Georgian Mathematical Journal Volume 8 (2001), Number 3, 469–492

BOUNDARY VARIATIONAL INEQUALITY APPROACH IN THE ANISOTROPIC ELASTICITY FOR THE SIGNORINI

PROBLEM

A. GACHECHILADZE AND D. NATROSHVILI

Abstract. The purpose of the paper is reducing the three-dimensional Signorini problem to a variational inequality which occurs on the two-dimen-sional boundary of a domain occupied by an elastic anisotropic body. The uniqueness and existence theorems for the solution of the boundary variational inequality are proved and a boundary element procedure together with an abstract error estimate is described for the Galerkin numerical approximation.

2000 Mathematics Subject Classification: 35J85, 74B05.

Key words and phrases: Signorini problem, anisotropic elasticity, variational inequalities.

1. Introduction

Signorini type problems in the elasticity theory are well studied (see [11], [12], [21], [7], [19], and the references therein). The main tool to investigate these problems is the theory of spatial variational inequalities. By this method the uniqueness and existence theorems are proved and the regularity properties of solutions are established in various functional spaces.

The purpose of the present paper is reducing of the three-dimensional Signorini problem to a variational inequality which occurs on the two-dimensional boundary of a domain occupied by the elastic anisotropic body under consideration. We will show that the spatial variational inequality (SVI) is equivalent to the boundary variational inequality (BVI) obtained. The uniqueness and existence theorems for the solution of BVI are proved, and a boundary element procedure together with an abstract error estimate is described for the Galerkin numerical approximation.

A similar approach in the isotropic case is considered in [16], [17], [15], where the Signorini problem is reduced to a system consisting of a BVI and a boundary singular integral equation (for related problems see also [9], [8] and the references therein).

2. Classical and Spatial Variational Inequality Formulations of the Signorini Problem

2.1. Let an elastic homogeneous anisotropic body in the natural configuration

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

occupies a bounded region \overline{Ω^{+}} of the three-dimensional space ℝ^{3}: \overline{Ω^{+}} = Ω^{+} ∪S, S = ∂Ω^{+}. For simplicity, we assume that the boundary S = ∂Ω^{+} is C^{∞}-smooth. Further, let Ω^{−} = ℝ^{3}\\overline{Ω^{+}}. The elastic coefficients C_{kjpq} satisfy the symmetry conditions

C_{kjpq} = C_{pqkj} = C_{jkpq}.

The stress tensor {σ_{kj}} and the strain tensor {ε_{kj}} are related by Hooke’s law

σ_{kj}(u) = C_{kjpq}ε_{pq}(u), ε_{kj}(u) = \frac{1}{2} (∂_{k}u_{j} + ∂_{j}u_{k}) ,

∂ = (∂_{1}, ∂_{2}, ∂_{3}) , ∂_{k} := \frac{∂}{∂x_{k}} ,

where u = (u_{1}, u_{2}, u_{3})^{⊤} is the displacement vector; here and in what follows the summation over the repeated indices is meant from 1 to 3 unless stated otherwise; the subscript ⊤ denotes transposition.

Define the symmetric bilinear forms:

E(u, v) = σ_{kj}(u)ε_{kj}(v) = C_{kjpq}ε_{pq}(u)ε_{kj}(v) = E(v, u),  (2.1)
a(u, v) = ∫_{Ω^{+}} E(u, v)dx = a(v, u) = ∫_{Ω^{+}} C_{kjpq} ∂_{k}u_{j} ∂_{p}v_{q} dx.  (2.2)

As usual, the quadratic form corresponding to the potential energy is supposed to be positive definite in the symmetric variables ε_{kj} = ε_{jk} (see, e.g., [12])

E(u, u) = σ_{kj}(u)ε_{kj}(u) = C_{kjpq}ε_{kj}(u)ε_{pq}(u)
≥ δ_{1} ε_{kj}(u) ε_{kj}(u), δ_{1} = const > 0.  (2.3)

Let the boundary S be divided into three disjoint open subsurfaces S_{1}, S_{2}, and S_{3}, where S_{k} ∩ S_{j} = ∅ for k ≠ j, S_{2} ≠ ∅, S_{3} ≠ ∅, \overline{S}_{1} ∪ \overline{S}_{2} ∪ \overline{S}_{3} = S, \overline{S}_{k} = S_{k} ∪ ∂S_{k}; for simplicity, we assume ∂S_{k}, k = 1, 2, 3, to be C^{∞}-smooth curves.

By T (∂, n)u we denote the stress vector acting on a surface element with the unit normal vector n = (n_{1}, n_{2}, n_{3}):

[ T (∂_{x}, n(x))u(x) ]_{k} = σ_{kj}(u)n_{j}(x) = C_{kjpq}n_{j}(x)∂_{q}u_{p}(x), k = 1, 2, 3. (2.4)

We note that throughout the paper we will use the following notation (when it causes no confusion):

(a) if all elements of a vector v = (v_{1}, …, v_{m})^{⊤} (a matrix N = [N_{kj}]_{m×n}) belong to one and the same space X, we will write v ∈ X (N ∈ X) instead of v ∈ X^{m} (N ∈ X_{m×n});

(b) if K : X_{1} ×X_{2} ×⋯×X_{m} → Y_{1} ×Y_{2} ×⋯×Y_{n} and X_{1} = X_{2} = ⋯ = X_{m}, Y_{1} = Y_{2} = ⋯ = Y_{n}, we will write K : X → Y instead of K : X^{m} → Y^{n};

(c) if a, b ∈ ℝ^{m}, then a · b := ∑_{k=1}^{m} a_{k}b_{k} denotes the usual scalar product in ℝ^{m}; (d) by ∥u; X∥ we denote the norm of the element u in the space X.

As usual, H^{α}(Ω^{+}), H_{loc}^{α}(Ω^{−}) and H^{α}(S) denote the Sobolev-Slobodetski (Bessel potential) spaces; here α is a real number (see, e.g., [22], [31], [32]). By H̃^{α}(S_{k}) we denote the subspace of H^{α}(S):

H̃^{α}(S_{k}) = \{ω : ω ∈ H^{α}(S), supp ω ⊂ \overline{S}_{k}\},

while H^{α}(S_{k}) denotes the space of restriction on S_{k} of functionals from H^{α}(S)

H^{α}(S_{k}) = {r_{S_{k}}f : f ∈ H^{α}(S)},

where r_{S_{k}} denotes the restriction operator on S_{k}.

2.2. The mathematical formulation of the typical Signorini problem reads as follows: Find the displacement vector u = (u_{1}, u_{2}, u_{3})^{⊤} ∈ H^{1}(Ω^{+}) by the following conditions:

A(∂)u(x) = 0 in Ω^{+},  (2.5)
[ T (∂_{x}, n(x))u(x) ]^{+} = g on S_{1},  (2.6)
[ u(x) ]^{+} = 0 on S_{2},  (2.7)

\begin{matrix}[ T (∂, n)u(x) ]^{+} − n [ T (∂, n)u(x) · n]^{+} = 0, \\ − [ u(x) · n]^{+} ≥ 0, \\ − [ T (∂, n)u(x) · n]^{+} ≥ 0, \\ [ T (∂, n)u(x) · n]^{+} [ u(x) · n]^{+} = 0, \end{matrix} \} on S_{3}, (2.8)

where A(∂) is a matrix differential operator of elastostatics

A(∂) = [ A_{kp}(∂) ]_{3×3}, A_{kp}(∂) = C_{kjpq}∂_{j}∂_{q},

the symbols [ ·]^{±}( [ ·] _{S}^{±}) denote limits (traces) on S from Ω^{±}, n = n(x) is the unit outward normal vector to S at the point x ∈ S.

