼ðu ; o Þ a ½H ðWqÞ to equation (2.3) satisfying the Dirichlet boundary condition

fU ðqÞgþ ¼ hðqÞ

ðqÞ 1=2 6 with h a ½H ðSqÞ , can be uniquely represented as a single layer potential (see [NGS1]) Z ðqÞ ðqÞ ðqÞ 1 ðqÞ ðqÞ ðqÞ 1 ðqÞ U ðxÞ¼V ð½H h ÞðxÞ¼ G ðx yÞð½H h ÞðyÞdyS; Sq frictionless contact problems for elastic hemitropic solids 279 where GðqÞ is the matrix of fundamental solutions of the operator LðqÞðqÞ and HðqÞ is the boundary integral operator generated by the single layer potential on ðqÞ the boundary Sq (see the explicit expression for G in [GGN1], [NGS1]): Z ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ þ H h ðxÞ¼ lim G ðz yÞh ðyÞdyS ¼fV ðh Þg : W C z!x A S q q Sq Note that the single layer potential operator V ðqÞ and the integral operator HðqÞ have the following mapping properties (see [NGS1]):

ðqÞ 1=2 6 1 6 V : ½H ðSqÞ !½H ðWqÞ ; ð2:12Þ ðqÞ 1=2 6 1=2 6 H : ½H ðSqÞ !½H ðSqÞ :

These operators are continuous. Moreover, the operator HðqÞ is continuously in- vertible and

ðqÞ 1 1=2 6 1=2 6 ð2:13Þ ½H : ½H ðSqÞ !½H ðSqÞ :

ðqÞ 1=2 6 For arbitrary h a ½H ðSqÞ there hold the following jump relations

ðqÞ ðqÞ ðqÞ ðqÞ þ 1 ðqÞ ðqÞ ð2:14Þ fT ðq; n ÞV ðh Þg ¼ð2 I6 þ K Þh on Sq; q ¼ 1; 2; where KðqÞ is a singular integral operator, Z ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ K h ðxÞ :¼ ½T ðq; n ÞG ðx yÞh ðyÞdyS: Sq Note that the operator

1 ðqÞ 1=2 6 1=2 6 ð2:15Þ 2 I6 þ K : ½H ðSqÞ !½H ðSqÞ is a continuous singular integral operator of normal type with zero index (for details see [NGS1]). Further, let us introduce the so-called Green’s operator for the Dirichlet problem

ðqÞ 1=2 6 1 6 G : ½H ðSqÞ !½H ðWqÞ deﬁned by the formula

ð2:16Þ GðqÞhðqÞ ¼ V ðqÞð½HðqÞ1hðqÞÞ:

ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ þ ðqÞ It is clear that L ðqÞðG h Þ¼0inWq and fG h g ¼ h on Sq. From the properties of the trace operator and mapping properties of the single layer potential operator it follows that there exist positive numbers C1 and C2 such that for ðqÞ 1=2 6 all h a ½H ðSqÞ

ðqÞ ðqÞ ðqÞ ðqÞ ð2:17Þ C kh k 1=2 6 a kG h k 1 6 a C kh k 1=2 6 : 1 ½H ðSqÞ ½H ðWqÞ 2 ½H ðSqÞ 280 a. gachechiladze, r. gachechiladze and d. natroshvili

Now we introduce the generalized Steklov-Poincare´ type operator

ð2:18Þ AðqÞhðqÞ :¼fT ðqÞðq; nðqÞÞðGðqÞhðqÞÞgþ ¼fT ðqÞðq; nðqÞÞV ðqÞð½HðqÞ1hðqÞÞgþ 1 ðqÞ ðqÞ 1 ðqÞ ¼ð2 I6 þ K Þ½H h :

ðqÞ Denote by L ðSqÞ the set of restrictions on Sq of rigid displacement vectors,

ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ > ð2:19Þ L ðSqÞ¼fw ðxÞ¼ð½a xþb ; a Þ ; x a Sqg; where aðqÞ and bðqÞ are arbitrary three-dimensional constant vectors. With the help 0 ðqÞ ðqÞ ðqÞ 1 ðqÞ of Green’s formula (1.17) with W ¼ Wq and U ¼ U ¼ U ¼ V ð½H h Þ, the relations (2.14), (2.18), (2.19) and the uniqueness theorem for the Dirichlet ðqÞ ðqÞ BVP, we infer that ker A ¼ L ðSqÞ. The properties of the Steklov-Poincare´ operator is described by the following lemma.