Equation (2.5) corresponds to the equilibrium state of the elastic body in question (with bulk forces equal to zero). Condition (2.6) describes that the body is subjected to assigned surface forces on S_{1}, while (2.7) shows that the body is fixed along the subsurface S_{2}. The unilateral Signorini conditions (“ambiguous boundary conditions” – due to the original terminology of Signorini [11]) mean that the body remains on or “above” the portion S_{3} of the boundary ∂Ω^{+} = S (the “upper” direction on S is defined by the outward normal n).

The first equality in (2.8) means that tangent stresses vanish on S_{3} (i.e., we have contact without friction with a rigid support along the S_{3}). The mechanical meaning of the last three conditions in (2.8) are described in detail, e.g., in [12], Part 2, Section 10.

We assume that the vector-function g in condition (2.6) belongs to the space L_{2}(S_{1}) = H^{0}(S_{1}).

Note that if u ∈ H^{1}(Ω^{+}) and A(∂)u ∈ L_{2}(Ω^{+}), then, in general, the limit [ T (∂, n)u] _{S}^{+} is defined as a functional of the class H^{− \frac{1}{2}} (S) defined by the duality relation (see, e.g., [22]),

⟨ [ T (∂, n)u(x) ] _{S}^{+}, [v]_{S}^{+}⟩_{S} := ∫_{Ω^{+}} A(∂)u · v dx + ∫_{Ω^{+}} E(u, v) dx,  (2.9)

where v = (v_{1}, v_{2}, v_{3})^{⊤} ∈ H^{1}(Ω^{+}); here ⟨·, ·⟩_{S} is the duality between the spaces H^{\frac{1}{2}} (S) and H^{− \frac{1}{2}} (S), which coincides with the usual [ L_{2}(S) ]^{3} scalar product for regular (in general, complex valued) vector-functions, i.e., if f, h ∈ [ L_{2}(S) ]^{3}, then

⟨f, h⟩_{S} = ∫_{S} f_{k}h_{k} dS =: (f, \overline{h})_{L_{2}(S)},

where the over-bar denotes complex conjugation.

Equation (2.9) can be interpreted as a Green formula for the operator A(∂). Due to the regularity results obtained in [19] if, in addition, g = [G]_{S_{1}}^{+}, where

G ∈ H^{1}(Ω^{+}), then all conditions in (2.8) can be understood in the usual classical sense (see also [12], Part 2, Section 10).

2.3. The above-formulated Signorini problem is equivalent to the following spatial variational inequality (see [12], [19]): Find u ∈ K such that

a(u, v − u) ≥ P (v − u),  ∀v ∈ K,  (2.10)

where the bilinear form a(u, v) is given by (2.2),

K = {u = (u_{1}, u_{2}, u_{3})^{⊤} ∈ H^{1}(Ω^{+}) : [n(x) · u(x)]^{+} ≤ 0 on S_{3}
and [u(x)]^{+} = 0 on S_{2}} ,  (2.11)

and the linear functional P is defined by the equation

P (v) = ⟨g, [v]^{+}⟩_{S_{1}} = ∫_{S_{1}} g · [v]^{+}dS  (2.12)

with g ∈ L_{2}(S_{1}).

In turn, the variational inequality (2.10) is equivalent to the minimization problem for the energy functional (see [12])

E(v) = 2^{−1} a(u, v) − ⟨g, [v]^{+}⟩_{S_{1}},  ∀v ∈ K.  (2.13)

Observe that the bilinear form a(·, ·) is coercive on the space

H^{1}(Ω^{+}; S_{2}) := {v ∈ H^{1}(Ω) : [v]^{+} = 0 on S_{2}}  (2.14)

since the measure of the subsurface S_{2} is positive (see, e.g., [29], [7]). Thus there exists a positive constant c_{0} such that

a(u, u) ≥ c_{0} ∥u; H^{1}(Ω^{+})∥^{2}, ∀u ∈ H^{1}(Ω^{+}; S_{2}).

Therefore the variational inequality (2.10) together with problem (2.5)–(2.8) and the minimization problem for functional (2.13) is uniquely solvable (see, e.g., [12], [7], [19], [13]).

In the subsequent sections, on the basis of the potential theory, we will equivalently reduce the spatial variational inequality (2.10) to a boundary variational inequality.

First we expose the mapping and coercive properties of the integral (pseudodifferential) operators.

3. Properties of Boundary Integral (Pseudodifferential) Operators

3.1. Single- and double-layer potentials and their properties. Let Γ(·) be the fundamental matrix of the operator A(∂)

A(∂)Γ(x) = δ(x)I,

where δ(·) is the Dirac distribution and I = [δ_{kj}]_{3×3} is the unit matrix (δ_{kj} is the Kronecker symbol). This matrix reads [Na1]

Γ(x) = F_{ξ→x}^{−1} [ A^{−1}(−iξ) ] = −\frac{1}{8π|x|} ∫_{0}^{2π}A^{−1}(aη)dφ,  (3.1)

where A^{−1}(−iξ) is the matrix inverse to A(−iξ), ξ ∈ ℝ^{3}\{0}, x ∈ ℝ^{3}\{0}, a = [a_{kj}]_{3×3} is an orthogonal matrix with the property a^{⊤}x = (0, 0, |x|)^{⊤}, η = (cos φ, sin φ, 0)^{⊤}; F_{ξ→x}^{−1} denotes the generalized inverse Fourier transform.

Further, let us introduce the single- and double-layer potentials

V (g)(x) = ∫_{S} Γ(x − y) g(y) dS_{y},  (3.2)
W (g)(x) = ∫_{S} [T (∂_{y}, n(y))Γ(x − y)]^{⊤} g(y) dS_{y},  (3.3)

where g = (g_{1}, g_{2}, g_{3})^{⊤} is a density vector and x ∈ ℝ^{3}\ S.

The properties of these potentials and the corresponding boundary integral (pseudodifferential) operators in the Hölder (C^{k+α}), Bessel potential (H_{p}^{s}) and Besov (B_{p,q}^{s}) spaces are studied in [20], [25], [2], [26], [27], [5], [6], [28] (see also [18], [3], [24], where the coerciveness of boundary operators and Lipschitz domains are considered).

In the sequel we need some results obtained in the above-cited papers and we recall them here for convenience.

Theorem 3.1 ([25], [2], [26]). Let k ≥ 0 be an integer and 0 < γ < 1. Then the operators

\begin{aligned}V & : C^{k+γ}(S) → C^{k+1+γ}(\overline{Ω^{±}}), \\ W & : C^{k+γ}(S) → C^{k+γ}(\overline{Ω^{±}}), \end{aligned}  (3.4)

are bounded.

For any g ∈ C^{k+γ}(S) and any x ∈ S

[V (g)(x)]^{±} = V (g)(x) = H g(x),
[T (∂_{x}, n(x))V (g)(x)]^{±} = [ ∓2^{−1}I + K] g(x)  (3.5)
[W (g)(x)]^{±} = [±2^{−1}I + K^{∗}] g(x),
[T (∂_{x}, n(x))W (g)(x)]^{+} = [T (∂_{x}, n(x))W (g)(x)]^{−} = L g(x), k ≥ 1,

where

H g(x) := ∫_{S} Γ(x − y) g(y) dS_{y},  (3.6)
K g(x) := ∫_{S} T (∂_{x}, n(x))Γ(y − x) g(y) dS_{y},  (3.7)
K^{∗} g(x) := ∫_{S} [T (∂_{y}, n(y))Γ(x − y)]^{⊤} g(y) dS_{y},  (3.8)
L g(x) := lim_{Ω^{±}∋z→x∈S} T (∂_{z}, n(x)) ∫_{S} [T (∂_{y}, n(y))Γ(y − z)]^{⊤} g(y) dS_{y}. (3.9)

Theorem 3.2 ([5]). Operators (3.4) can be extended by continuity to the bounded mappings

V : H^{s}(S) → H^{s+1+ \frac{1}{2}} (\overline{Ω^{+}}) [H^{s}(S) → H_{loc}^{s+1+ \frac{1}{2}} (Ω^{−})],
W : H^{s}(S) → H^{s+ \frac{1}{2}} (\overline{Ω^{+}}) [H^{s}(S) → H_{loc}^{s+ \frac{1}{2}} (Ω^{−})],

with s ∈ ℝ. The jump relations (3.5) on S remain valid for the extended operators in the corresponding functional spaces.

Theorem 3.3 ([25], [5]). Let k ≥ 0 be an integer, 0 < γ < 1, and s ∈ ℝ. Then the operators

H : C^{k+γ}(S) → C^{k+1+γ}(S),
: H^{s}(S) → H^{s+1}(S),  (3.10)
±2^{−1}I + K, ±2^{−1}I + K^{∗} : C^{k+γ}(S) → C^{k+γ}(S),
: H^{s}(S) → H^{s}(S),  (3.11)
L : C^{k+1+γ}(S) → C^{k+γ}(S),
: H^{s+1}(S) → H^{s}(S)  (3.12)

are bounded.

Moreover,

(i) the operators ±2^{−1}I + K and ±2^{−1}I + K^{∗} are mutually adjoint singular integral operators of normal type with the index equal to zero. The operators H, 2^{−1}I + K and 2^{−1}I + K^{∗} are invertible. The inverse of H

H^{−1} : C^{k+1+γ}(S) → C^{k+γ}(S) [H^{s+1}(S) → H^{s}(S)]

is a singular integro-differential operator;

(ii) the L is a singular integro-differential operator and the following equalities hold in appropriate functional spaces:

K^{∗}H = HK, LK^{∗} = KL, HL = −4^{−1}I + (K^{∗})^{2}, LH = −4^{−1}I + K^{2}; (3.13)

(iii) The operators −H and L are self-adjoint and non-negative elliptic pseudodifferential operators with the index equal to zero:

⟨−Hh, h⟩_{S} ≥ 0,  ⟨Lg, g⟩_{S} ≥ 0,  (3.14)
∀h ∈ C^{γ}(S), ∀g ∈ C^{1+γ}(S), [ ∀h ∈ H^{− \frac{1}{2}} (S), ∀g ∈ H^{\frac{1}{2}} (S) ] ,

with equality only for h = 0 and for

g = [a × x] + b,  (3.15)

respectively; here a, b ∈ ℝ^{3} are arbitrary constant vectors and [· × ·] denotes the cross product of two vectors;

(iv) a general solution of the homogeneous equations [−2^{−1}I + K^{∗}]g = 0 and Lg = 0 is given by (3.15), i.e.,

ker L = ker (−2^{−1}I + K^{∗}) and
dim ker L = dim coker L = dim ker (−2^{−1}I + K^{∗})
= dim coker (−2^{−1}I + K^{∗}) = 6.

Theorem 3.4 ([5], citeNa1). Let u be a solution of the homogeneous equation A(D)u = 0 in Ω^{±}. Then

W ( [ u]^{±}) − V ( [ T u]^{±}) = \{ \begin{matrix}±u(x), & x ∈ Ω^{±}, \\ 0, & x ∈ Ω^{∓}, \end{matrix}

where either u ∈ C^{k+γ}(\overline{Ω^{±}}) with k ≥ 1, 0 < γ < 1, or u ∈ H^{1}(Ω^{+}) [ H_{loc}^{1}(Ω^{−}), u(x) → 0 as |x| → +∞] .

We also need some additional properties of the above-introduced operators, which will be proved below.

Theorem 3.5. Let f ∈ H^{\frac{1}{2}} (S) and h ∈ H^{− \frac{1}{2}} (S) satisfy the condition

(−2^{−1}I + K^{∗}) f = Hh, i.e., h = H^{−1} (−2^{−1}I + K^{∗}) f. (3.16)

Then there exists a unique vector-function u ∈ H^{1} (Ω^{+}) such that

A (∂) u(x) = 0 in Ω^{+},
[ u]^{+} = f and [ T u]^{+} = h on S = ∂Ω^{+}.

Moreover,

Lf = (2^{−1}I + K) h, i.e., h = (2^{−1}I + K)^{−1} Lf.  (3.17)

Proof. We put

u (x) = W (f)(x) − V (h) (x) , x ∈ Ω^{±}.

Clearly, u ∈ H^{1}(Ω^{+}) ∩ H_{loc}^{1}(Ω^{−}) and A (∂) u(x) = 0 in Ω^{+}. Due to Theorem 3.2 and equality (3.16) we have

[ u]^{−} = (−2^{−1}I + K^{∗}) f − Hh = 0.

Therefore, with the help of the uniqueness theorem for the exterior Dirichlet boundary value problem (in Ω^{−}) for the operator A(∂), we conclude that u(x) = 0 in Ω^{−}. Thus

W (g)(x) − V (h)(x) = \{ \begin{matrix}u(x) & for & x ∈ Ω^{+}, \\ 0 & for & x ∈ Ω^{−}. \end{matrix}

Whence, applying again Theorem 3.2, we get

[ u]^{+} − [ u]^{−} = f and [ T u]^{+} − [ T u]^{−} = h on S,

i.e.,

[ u]^{+} = f and [ T u]^{+} = h.

The uniqueness of u can be easily shown by the uniqueness theorem for the Dirichlet problem.

To complete the proof we note that (3.16) and Theorem 3.3, (ii) imply

HLf = (2^{−1}I + K^{∗}) (−2^{−1}I + K^{∗}) f = (2^{−1}I + K^{∗}) Hh = H (2^{−1}I + K) h,

from which (3.17) follows since H is an invertible operator.

Further, we discuss in detail the coercive properties of the bilinear forms in (3.14) and the operators related to them. Note that similar problems for the isotropic case and the Laplace equation are studied in [18], [3], [24], Ch.10, [4], Ch. XI, § 4 with the help of Korn’s inequalities. Here we treat the general anisotropic case and prove the coerciveness of the corresponding operators by means of different arguments.

3.2. Coercive properties of the boundary bilinear forms. We start by the following simple lemma.

Lemma 3.6. Let ⟨·, ·⟩_{S} be the bracket of duality (bilinear form) between the dual pair H^{r}(S) and H^{−r}(S), ∀r ∈ ℝ, and (·, ·)_{H^{−r}(S)} denote a scalar product in H^{−r}(S).

There exists a linear bounded bijective operator

P_{2r} : H^{r}(S) → H^{−r}(S)  (3.18)

and positive constants C_{1} and C_{2}, such that

⟨f, \overline{g}⟩_{S} = (P_{2r}f , g)_{H^{−r}(S)} , ∀f ∈ H^{r}(S), ∀g ∈ H^{−r}(S),  (3.19)

and

C_{1} ∥f; H^{r}(S)∥ ≤ ∥P_{2r}f; H^{−r}(S)∥ ≤ C_{2} ∥f; H^{r}(S)∥, ∀f ∈ H^{r}(S). (3.20)

Moreover,

⟨f, P_{2r}\overline{f}⟩_{S} ≥ 0, ∀f ∈ H^{r}(S),  (3.21)

with the equality only for f = 0.