Lemma 1=2 6 ~ 1=2 6 2.1. Let h; h a ½H ðSqÞ and g a ½H ðSq Þ , where Sq is an open proper part of the boundary Sq ¼ qWq. Then

ðqÞ ðqÞ (a) 3A h; h4Sq ¼ 3A h; h4Sq ; ðqÞ 1=2 6 1=2 6 (b) A : ½H ðSqÞ !½H ðSqÞ is a continuous operator; ðqÞ 2 2 (c) 3A h; h4 b C1khk 1=2 6 C2khk 6 ; Sq ½H ðSqÞ ½L2ðSqÞ ðqÞ 2 (d) 3A g; g4 b Ckgk 1=2 6 ; Sq ½H ðSqÞ ðqÞ 2 (e) 3A h; h4 b Ckh Phk 1=2 6 ; Sq ½H ðSqÞ where P is the orthogonal projection (in the sense of L2ðSqÞ) of the space 1=2 6 ðqÞ ½H ðSqÞ onto the space L ðSqÞ; the positive constants C, C1 and C2 depend on the material parameters and on the geometry of the surface Sq and do not depend on h and g.

1=2 6 ðqÞ Proof. Let h; h a ½H ðSqÞ . Taking into account that the vector G h solves the homogeneous equation LðqÞðqÞGðqÞh ¼ 0, from Green’s formula (1.17) we get:

ðqÞ ðqÞ ðqÞ ðqÞ þ ðqÞ þ 3A h; h4Sq ¼ 3fT ðq; n ÞðG hÞg ; fG hg 4Sq ¼ BðqÞðGðqÞh; GðqÞhÞ¼BðqÞðGðqÞh; GðqÞhÞ ðqÞ ðqÞ ðqÞ þ ðqÞ þ ðqÞ ¼ 3fT ðq; n ÞðG hÞg ; fG hg 4Sq ¼ 3A h; h4Sq ; whence the item (a) follows. The item (b) is evident, since AðqÞ is the composition of the continuous oper- ðqÞ 1 1 ðqÞ ators ½H and 2 I6 þ K (see (2.14) and (2.15)). 1=2 6 To prove the item (c) we proceed as follows. For arbitrary h a ½H ðSqÞ with the help of (1.16) we derive frictionless contact problems for elastic hemitropic solids 281

ðqÞ ðqÞ ðqÞ ðqÞ 1 ðqÞ ðqÞ 1 3A h; h4Sq ¼ B ðV ð½H hÞ; V ð½H hÞÞ ðqÞ ðqÞ 1 2 ðqÞ ðqÞ 1 2 b c1kV ð½H hÞk 1 6 c2kV ð½H hÞk 6 : ½H ðWqÞ ½L2ðWqÞ

By formulas (2.16) and (2.17) we have

ðqÞ ðqÞ 1 kV ð½H hÞk 1 6 b C khk 1=2 6 : ½H ðWqÞ 1 ½H ðSqÞ

6 1=2 6 On the other hand, since ½L2ðSqÞ is compactly embedded into ½H ðSqÞ ,by virtue of continuity of the operators (2.12) and (2.13) we have:

ðqÞ ðqÞ 1 ðqÞ 1 kV ð½H hÞk 6 a C k½H hk 3=2 6 ½L2ðWqÞ 1 ½H ðSqÞ a C khk 1=2 6 a C khk 6 2 ½H ðSqÞ 3 ½L2ðSqÞ

with some positive constants C1 , C2 and C3 independent of h. So, ﬁnally we obtain that

ðqÞ 2 2 2 2 3A h; h4 b c1C khk 1=2 6 c2ðC Þ khk 6 ; Sq 1 ½H ðSqÞ 3 ½L2ðSqÞ which proves the item (c). Now the item (e) follows from (c) and the nonnegativity property of the operator AðqÞ, while (d) follows from (e). The lemma is proved. r

Further, we introduce the vector function space Hl and the convex set of vector functions Kj0 :

l ð1Þ ð2Þ > ðqÞ ðqÞ ðqÞ > l 6 H ¼fg ¼ðg ; g Þ : g ¼ðj ; c Þ a ½H ðSqÞ ; q ¼ 1; 2g; l a R; 1=2 ðqÞ ð1Þ ð1Þ ð2Þ ð2Þ þ K g a H : r D g 0; r j n j n a j ; j0 ¼f Sq ¼ Sc f þ g 0g

where j0 is deﬁned by formula (2.11). It is clear that the set Kj0 is convex and 1=2 closed in the space H . On the convex closed set Kj0 we consider the following ð1Þ ð2Þ > a variational inequality: Find h0 ¼ðh0 ; h0 Þ Kj0 , such that the boundary variational inequality

X2 X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ð2:20Þ 3 h ; h h 4 b 3r N C ; r N ðh h Þ4 N A 0 0 Sq Sq Sq 0 Sq q¼1 q¼1

ð1Þ ð2Þ > is fulﬁlled for all h ¼ðh ; h Þ a Kj0 :

3. Equivalence of problem ðA0Þ and variational inequality (2.20)

In this section we prove the equivalence of the boundary variational inequality (2.20) and contact problem ðA0Þ. 282 a. gachechiladze, r. gachechiladze and d. natroshvili

Theorem ð1Þ ð2Þ > a 3.1. If h0 ¼ðh0 ; h0 Þ Kj0 is a solution of the boundary variational ð1Þ ð2Þ > ðqÞ ðqÞ ðqÞ 1 6 inequality (2.20), then U ¼ðU ; U Þ with U ¼ G h0 a ½H ðWqÞ is a so- ð1Þ ð2Þ > lution of Problem ðA0Þ, and vice versa, if U ¼ðU ; U Þ is a solution of Prob- ð1Þ ð2Þ > ðqÞ ðqÞ þ lem ðA0Þ, then h0 ¼ðh0 ; h0 Þ with h0 ¼fU g is a solution of the boundary variational inequality (2.20).

Proof. ð1Þ ð2Þ > a ðqÞ ðqÞ ðqÞ > Let h0 ¼ðh0 ; h0 Þ Kj0 with h0 ¼ðQ0 ; c0 Þ be a solution of the variational inequality (2.20). We can show that then the vector-function U ¼ ð1Þ ð2Þ > ðqÞ ðqÞ ðqÞ ðU ; U Þ , where U ¼ G h0 , is a solution of Problem ðA0Þ. Here the Green operator GðqÞ is defined by formula (2.16). Taking into account the defini- tion of the operator GðqÞ it is not di‰cult to show that the conditions (2.3), (2.4), and (2.6) are fulfilled. Let

ð1Þ ð2Þ > ~ 1=2 N 6 ~ 1=2 N 6 f ¼ðf ; f Þ a ½H ðS1 Þ ½H ðS2 Þ be an arbitrary vector and substitute h0 e f instead of h in (2.20) to obtain:

X2 X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ 3r N h ; r N f 4 N ¼ 3r N C ; r N f 4 N : Sq A 0 Sq Sq Sq Sq Sq q¼1 q¼1

Whence the equallity

ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ þ ðqÞ r N h r N T q; n G h r N C Sq A 0 ¼ Sq f ð Þð 0 Þg ¼ Sq

N almost everywhere on Sq follows, i.e., conditions (2.5) are fulﬁlled. Therefore we can rewrite (2.20) as follows:

X2 3 ðqÞ ðqÞ ðqÞ ðqÞ 4 ð3:1Þ rSc A h0 ; rSc ðh h0 Þ Sc q¼1

X2 3 ðqÞ ðqÞ þ ðqÞ ðqÞ 4 ¼ ½ rSc fT U g ; rSc ðQ Q0 Þ Sc q¼1 3 ðqÞ ðqÞ þ ðqÞ ðqÞ 4 þ rSc fM U g ; rSc ðc c0 Þ Sc b 0 E ð1Þ ð2Þ > ðqÞ ðqÞ ðqÞ > 1=2 6 h ¼ðh ; h Þ a Kj0 ; h ¼ðQ ; c Þ a ½H ðSqÞ :

ð1Þ ð2Þ > ð1Þ ð1Þ ð1Þ > ð2Þ ð2Þ ð2Þ > Let h ¼ðh ; h Þ , h ¼ðQ0 ; c0 e jÞ , h ¼ðQ0 ; c0 Þ , where j a ~ 1=2 3 ½H ðScÞ is an arbitrary vector-function. Clearly, h a Kj0 , and from (3.1) we ﬁnd

ð1Þ ð1Þ þ E ~ 1=2 3 3rSc fM U g ; j4Sc ¼ 0 j a ½H ðScÞ ;

ð1Þ ð1Þ þ i.e., rSc fM U g ¼ 0. Analogously, we ﬁnd that the equality

ð2Þ ð2Þ þ rSc fM U g ¼ 0 frictionless contact problems for elastic hemitropic solids 283 is fulﬁlled which together with the latter one results in (2.10). Taking into account the above equalities, we can rewrite (3.1) as

X2 3 ðqÞ ðqÞ þ ðqÞ ðqÞ 4 rSc fT U g ; rSc ðQ Q0 Þ Sc b 0 ð3:2Þ q¼1 E ð1Þ ð2Þ > h ¼ðh ; h Þ a Kj0 :

Let now h ¼ðhð1Þ; hð2ÞÞ> a H1=2 be such that hðqÞ ¼ðQðqÞ; cðqÞÞ>,

ðqÞ ðqÞ ð2Þ ð2Þ ð1Þ ð1Þ ð1Þ ð1Þ c ¼ c0 ; q ¼ 1; 2; Q ¼ Q0 ; Q n ¼ Q0 n

ð1Þ ð1Þ ~ 1=2 3 and Qt ¼ Q0t e j, where j a ½H ðScÞ is again an arbitrary vector function.