Proof. First we consider the case of real functions. Let f ∈ H^{r}(S). Then

|⟨f, g⟩_{S}| ≤ C∥f; H^{r}(S)∥ ∥g; H^{−r}(S)∥, ∀g ∈ H^{−r}(S),

where C is independent of f and g, i.e., ⟨f, ·⟩_{S} is the linear bounded functional on H^{−r}(S) and due to the Riesz theorem there exists a unique element F ∈ H^{−r}(S) such that

⟨f, \overline{g}⟩_{S} = (F, g)_{H^{−r}(S)} , ∀g ∈ H^{−r}(S).  (3.22)

We put F = P_{2r}f. It is evident that the mapping P_{2r} : H^{r}(S) → H^{−r}(S) is linear and injective. Now we show that it is surjective, i.e., for arbitrary F ∈ H^{−r}(S) there exists unique f ∈ H^{−r}(S) such that equality (3.22) holds.

Let Λ_{r} be an equivalent lifting (i.e., order reducing) pseudodifferential operator of order −r (see, e.g., [22], [10], [14], [1]):

Λ_{r} : H^{0}(S) → H^{r}(S).

This mapping is an isomorphism. Then the adjoint operator (with respect to the duality bracket) Λ_{r}^{∗} is also an equivalent lifting operator

Λ_{r}^{∗} : [ H^{r}(S) ]^{∗} → [ H^{0}(S) ]^{∗}, i.e., Λ_{r}^{∗} : H^{−r}(S) → H^{0}(S).

Obviously, there exist positive constants a_{1}, a_{2}, b_{1}, and b_{2} such that

a_{1} ∥f; H^{0}(S)∥ ≤ ∥Λ_{r}f; H^{r}(S)∥ ≤ a_{2} ∥f; H^{0}(S)∥, ∀f ∈ H^{0}(S),
b_{1} ∥g; H^{−r}(S)∥ ≤ ∥Λ_{r}^{∗}g; H^{0}(S)∥ ≤ b_{2} ∥g; H^{−r}(S)∥, ∀g ∈ H^{−r}(S).

Here a_{j} and b_{j} do not depend on f and g, respectively. Note that

B(f̃, g) := ⟨\overline{Λ_{r}}Λ_{r}^{∗}f̃, g⟩_{S}

is a bilinear form on H^{−r}(S) × H^{−r}(S).

Consider the equation

B(f̃, g) = (F, g)_{H^{−r}(S)} , ∀g ∈ H^{−r}(S),  (3.23)

where F ∈ H^{−r}(S) is some given element and f̃ ∈ H^{−r}(S) is the sought for element. Obviously, the linear functional in the right-hand side of (3.23) is bounded on H^{−r}(S).

It is also evident that B(·, ·) is continuous:

|B(f̃, g)| = |⟨\overline{Λ_{r}}Λ_{r}^{∗}f̃, g⟩_{S}| ≤ C ∥\overline{Λ_{r}}Λ_{r}^{∗}f̃; H^{r}(S)∥ ∥g; H^{−r}(S)∥
≤ c a_{2} b_{2} ∥f̃; H^{−r}(S)∥ ∥g; H^{−r}(S)∥.

Moreover, B(·, ·) is coercive:

B(f̃, f̃) = ⟨\overline{Λ_{r}}Λ_{r}^{∗}f̃, f̃⟩_{S} = ⟨Λ_{r}^{∗}f̃, \overline{Λ_{r}^{∗}}f̃⟩_{S} = (Λ_{r}^{∗}f̃, Λ_{r}^{∗}f̃)_{H^{0}(S)}
= ∥Λ_{r}^{∗}f; H^{0}(S)∥^{2} ≥ b_{1}^{2} ∥f̃; H^{−r}(S)∥^{2}.

Due to the Lax–Milgram theorem then there exists a unique solution f̃ ∈ H^{−r}(S) of equation (3.23), i.e.,

⟨\overline{Λ_{r}}Λ_{r}^{∗}f̃, g⟩_{S} = (F, g)_{H^{−r}(S)} , g ∈ H^{−r}(S),

which proves the surjectivity of the operator P_{2r} since \overline{Λ_{r}}Λ_{r}^{∗}f̃ ∈ H^{−r}(S). Thus mapping (3.18) is bijective. Inequality (3.20) then follows from the Banach theorem on inverse operators.

Now let f = f_{1} + if_{2} and g = g_{1} + ig_{2}, where f_{j} ∈ H^{r}(S) and g_{j} ∈ H^{−r}(S) are real functions.

Due to the results just proven we have

⟨f, \overline{g}⟩_{S} = ⟨f_{1} + if_{2}, g_{1} − ig_{2}⟩_{S} = ⟨f_{1}, g_{1}⟩_{S} + i⟨f_{2}, g_{1}⟩_{S} − i⟨f_{1}, g_{2}⟩_{S}
−i^{2}⟨f_{2}, g_{2}⟩_{S} = (P_{2r}f_{1}, g_{1})_{H^{−r}(S)} + i (P_{2r}f_{2}, g_{1})_{H^{−r}(S)}
−i (P_{2r}f_{1}, g_{2})_{H^{−r}(S)} − i^{2} (P_{2r}f_{2}, g_{2})_{H^{−r}(S)} = (P_{2r}f, g)_{H^{−r}(S)} ,

whence (3.20) and (3.21) follow.

Lemma 3.7. There exists a positive constant C_{3} such that

⟨−Hφ, \overline{φ}⟩_{S} ≥ C_{3} ∥φ; H^{− \frac{1}{2}} (S)∥^{2}, ∀φ ∈ H^{− \frac{1}{2}} (S).

Proof. Due to Lemma 3.6

⟨−Hφ, \overline{φ}⟩_{S} = (−P_{1}Hφ, φ)_{H^{− \frac{1}{2}} (S)} , ∀φ ∈ H^{− \frac{1}{2}} (S).

Moreover, the mapping

−P_{1}H : H^{− \frac{1}{2}} (S) → H^{− \frac{1}{2}} (S)

is an isomorphism and

(−P_{1}Hφ, φ)_{H^{− \frac{1}{2}} (S)} = ⟨−Hφ, \overline{φ}⟩_{S} ≥ 0

for an arbitrary (complex-valued) function φ ∈ H^{− \frac{1}{2}} (S). Evidently, −P_{1}H is a self-adjoint positive operator (see, e.g., [30], Theorem 12.32).

By the square root theorem (see, e.g., [30], Theorem 12.33, [23], Ch.7, §3, Theorem 2) we conclude that there exists a unique positive linear bounded bijective operator

Q : H^{− \frac{1}{2}} (S) → H^{− \frac{1}{2}} (S)

such that Q^{2} = −P_{1}H. Consequently, Q is self-adjoint and invertible, and

e_{1} ∥φ; H^{− \frac{1}{2}} (S)∥ ≤ ∥Qφ; H^{− \frac{1}{2}} (S)∥ ≤ e_{2} ∥φ; H^{− \frac{1}{2}} (S)∥

with some positive constants e_{1} and e_{2}. Therefore

⟨−Hφ, \overline{φ}⟩_{S} = (−P_{1}Hφ, φ)_{H^{− \frac{1}{2}} (S)}
= (Q^{2}φ, φ)_{H^{− \frac{1}{2}} (S)} = ∥Qφ; H^{− \frac{1}{2}} (S)∥^{2} ≥ e_{1}^{2} ∥φ; H^{− \frac{1}{2}} (S)∥^{2},

which completes the proof.