Clearly, h a Kj0 , and (3.2) yields

ð1Þ ð1Þ þ rSc fðT U Þtg ¼ 0:

Quite analogously, we ﬁnd that the equality

ð2Þ ð2Þ þ rSc fðT U Þtg ¼ 0 is fulﬁlled, and hence the validity of condition (2.9) is proved. 1=2 Further, let j a H~ ðScÞ be an arbitrary scalar function and choose h ¼ ðhð1Þ; hð2ÞÞ> a H1=2 as follows

ðqÞ ðqÞ ðqÞ > h ¼ðQ ; c0 Þ ; q ¼ 1; 2; ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ Q ¼ Q0 þ n j; Q ¼ Q0 n j:

Then it can be easily seen that h a Kj0 and from (3.2) it follows that

ð1Þ ð1Þ þ ð2Þ ð2Þ þ E ~ 1=2 3rSc fðT U Þnð1Þ g rSc fðT U Þnð2Þ g ; rSc j4Sc b 0 j a H ðScÞ:

Whence we can conclude that the ﬁrst condition in (2.7) is fulﬁlled. ðqÞ ðqÞ ð2Þ ð2Þ ð1Þ ð1Þ ð1Þ ð1Þ Let now c ¼ c0 , q ¼ 1; 2, Q ¼ Q0 and Q n ¼ Q0 n þ j, where 1=2 j a H~ ðScÞ is an arbitrary scalar function satisfying the condition j a 0: Then h a Kj0 and

ð1Þ ð1Þ þ E ~ 1=2 3rSc fðT U Þnð1Þ g ; rSc j4Sc b 0 j a H ðScÞ; j a 0; i.e., conditions (2.7) are fulﬁlled completely. It remains to prove the condition (2.8). Taking into account all the above obtained relations, from (3.2) it follows that for all h a Kj0

3 ð1Þ ð1Þ þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ 4 ð3:3Þ rSc fðT U Þnð1Þ g ; rSc ½ðQ Q0 Þn þðQ Q0 Þn Sc b 0: 284 a. gachechiladze, r. gachechiladze and d. natroshvili

ð1Þ ð2Þ > 1=2 ðqÞ ðqÞ ðqÞ > 1=2 6 Let now h ¼ðh ; h Þ a H , h ¼ðQ ; c Þ a ½H~ ðScÞ , q ¼ 1; 2, ð1Þ ð1Þ ð2Þ ð2Þ rSc Q n ¼ j0 and rSc Q n ¼ 0: Then (3.3) implies that

3 ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ 4 ð3:4Þ rSc fðT U Þnð1Þ g ; j0 rSc ðQ0 n þ Q0 n Þ Sc b 0:

ðqÞ ðqÞ ðqÞ > 1=2 6 Analogously, if h ¼ðQ ; c Þ a ½H~ ðScÞ , q ¼ 1; 2, and

ð1Þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ rSc Q n ¼ 2rSc Q0 n j0; rSc Q n ¼ 2rSc Q0 n ; then (3.3) yields

3 ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ 4 ð3:5Þ rSc fðT U Þnð1Þ g ; rSc ðQ0 n þ Q0 n Þj0 Sc b 0:

The inequalities (3.4) and (3.5) are equivalent to the condition (2.8). Conse- quently, the ﬁrst part of Theorem 3.1 is proved. ðqÞ ðqÞ ðqÞ > 1 6 Assume now that U ¼ðu ; o Þ a ½H ðWqÞ , q ¼ 1; 2, and U ¼ ð1Þ ð2Þ > ðU ; U Þ is a solution of Problem ðA0Þ. We set

ðqÞ ðqÞ ðqÞ > ðqÞ þ h0 ¼ðQ0 ; c0 Þ :¼fU g ; q ¼ 1; 2;

ðqÞ ðqÞ þ ðqÞ ðqÞ þ ð1Þ ð2Þ > i.e., Q0 ¼fu g and c0 ¼fo g : Then since U ¼ðU ; U Þ is a solution of Problem ðA0Þ, by virtue of conditions (2.4) and (2.6) it is clear that h0 ¼ ð1Þ ð2Þ > a ðqÞ ðh0 ; h0 Þ Kj0 and due to the deﬁnition of the operator G (see (2.16)) the ðqÞ solution U in Wq can be represented uniquely by the formula

ðqÞ ðqÞ ðqÞ U ¼ G h0 ; q ¼ 1; 2:

Taking into account the boundary conditions of Problem ðA0Þ and the deﬁnition ð1Þ ð2Þ > ðqÞ of the Steklov-Poincare´ operator, for every h ¼ðh ; h Þ a Kj0 with h ¼ ðQðqÞ; cðqÞÞ>, q ¼ 1; 2, we have

X2 3 ðqÞ ðqÞ ðqÞ ðqÞ4 A h0 ; h h0 Sq q¼1

X2 3 ðqÞ ðqÞ ðqÞ ðqÞ þ ðqÞ ðqÞ4 ¼ fT ðq; n ÞG h0 g ; h h0 Sq q¼1

X2 ðqÞ ðqÞ ðqÞ ¼ 3r N C ; r N ðh h Þ4 N Sq Sq 0 Sq q¼1

X2 3 ðqÞ ðqÞ ðqÞ ðqÞ þ ðqÞ ðqÞ 4 þ rSc fT ðq; n ÞG h0 g ; rSc ðh h0 Þ Sc q¼1 frictionless contact problems for elastic hemitropic solids 285

X2 ðqÞ ðqÞ ðqÞ ¼ 3r N C ; r N ðh h Þ4 N Sq Sq 0 Sq q¼1

X2 3 ðqÞ ðqÞ þ ðqÞ ðqÞ ðqÞ ðqÞ 4 þ rSc fðT U ÞnðqÞ g ; rSc ðQ n Q0 n Þ Sc q¼1

X2 ðqÞ ðqÞ ðqÞ ¼ 3r N C ; r N ðh h Þ4 N Sq Sq 0 Sq q¼1

X2 3 ð1Þ ð1Þ þ ð1Þ ð1Þ ð1Þ ð1Þ 4 þ rSc fðT U Þnð1Þ g ; rSc ðQ n Q0 n Þ Sc q¼1

X2 3 ð1Þ ð1Þ þ ð2Þ ð2Þ ð2Þ ð2Þ 4 þ rSc fðT U Þnð1Þ g ; rSc ðQ n Q0 n Þ Sc q¼1

X2 ðqÞ ðqÞ ðqÞ ¼ 3r N C ; r N ðh h Þ4 N Sq Sq 0 Sq q¼1

ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ 3rSc fðT U Þnð1Þ g ; rSc ðQ n þ Q n Þj04Sc 3 ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ 4 rSc fðT U Þnð1Þ g ; rSc ðQ0 n þ Q0 n Þj0 Sc X2 ðqÞ ðqÞ ðqÞ ¼ 3r N C ; r N ðh h Þ4 N Sq Sq 0 Sq q¼1

ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ 3rSc fðT U Þnð1Þ g ; rSc ðQ n þ Q n Þj04Sc : ð1Þ ð2Þ > Since h ¼ðh ; h Þ a Kj0 and the condition (2.7) is fulﬁlled, the second term of the last equality is nonnegative, and hence

X2 X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ 3 h ; h h 4 b 3r N C ; r N ðh h Þ4 N : A 0 0 Sq Sq Sq 0 Sq q¼1 q¼1 This completes the proof of the theorem. r

4. Existence results Here we show the existence of a solution to the boundary variational inequality

(2.20). To this end, on the convex closed set Kj0 we consider the functional X2 X2 1 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ JðhÞ¼ 3A h ; h 4S 3rS N C ; rS N h 4S N 2 q q q q ð4:1Þ q¼1 q¼1 E ð1Þ ð2Þ > h ¼ðh ; h Þ a Kj0 : 286 a. gachechiladze, r. gachechiladze and d. natroshvili

It can be easily shown that in view of the symmetry property of the operator AðqÞ (see Lemma 2.1(a)), the existence of solutions of the variational inequality (2.20) is equivalent to the existence of the minimizing element in the set Kj0 of the functional (4.1), i.e., the variational inequality (2.20) is equivalent to the following minimizing problem: Find h0 a Kj0 such that

ð4:2Þ Jðh0Þ¼ inf JðhÞ: A h Kj0

The functional (4.1) is continuous and convex. Let us prove that the functional J is coercive on the set Kj0 , i.e., JðhÞ!þl, when h a Kj0 and

X2 2 ðqÞ 2 khk 1=2 :¼ kh k 1=2 6 !þl: H ½H ðSqÞ q¼1

ðqÞ Since the operator A is coercive on the set Kj0 (see Lemma 2.1(d)), the coer- civeness of the functional J in the above mentioned sense follows from the following obvious estimate

X2 X2 ðqÞ 2 ðqÞ JðhÞ b C kh k 1=2 6 C1 kh k 1=2 6 ½H ðSqÞ ½H ðSqÞ q¼1 q¼1 E ð1Þ ð2Þ > h ¼ðh ; h Þ a Kj0 ; where the positive constants C and C1 do not depend on h. The general theory of variational inequalities (see [GLT1], [Fi1]) makes it pos- sible to conclude that the problem (4.2) is uniquely solvable. Finally, we arrive at the following assertion.