We denote by X(Ω^{+}) the space of rigid displacements in Ω^{+}, and by X(S) = X(∂Ω^{+}) the space of their restrictions on S = ∂Ω^{+} (see (3.15)). Note that dimX(Ω^{+}) =dimX(S) = 6. Let Ψ(S) = {χ^{j}}_{j=1}^{6} be the orthonormal (in the H^{0}(S)-sense) basis in X(S):

(χ^{j}, χ^{k})_{H^{0}(S)} = ⟨χ^{j}, χ^{k}⟩_{S} = δ_{kj},

where δ_{kj} is Kronecker’s symbol. Clearly, the system {χ^{j}}_{j=1}^{6} can be obtained by the orthonormalization (in the H^{0}(S)-sense) procedure of the following basis of X(S):

ν^{(1)} = (1, 0, 0), ν^{(2)} = (0, 1, 0), ν^{(3)} = (0, 0, 1),
ν^{(4)} = (x_{3}, 0, −x_{1}), ν^{(5)} = (−x_{2}, x_{3}, 0), ν^{(6)} = (0, −x_{3}, x_{2}).

Further, let r ∈ ℝ and

H_{∗}^{r}(S) := \{φ : φ ∈ H^{r}(S), ⟨φ, χ^{j}⟩_{S} = 0, j = \overline{1, 6}\}.  (3.24)

It is evident that H_{∗}^{r}(S) is a Hilbert space with the scalar product induced by (·, ·)_{H^{r}(S)}.

Lemma 3.8. L(H^{\frac{1}{2}} (S)) = H_{∗}^{− \frac{1}{2}} (S), where the operator L is given by (3.9).

Proof. The Green identity

∫_{Ω^{+}} [ Au · v − u · Av] dx = ∫_{S} { [ T u]^{+} · [ v]^{+} − [ u]^{+} · [ T v]^{+}} dS

with u = W (φ) and ∀v ∈ X(Ω^{+}) yields

∫_{S} Lφ · χ^{j}dS = ⟨Lφ, χ^{j}⟩_{S} = 0, ∀φ ∈ H^{\frac{1}{2}} (S), j = \overline{1, 6}.

Therefore

L : H^{\frac{1}{2}} (S) → H_{∗}^{− \frac{1}{2}} (S), i.e., L(H^{\frac{1}{2}} (S)) ⊂ H_{∗}^{− \frac{1}{2}} (S).

On the other hand, the equation

Lφ = f, f ∈ H_{∗}^{− \frac{1}{2}} (S),  (3.25)

is solvable, and a solution φ ∈ H^{\frac{1}{2}} (S) can be represented in the form ([25], [5])

φ = φ_{0} + ∑_{j=1}^{6} c_{j}χ^{j}, φ_{0} ∈ H^{\frac{1}{2}} (S),

where φ_{0} is some particular solution of the above non-homogeneous equation and c_{j} are arbitrary constants. In addition, if we require that φ ∈ H∗^{\frac{1}{2}} (S), then (3.25) is uniquely solvable and the solution reads

φ = φ_{0} − ∑_{j=1}^{6}⟨φ_{0}, χ^{j}⟩_{S} χ^{j}.  (3.26)

It can be easily shown that the right-hand side in (3.26) does not depend on the choice of the particular solution φ_{0} since the homogeneous version of equation (3.25) (with f = 0) possesses only the trivial solution in the space H∗^{\frac{1}{2}} (S).

Corollary 3.9. The operator

L : H∗^{\frac{1}{2}} (S) → H_{∗}^{− \frac{1}{2}} (S)

is an isomorphism, and the inequality

c_{1}^{∗} ∥φ; H^{\frac{1}{2}} (S)∥ ≤ ∥Lφ; H^{− \frac{1}{2}} (S)∥ ≤ c_{2}^{∗} ∥φ; H^{\frac{1}{2}} (S)∥,  (3.27)

holds for all φ ∈ H∗^{\frac{1}{2}} (S) with some positive constants c_{1}^{∗} and c_{2}^{∗} independent of φ.

Lemma 3.10. There exists a positive constant c_{3}^{∗} such that

⟨Lφ, \overline{φ}⟩_{S} ≥ c_{3}^{∗} ∥φ; H^{\frac{1}{2}} (S)∥^{2}, ∀φ ∈ H∗^{\frac{1}{2}} (S).

Proof. Let

L̃φ := Lφ + ∑_{j=1}^{6}⟨φ, χ^{j}⟩_{S}χ^{j}.

Note that L̃φ = Lφ if and only if φ ∈ H∗^{\frac{1}{2}} (S).

Evidently,

⟨L̃φ, \overline{φ}⟩_{S} = ⟨Lφ, \overline{φ}⟩_{S} + ∑_{j=1}^{6} |⟨φ, χ^{j}⟩_{S}|^{2}, ∀φ ∈ H^{\frac{1}{2}} (S),

and ⟨L̃φ, \overline{φ}⟩_{S} = 0 implies φ = 0, due to Theorem 3.3, (iii) and (iv). This also shows that the homogeneous equation

L̃φ = 0, φ ∈ H^{\frac{1}{2}} (S),

possesses only the trivial solution, and since Ind L̃ = Ind L = 0, we conclude that the mapping

L̃ : H^{\frac{1}{2}} (S) → H^{− \frac{1}{2}} (S)

is an isomorphism.

Let Λ_{\frac{1}{2}} be an equivalent lifting operator

Λ_{\frac{1}{2}} : H^{0}(S) → H^{\frac{1}{2}} (S),

and construct the operator \overline{Λ∗_{\frac{1}{2}}} L̃Λ_{\frac{1}{2}} , where Λ∗_{\frac{1}{2}} is the operator adjoint to Λ_{\frac{1}{2}} with

respect to the duality bracket. Therefore the isomorphism Λ∗_{\frac{1}{2}} : H^{− \frac{1}{2}} (S) → H^{0}(S) is also an equivalent lifting operator.

It is obvious that the mapping

\overline{Λ∗_{\frac{1}{2}}} L̃Λ_{\frac{1}{2}} : H^{0}(S) → H^{0}(S)  (3.28)

is an isomorphism. Moreover,

⟨\overline{Λ∗_{\frac{1}{2}}} L̃Λ_{\frac{1}{2}} φ, \overline{φ}⟩_{S} = ⟨L̃Λ_{\frac{1}{2}} φ, \overline{Λ_{\frac{1}{2}} φ}⟩_{S} ≥ 0, ∀φ ∈ L_{2}(S) = H^{0}(S),

with equality only for φ = 0. Thus (3.28) defines a positive invertible operator. Further, due to the square root theorem, there exists a positive (self-adjoint) invertible operator Q̃ : H^{0}(S) → H^{0}(S) such that

\overline{Λ∗_{\frac{1}{2}}} L̃Λ_{\frac{1}{2}} = Q̃^{2}.

Therefore, with the help of the self-adjointness of the operator Q̃ and invoking the Banach theorem on inverse operator, we get

⟨\overline{Λ∗_{\frac{1}{2}}} L̃Λ_{\frac{1}{2}} φ, \overline{φ}⟩_{S} = ⟨Q̃^{2}φ, \overline{φ}⟩_{S} = (Q̃^{2}φ, φ)_{H^{0}(S)}

= ∥Q̃φ; H^{0}(S)∥^{2} ≥ c_{4}∥φ; H^{0}(S)∥^{2}, ∀φ ∈ H^{0}(S). (3.29)

Note that for arbitrary ψ ∈ H^{\frac{1}{2}} (S) there exists unique φ ∈ H^{0}(S) such that Λ_{\frac{1}{2}} φ = ψ, i.e., ψ = Λ−1_{\frac{1}{2}}φ and ∥φ; H^{0}(S)∥ ≥ c_{5}∥ψ; H^{\frac{1}{2}} (S)∥, where c_{5} is a positive constant independent of φ and ψ. Consequently, by virtue of (3.29) we derive

⟨L̃ψ, \overline{ψ}⟩_{S} = ⟨L̃Λ_{\frac{1}{2}} φ, \overline{Λ_{\frac{1}{2}} φ}⟩_{S} = ⟨\overline{Λ∗_{\frac{1}{2}}} L̃Λ_{\frac{1}{2}} φ, \overline{φ}⟩_{S}

≥ c_{4} ∥φ; H^{0}(S)∥^{2} ≥ c_{4}c_{5}^{2} ∥ψ; H^{\frac{1}{2}} (S)∥^{2},

which completes the proof since L̃ψ = Lψ, ∀ψ ∈ H∗^{\frac{1}{2}} (S).

Below we need some mapping and coercive properties of the operator

M := L − (−2^{−1}I + K) H^{−1} (−2^{−1}I + K^{∗}) .  (3.30)

Applying equalities (3.13) we easily transform (3.30) to obtain

M = H^{−1} (−2^{−1}I + K^{∗}) = (−2^{−1}I + K) H^{−1}.  (3.31)

Corollary 3.11. The operator

M : H^{\frac{1}{2}} (S) → H^{− \frac{1}{2}} (S)

is a bounded, positive, formally self-adjoint (with respect to the duality bracket), elliptic pseudodifferential operator of order 1, and

kerM = kerL = ker (−2^{−1}I + K^{∗}) , Ind M = 0.

Moreover,

M (H^{\frac{1}{2}} (S)) = H_{∗}^{− \frac{1}{2}} (S).

The mapping

M : H∗^{\frac{1}{2}} (S) → H_{∗}^{− \frac{1}{2}} (S)

is an isomorphism, and the inequalities

c′_{1} ∥φ; H^{\frac{1}{2}} (S)∥ ≤ ∥Mφ; H^{− \frac{1}{2}} (S)∥ ≤ c′_{2} ∥φ; H^{\frac{1}{2}} (S)∥, ∀φ ∈ H∗^{\frac{1}{2}} (S),

⟨Mφ, \overline{φ}⟩_{S} ≥ c′_{3} ∥φ; H^{\frac{1}{2}} (S)∥^{2}, ∀φ ∈ H∗^{\frac{1}{2}} (S),

hold with some positive constants c′_{1}, c′_{2}, and c′_{3} independent of φ.

Proof. It is a ready consequence of (3.30), (3.31), Theorem 3.3, Corollary 3.9, and Lemma 3.10.

4. Reduction to BVI. Existence and Uniqueness Results

Let u ∈ K be the unique solution of the SVI (2.10). Due to Theorems 3.4 and 3.1 we have the following Steklov-Poincaré relations connecting the Dirichlet and Neumann boundary data on S = ∂Ω^{+} of the vector u:

L[u]^{+} = (2^{−1}I + K) [T u]^{+}, (−2^{−1}I + K^{∗}) [u]^{+} = H[T u]^{+}.  (4.1)

These equalities imply

[T u]^{+} = L[u]^{+} − (−2^{−1}I + K) [T u]^{+}
= L[u]^{+} − (−2^{−1}I + K) H^{−1} (−2^{−1}I + K^{∗}) [u]^{+} = M[u]^{+}, (4.2)

where M is defined by (3.30).

The Green formula (2.9) with the vector u and arbitrary v ∈ H^{1}(Ω^{+}) can be rewritten as

∫_{Ω^{+}} E(u, v)dx = ⟨ [ T (∂, n)u(x) ]^{+}, [v]^{+}⟩_{S} = ⟨M[u]^{+}, [v]^{+}⟩_{S},  (4.3)

whence by virtue of (2.2)

a(u, v) = ⟨M[u]^{+}, [v]^{+}⟩_{S}.  (4.4)

Substituting (4.4) into (2.10) leads to the BVI: Find φ ∈ K̃(S) such that

⟨Mφ, ψ − φ⟩_{S} ≥ ∫_{S_{1}} g · (ψ − φ)dS, ∀ψ ∈ K̃(S),  (4.5)

where, in our case, g ∈ L_{2}(S_{1}) is a given function (see(2.6)),

φ = [u]^{+}, ψ = [v]^{+},  (4.6)
K̃(S) := \{f : f ∈ H^{\frac{1}{2}} (S), n · f ≤ 0 on S_{3}, f = 0 on S_{2}\}. (4.7)

Obviously, the closed convex cone K̃(S) coincides with the space of traces (on S) of functions from K (see (2.11)).

In what follows we show that the BVI (4.5) is equivalent to the SVI (2.10) in the following sense. If u is a solution of the SVI (2.10), then [u]^{+} = φ solves the BVI (4.5) which has just been proved.

Vice versa, if φ is a solution of the BVI (4.5), then the vector-function

u(x) = W (φ) (x) − V (h) (x), x ∈ Ω^{+},  (4.8)

where (see (3.31))

h = H^{−1} (−2^{−1}I + K^{∗}) φ = (−2^{−1}I + K) H^{−1}φ = Mφ  (4.9)

solves the SVI (2.10). To prove this, we show that the vector-function (4.8) meets conditions (2.5)–(2.8). Since φ ∈ H^{\frac{1}{2}} (S) and h ∈ H^{− \frac{1}{2}} (S), it is evident that u ∈ H^{1}(Ω^{+}) by Theorem 3.2 and A(∂)u = 0 in Ω^{+}. In accord with Theorem 3.5 (see the proof of Theorem 3.5) we have φ = [u]_{S}^{+} and h = [T u]_{S}^{+} = Mφ.

Obviously, condition (2.7) holds since φ ∈ K̃.

If in (4.5) we put ψ = φ ± ψ̃ ∈ K̃, where ψ̃ ∈ H̃^{\frac{1}{2}} (S_{1}), we arrive at the equation

⟨Mφ, ψ̃⟩_{S} = ⟨g, ψ̃⟩_{S_{1}} ∀ψ̃ ∈ H̃^{\frac{1}{2}} (S_{1}),

whence Mφ = g on S_{1} follows, i.e., condition (2.6) holds as well.

Further, if in (4.5) we put ψ = φ ± ω ∈ K̃, where ω ∈ H̃^{\frac{1}{2}} (S_{3}) and ω · n = 0, we get

⟨Mφ, ω⟩_{S} = ⟨g, ω⟩_{S_{3}}, ∀ω ∈ H̃^{\frac{1}{2}} (S_{3}), ω · n = 0,

which implies Mφ − n(n · Mφ) = 0 on S_{3}. Thus, the first condition in (2.8) holds automatically due to the inclusion φ ∈ K̃.

Let us set ψ = φ−n ν, where n is the outward normal vector and ν ∈ H̃^{\frac{1}{2}} (S_{3}) is a non-negative scalar function .