Theorem ðqÞ ~ 1=2 N 6 4.1. Let C a ½H ðSq Þ ,q¼ 1; 2. Then the variational inequality ðqÞ 1=2 6 ðqÞ ðqÞ ðqÞ (2.20) has a unique solution h0 a ½H ðSqÞ , and U ¼ G h0 will be a unique solution of Problem ðA0Þ.

Remark ðqÞ ~ 1=2 N 6 ðqÞ 6 4.1. Let C a ½H ðSq Þ and G a ½L2ðWqÞ , q ¼ 1; 2: Then ð1Þ ð1Þ Problem ðAÞ has a unique solution representable in the form ðU þ U0 ; ð2Þ ð2Þ > ð1Þ ð2Þ > U þ U Þ , where a pair ðU ; U Þ is a unique solution of Problem ðA0Þ ðqÞ 0 and U0 , q ¼ 1; 2, are solutions of the auxiliary problems (2.2).

5. The semicoercive case

5.1. Formulation of the problem

D j N A ðqÞ 6 ðqÞ ~ 1=2 N 6 Let Sq ¼ , Sq Sc ¼ Sq, G a ½L2ðWqÞ and C a ½H ðSq Þ , q ¼ 1; 2: Consider the so-called semicoercive contact problem. frictionless contact problems for elastic hemitropic solids 287

Problem ðBÞ: Find a pair of vector-functions U ¼ðU ð1Þ; U ð2ÞÞ> a H1 with U ðqÞ being a weak solution of the equation

ð5:1Þ LðqÞðqÞU ðqÞ þ GðqÞ ¼ 0; q ¼ 1; 2;

N in the domain Wq, and satisfying the boundary conditions on Sq

ðqÞ ðqÞ ðqÞ þ ðqÞ r N T q; n U r N C ; q 1; 2; Sq f ð Þ g ¼ Sq ¼ and the contact conditions on Sc:

ð1Þ ð1Þ ð2Þ ð2Þ þ rSc fu n þ u n g a 0; ð1Þ ð1Þ þ ð2Þ ð2Þ þ rSc fðT U Þnð1Þ g ¼ rSc fðT U Þnð2Þ g a 0; ð1Þ ð1Þ þ ð1Þ ð1Þ ð2Þ ð2Þ þ 3rSc fðT U Þnð1Þ g ; rSc fu n þ u n g 4Sc ¼ 0; ðqÞ ðqÞ þ rSc fðT U Þtg ¼ 0; q ¼ 1; 2; ðqÞ ðqÞ þ rSc fM U g ¼ 0; q ¼ 1; 2:

To reduce this problem to a boundary variational inequality, we have to reduce the inhomogeneous equation (5.1) to the homogeneous one. To this end, let us consider the auxiliary problem: Find a pair of vector-functions U0 ¼ ð1Þ ð2Þ > 1 ðqÞ ðqÞ ðqÞ > ðU0 ; U0 Þ a H with U0 ¼ðu0 ; o0 Þ being a weak solution of equation (5.1) and satisfying the boundary conditions:

ðqÞ ðqÞ ðqÞ þ ðqÞ ðqÞ þ N rS fT ðq; n ÞU0 g ¼ 0; rSc fu0 n g ¼ 0; ð5:2Þ q ðqÞ ðqÞ þ ðqÞ ðqÞ þ rSc fðT U0 Þtg ¼ 0; rSc fM U0 g ¼ 0; q ¼ 1; 2:

It is known that this problem has a unique solution when Sc, a part of the boundary Sq, is neither rotational nor ruled surface (see [GGN1]). Then, if the ð1Þ ð2Þ > ðqÞ 1 6 pair W ¼ðW ; W Þ with W a ½H ðWqÞ , q ¼ 1; 2, is a solution of Prob- ðqÞ lem ðBÞ and U0 is a solution of the auxiliary problem (5.2), then the di¤erence ðqÞ ðqÞ ðqÞ U ¼ W U0 will be a solution of the following problem.