From (4.5) then it follows

⟨Mφ, −nν⟩_{S} ≥ 0, i.e., ⟨−n · Mφ, ν⟩_{S} ≥ 0, ∀ν ∈ H̃^{\frac{1}{2}} (S_{3}), ν ≥ 0.

This shows that the third inequality in (2.10) holds.

Now, let h be a scalar function with the properties

0 ≤ h(x) ≤ 1, h ∈ C^{1}(S), supp h ⊂ \overline{S_{3}},

and put ψ = [1 + t h(x)]φ ∈ K̃ with t ∈ (−1, 1). From inequality (4.5) then we get

⟨Mφ, h(x)φ⟩_{S} = 0,

which can be rewritten as

⟨n · Mφ, h(n · φ)⟩_{S} = 0  (4.10)

due to the equation Mφ = n (n · Mφ) on S_{3}.

Since h is an arbitrary function with the above-mentioned properties, from (4.10) we conclude that the fourth condition in (2.8) also holds. The above arguments prove that the vector u defined by (4.8) meets conditions (2.5)–(2.8) and therefore it solves the SVI (2.10).

Thus, there holds

Lemma 4.1. The SVI (2.10) is equivalent (in the above-mentioned sense) to the following BVI: Find φ ∈ K̃ such that

⟨Mφ, ψ − φ⟩_{S} ≥ ∫_{S_{1}} g · (ψ − φ)dS, ∀ψ ∈ K̃(S),  (4.11)

with given g ∈ L_{2}(S_{1}).

From the existence and uniqueness theorems for the SVI (2.10) it follows that BVI (4.11) is also uniquely solvable. Observe that to develop the Galerkin method for approximation of solutions by means of the boundary element procedure and to obtain the corresponding abstract error estimate we need the coercive property of the pseudodifferential operator M on the cone K̃(S). This property is also sine–qua–non to study the well–posedness of the BVI (4.11) independently (without invoking the mentioned SVI) on the basis of the theory of abstract variational inequalities in Hilbert spaces.

Note that Corollary 3.11 proves the coercivity of the operator M on the space H∗^{\frac{1}{2}} (S). But, in general, the K̃(S) is not a subset of H∗^{\frac{1}{2}} (S). However, there holds

Lemma 4.2. The bilinear form ⟨Mφ, ψ⟩_{S} is bounded and coercive on the

space H̃∗^{\frac{1}{2}} (S\\overline{S_{2}}) × H̃∗^{\frac{1}{2}} (S\\overline{S_{2}}):

⟨Mφ, ψ⟩_{S} ≤ κ_{1} ∥φ; H^{\frac{1}{2}} (S)∥ ∥ψ; H^{\frac{1}{2}} (S)∥,
⟨Mφ, \overline{φ}⟩_{S} ≥ κ_{2} ∥φ; H^{\frac{1}{2}} (S)∥^{2}

with positive κ_{1} and κ_{2} independent of φ and ψ.

Proof. Step 1. It is evident that for any vector-function φ ∈ H^{\frac{1}{2}} (S) we have the unique representation

φ(x) = φ^{(1)}(x) + φ^{(0)}(x),  (4.12)

where

φ^{(1)}(x) = φ(x) − ∑_{j=1}^{6} c_{j}(φ)χ^{(j)}(x), c_{j}(φ) = ⟨φ, χ^{(j)}⟩_{S},  (4.13)

φ^{(0)}(x) = ∑_{j=1}^{6} c_{j}(φ)χ^{(j)}(x);  (4.14)

here {χ^{(j)}}_{j=1}^{6} is the above-introduced H^{0}(S)-orthogonal basis in the six-di-mensional space X(S) (traces on S of rigid displacements, i.e., vectors of type (3.15)). Clearly,

φ^{(1)} ∈ H∗^{\frac{1}{2}} (S), φ^{(0)} ∈ X(S) = ker M ⊂ H^{\frac{1}{2}} (S).  (4.15)

Note that

|c_{j}(φ)| = |⟨φ, χ^{(j)}⟩_{S}| ≤ κ_{j} ∥φ; H^{\frac{1}{2}} (S)∥, j = \overline{1, 6}, φ ∈ H^{\frac{1}{2}} (S), (4.16)

with the constant κ_{j} independent of φ.

We put

l_{j}(φ) = ∫_{S_{2}} φ · χ^{(j)}dS.  (4.17)

It is easy to see that if φ ∈ X(S), i.e., φ(x) = ∑_{j=1}^{6} c_{j}(φ)χ^{(j)}(x), and l_{k}(φ) =

0, k = \overline{1, 6}, then φ(x) = 0. Indeed, from these conditions we derive

0 = ∑_{k=1}^{6} c_{k}(φ)l_{k}(φ) = ∫_{S_{2}} φ · φ dS = ∫_{S_{2}} |φ|^{2}dS,

whence φ = 0 on S_{2} and, consequently, φ = 0 on S (if a vector of rigid displacements vanishes at three points which do not belong to the same straight line, then it is identically zero in ℝ^{3}, that is, the vectors a and b in (3.15) vanish). This implies c_{j}(φ) = 0, j = \overline{1, 6}.

Step 2. Let us introduce a new norm in H^{\frac{1}{2}} (S):

|||φ||| = ∥φ∥_{∗} + ∥φ∥_{∗∗}, ∥φ∥_{∗} = ∥φ^{(1)}; H^{\frac{1}{2}} (S)∥, ∥φ∥_{∗∗} = ∑_{j=1}^{6} |l_{j}(φ)|, (4.18)

where φ^{(1)} and l_{j}(φ) are given by (4.13) and (4.17),

Note that ∥ · ∥_{∗} and ∥ · ∥_{∗∗} represent semi-norms in H^{\frac{1}{2}} (S), which admits the following estimates:

∥φ∥_{∗} = ∥φ^{(1)}; H^{\frac{1}{2}} (S)∥ = ∥φ − ∑_{j=1}^{6} c_{j}(φ)χ^{(j)}; H^{\frac{1}{2}} (S)∥

≤ ∥φ; H^{\frac{1}{2}} (S)∥ + ∑_{j=1}^{6} c_{j}(φ)∥χ^{(j)}; H^{\frac{1}{2}} (S)∥

≤ M_{1}∥φ; H^{\frac{1}{2}} (S)∥,  (4.19)

M_{1} = 1 + ∑_{j=1}^{6} κ_{j}∥χ^{(j)}; H^{\frac{1}{2}} (S)∥, ∀φ ∈ H^{\frac{1}{2}} (S),

∥φ∥_{∗∗} = ∑_{j=1}^{6} |∫_{S_{2}} φ · χ^{(j)}dS| ≤ ∑_{j=1}^{6} ∥φ; H^{0}(S)∥ ∥χ^{(j)}; H^{0}(S)∥

= 6 ∥φ; H^{0}(S)∥ ≤ 6 ∥φ; H^{\frac{1}{2}} (S)∥, ∀φ ∈ H^{\frac{1}{2}} (S),  (4.20)

where the positive constant M_{1} does not depend on φ (see (4.16)).

Further, we show that the seminorm ∥ · ∥_{∗∗} is a norm in X(S), i.e., if φ(x) =

∑_{k=1}^{6} c_{k}χ^{(k)}(x) and ∥φ∥_{∗∗} = 0, then φ = 0 on S. Indeed, these conditions yield

∥φ∥_{∗∗} = ∑_{j=1}^{6} | ∫_{S_{2}} ∑_{k=1}^{6} c_{k}χ^{(k)}(x) · χ^{(j)}(x)dS | = 0,

i.e.,

∫_{S_{2}} ∑_{k=1}^{6} c_{k}χ^{(k)}(x) · χ^{(j)}(x)dS = 0.