ðqÞ 1 6 Problem ðB0Þ: Find vector-functions U a ½H ðWqÞ , q ¼ 1; 2, which are weak solutions of the homogeneous equations

ðqÞ ðqÞ L ðqÞU ¼ 0inWq; q ¼ 1; 2;

N and satisfy the boundary conditions on Sq

ðqÞ ðqÞ ðqÞ þ ðqÞ r N T q; n U r N C ; q 1; 2; Sq f ð Þ g ¼ Sq ¼ and the contact conditions on Sc 288 a. gachechiladze, r. gachechiladze and d. natroshvili

ð1Þ ð1Þ ð2Þ ð2Þ þ rSc fu n þ u n g a 0; ð1Þ ð1Þ þ ð1Þ ð2Þ ð2Þ þ ð2Þ rSc fðT U Þnð1Þ g þ j0 ¼ rSc fðT U Þnð2Þ g þ j0 a 0; 3 ð1Þ ð1Þ þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ þ4 rSc fðT U Þnð1Þ g þ j0 ; rSc fu n þ u n g Sc ¼ 0; ðqÞ ðqÞ þ rSc fðT U Þtg ¼ 0; q ¼ 1; 2; ðqÞ ðqÞ þ rSc fM U g ¼ 0; q ¼ 1; 2; where

ðqÞ ðqÞ ðqÞ þ j0 ¼ rSc fðT U0 ÞnðqÞ g ; q ¼ 1; 2:

Consider a convex closed set

K ¼fh ¼ðhð1Þ; hð2ÞÞ> a H1=2 : hðqÞ ¼ðQðqÞ; cðqÞÞ>; ð1Þ ð1Þ ð2Þ ð2Þ rSc fQ n þ Q n g a 0g and formulate the following boundary variational inequality: ð1Þ ð2Þ > Find h0 ¼ðh0 ; h0 Þ a K such that the inequality

X2 X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ð5:3Þ 3 h ; h h 4 b ½3r N C ; r N ðh h Þ4 N A 0 0 Sq Sq Sq 0 Sq q¼1 q¼1 3 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ 4 j0 ; rSc fQ n Q0 n g Sc is fulﬁlled for all h ¼ðhð1Þ; hð2ÞÞ> a K: As in the coercive case, one can show that inequality (5.3) is equivalent to Problem ðB0Þ. Theorem ð1Þ ð2Þ > 5.1. Let h0 ¼ðh0 ; h0 Þ a K be a solution of the variational in- ðqÞ equality (5.3) and G denote the Green operator for the Dirichlet problem in Wq. ð1Þ ð2Þ > ðqÞ ðqÞ ðqÞ Then U ¼ðU ; U Þ with U ¼ G h0 solves Problem ðB0Þ, and vice versa, ð1Þ ð2Þ > ð1Þ ð2Þ > if U ¼ðU ; U Þ is a solution of Problem ðB0Þ, then h0 ¼ðh0 ; h0 Þ with hðqÞ U ðqÞ þ solves the variational inequality (5.3). 0 ¼f gSq

The proof of this theorem is quite similar to that of Theorem 3.1 and we omit it here. To prove the existence of a solution to the variational inequality (5.3) we proceed as follows. Let

ð1Þ ð2Þ > ðqÞ ðqÞ L :¼fw ¼ðw ; w Þ : w a L ðSqÞðq ¼ 1; 2Þg;

ðqÞ where L ðSqÞ is deﬁned by (2.19), and let

R :¼ K B L ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ¼fw a L : ð½a xþb Þn þð½a xþb Þn a 0 for x a Scg: frictionless contact problems for elastic hemitropic solids 289

ð1Þ ð2Þ Since n ¼n at the points of Sc, we have

ð1Þ ð2Þ > ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ R ¼fw ¼ðw ; w Þ a L : ð½ða a Þxþb b Þn a 0; x a Scg:

ð1Þ ð2Þ > Let now h0 ¼ðh0 ; h0 Þ a K be a solution of inequality (5.3). Clearly, h0 þ w a K for w a R. Substitute h0 þ w into (5.3) instead of h and take into ac- ðqÞ ðqÞ count that ker A ¼ L ðSqÞ. We get the inequality

X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ð5:4Þ ½3r N C ; r N w 4 N 3j ; r f½a xþb gn 4 a 0 Sq Sq Sq 0 Sc Sc q¼1 which should hold for all w a R. Thus we have obtained that (5.4) is the necessary condition for the existence of a solution of the variational inequality (5.3). Fur- ther, let

ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ R :¼fw a L : ð½ða a Þxþb b Þn ¼ 0; x a Scg and assume that inequality (5.4) is fulﬁlled in a strong sense, i.e., equality (5.4) holds if and only if w a R. Due to the general theory of variational inequalities (see [Fi1], [GLT1]), the condition (5.4) becomes su‰cient for the solvability of inequality (5.3). Consider the uniqueness question of a solution of (5.3). Suppose ð1Þ ð2Þ > ~ ~ð1Þ ~ð2Þ > that h0 ¼ðh0 ; h0 Þ a K and h0 ¼ðh0 ; h0 Þ a K are two solutions of the variational inequality (5.3). Then

X2 X2 ðqÞ ðqÞ ~ðqÞ ðqÞ ðqÞ ~ðqÞ ðqÞ ð5:5Þ 3 h ; h h 4 b ½3r N C ; r N ðh h Þ4 N A 0 0 0 Sq Sq Sq 0 0 Sq q¼1 q¼1 3 ðqÞ ~ðqÞ ðqÞ ðqÞ ðqÞ 4 j0 ; rSc fQ0 n Q0 n g Sc and

X2 X2 ðqÞ~ðqÞ ðqÞ ~ðqÞ ðqÞ ðqÞ ~ðqÞ ð5:6Þ 3 h ; h h 4 b ½3r N C ; r N ðh h Þ4 N A 0 0 0 Sq Sq Sq 0 0 Sq q¼1 q¼1 3 ðqÞ ðqÞ ðqÞ ~ðqÞ ðqÞ 4 j0 ; rSc fQ0 n Q0 n g Sc :

Adding the above inequalities and taking into account the positive deﬁniteness of the operators AðqÞ, q ¼ 1; 2, we obtain

3 ðqÞ ðqÞ ~ðqÞ ðqÞ ~ðqÞ4 A ðh0 h0 Þ; h0 h0 Sq ¼ 0; q ¼ 1; 2:

Whence we can write

ðqÞ ~ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ > h0 h0 ¼ w ðxÞ¼ð½a xþb ; a Þ ; x a Sq; 290 a. gachechiladze, r. gachechiladze and d. natroshvili

ðqÞ ðqÞ since ker A ¼ L ðSqÞ. Consequently, in view of the symmetry property of ðqÞ 3 ðqÞ ðqÞ ðqÞ4 the operator A (see Lemma 2.1(a)), we have A h0 ; w Sq ¼ 0 and 3 ðqÞ~ðqÞ ðqÞ4 A h0 ; w Sq ¼ 0. Therefore from (5.5) and (5.6) we conclude that

X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ½3r N C ; r N w 4 N 3j ; r ð½a xþb ; a Þn 4 b 0 Sq Sq Sq 0 Sc Sc q¼1 and

X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ½3r N C ; r N w 4 N 3j ; r ð½a xþb ; a Þn 4 a 0; Sq Sq Sq 0 Sc Sc q¼1 i.e.,

X2 ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ð5:7Þ ½3r N C ; r N w 4 N 3j ; r ð½a xþb ; a Þn 4 ¼0: Sq Sq Sq 0 Sc Sc q¼1

Thus we arrive at the following

Theorem D j ðqÞ ~ 1=2 N 6 5.2. Let Sq ¼ and C a ½H ðSq Þ ,q¼ 1; 2: Then the inequality (5.4) is the necessary condition for the existence of a solution of Problem ðB0Þ. If the condition (5.4) is fulﬁlled in a strong sense, i.e., the equality (5.4) holds true if and only if w a R , then Problem ðB0Þ is solvable. Two solutions of Problem ðB0Þ may di¤er from each other by a vector of rigid displacement for which the condition ðqÞ 6 (5.7) is fulﬁlled. Solutions of Problem ðBÞ with G a ½L2ðWqÞ ,q¼ 1; 2, are rep- ð1Þ ð1Þ ð2Þ ð2Þ > ð1Þ ð2Þ > resented then in the form ðU þ U0 ; U þ U0 Þ , where a pair ðU ; U Þ is ðqÞ a solution of Problem ðB0Þ and U0 ,q¼ 1; 2, are solutions of the auxiliary problems (5.2).

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Received 09 March 2012, and in revised form 12 May 2012.

A. Gachechiladze Department of Mathematics Georgian Technical University 77 M. Kostava St., Tbilisi 0175 Republic of Georgia [email protected] R. Gachechiladze A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University 2 University St., Tbilisi 0186, Republic of Georgia [email protected] D. Natroshvili Department of Mathematics Georgian Technical University 77 M. Kostava St., Tbilisi 0175 Republic of Georgia I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University 2 University St., Tbilisi 0186 Republic of Georgia [email protected]