Hence

∫_{S_{2}} (∑_{k=1}^{6} c_{k} χ^{(k)}) · (∑_{j=1}^{6} c_{j} χ^{(j)}) dS = ∫_{S_{2}} |φ|^{2}dS = 0,

and, consequently, φ = 0 on S (that is, c_{k} = 0, k = \overline{1, 6}). This proves that ∥ · ∥_{∗∗} is a norm in X(S).

Since X(S) is a six-dimensional space, we have the estimate

m_{0} ∥φ; H^{\frac{1}{2}} (S)∥ ≤ ∥φ∥_{∗∗}, ∀φ ∈ X(S),  (4.21)

with some constant m_{0} > 0 independent of φ, due to the equivalence of all norms in finite-dimensional spaces.

Step 3. Here we show the equivalence of the norms ||| · ||| and ∥ ·; H^{\frac{1}{2}} (S)∥ in H^{\frac{1}{2}} (S). On the one hand, by virtue of (4.18), (4.19), and (4.20)

|||φ||| ≤ M ∥φ; H^{\frac{1}{2}} (S)∥, φ ∈ H^{\frac{1}{2}} (S)

with M = 6 + M_{1}.

On the other hand, with the help of (4.18), (4.20) and (4.21) we derive

|||φ||| = ∥φ∥_{∗} + ∥φ∥_{∗∗} ≥ ∥φ^{(1)}; H^{\frac{1}{2}} (S)∥ + \frac{1}{12} ∥φ^{(1)} + φ^{(0)}∥_{∗∗}

≥ ∥φ^{(1)}; H^{\frac{1}{2}} (S)∥ + \frac{1}{12} ∥φ^{(0)}∥_{∗∗} − \frac{1}{12} ∥φ^{(1)}∥_{∗∗} ≥ \frac{1}{2} ∥φ^{(1)}; H^{\frac{1}{2}} (S)∥

+\frac{m_{0}}{12} ∥φ^{(0)}; H^{\frac{1}{2}} (S)∥ ≥ m \{∥φ^{(1)}; H^{\frac{1}{2}} (S)∥ + ∥φ^{(0)}; H^{\frac{1}{2}} (S)∥\}

= m ∥φ; H^{\frac{1}{2}} (S)∥, ∀φ ∈ H^{\frac{1}{2}} (S),

where the constant m =min\{\frac{1}{2}, \frac{m_{0}}{12} \} > 0 is independent of φ.

Thus there exists positive constants m and M such that

m ∥φ; H^{\frac{1}{2}} (S)∥ ≤ |||φ||| ≤ M∥φ; H^{\frac{1}{2}} (S)∥, ∀φ ∈ H^{\frac{1}{2}} (S).  (4.22)

Step 4. Here we complete the proof of the lemma.

Let ∀φ ∈ H̃^{\frac{1}{2}} (S\\overline{S_{2}}) ⊂ H^{\frac{1}{2}} (S). Applying the self-adjointness of the operator M together with Corollary 3.11 and relations (4.15) and (4.22) we proceed as follows:

⟨Mφ, φ⟩_{S} = ⟨M (φ^{(1)} + φ^{(0)}) , φ^{(1)} + φ^{(0)}⟩_{S} = ⟨Mφ^{(1)}, φ^{(1)} + φ^{(0)}⟩_{S}
= ⟨Mφ^{(1)}, φ^{(1)}⟩_{S} + ⟨Mφ^{(1)}, φ^{(0)}⟩_{S} = ⟨Mφ^{(1)}, φ^{(1)}⟩_{S} + ⟨φ^{(1)}, Mφ^{(0)}⟩_{S}
= ⟨Mφ^{(1)}, φ^{(1)}⟩_{S} ≥ c′_{3} ∥φ^{(1)}; H^{\frac{1}{2}} (S)∥^{2} = c′_{3} \{∥φ^{(1)}∥_{∗} + ∥φ∥_{∗∗}\}^{2}
= c′_{3} |||φ|||^{2} ≥ κ_{2} ∥φ; H^{\frac{1}{2}} (S)∥^{2}, κ_{2} = c′_{3}m^{2},

with the constant κ_{2} > 0 independent of φ.

The boundedness of the bilinear form ⟨Mφ, ψ⟩_{S} is a trivial consequence of Corollary 3.11.

Now, let us recall the well-known theorem concerning an abstract variational inequality in a Hilbert space (see, e.g., [13], Ch.1, Theorems 2.1 and 2.2).

Theorem 4.3. Let V_{0} be a closed convex subset of a Hilbert space V, F be a linear bounded functional on V, and B(·, ·) be a coercive bilinear form on V ×V. Then the problem: Find u ∈ V_{0} such that

B(u, v − u) ≥ F (v − u), ∀v ∈ V_{0},

possesses a unique solution.

From this theorem along with Lemma 4.2 we get the following assertion.

Theorem 4.4. The BVI (4.11) possesses a unique solution φ satisfying the estimate

∥φ; H^{\frac{1}{2}} (S)∥ ≤ κ_{2}^{−1}∥g; L_{2}(S_{1})∥  (4.23)

with the same κ_{2} as in Lemma 4.2.

Proof. The first part of the theorem immediately follows from Theorem 4.3, since the cone K̃(S) is a closed convex set of the Hilbert space H∗^{\frac{1}{2}} (S\\overline{S_{2}}) and the linear functional defined by the right-hand side expression in (4.23) is bounded

on H̃∗^{\frac{1}{2}} (S\\overline{S_{2}}):

| ∫_{S_{1}} g · ψ dS | ≤ ∥g; L_{2}(S_{1})∥ ∥ψ; H^{\frac{1}{2}} (S_{1})∥

≤ ∥g; L_{2}(S_{1})∥ ∥ψ; H^{\frac{1}{2}} (S)∥ ∀ψ ∈ H∗^{\frac{1}{2}} (S\\overline{S_{2}}). (4.24)

To prove (4.23), we proceed as follows. We put ψ = 2φ and ψ = 0 in (4.11) to obtain the equality

⟨Mφ, φ⟩_{S} = ∫_{S_{1}} g · φ dS.

Further, applying the coercivity of the operator M (see Lemma 4.2) and relation (4.24) we arrive at inequality (4.23).

Remark 4.5. Note that we can extend the domain of the definition of the linear functional in the right-hand side of (4.11) with respect to g. In fact, instead of (4.11) we can consider the variational inequality

⟨Mφ, ψ − φ⟩_{S} ≥ ⟨g, ψ − φ⟩_{S_{1}}, ∀ψ ∈ K̃(S),  (4.25)

where ⟨·, ·⟩_{S_{1}} is the duality pairing between either the spaces H^{− \frac{1}{2}} (S_{1}) and H̃^{\frac{1}{2}} (S_{1}) if ∂S_{1} ∩ ∂S_{3} = ∅, or H̃^{− \frac{1}{2}} (S_{1}) and H^{\frac{1}{2}} (S_{1}) if ∂S_{1} ∩ ∂S_{3} ≠ ∅.

In this case a theorem similar to Theorem 4.4 holds with the corresponding estimate (instead of (4.23))

∥φ; H^{\frac{1}{2}} (S)∥ ≤ \{ \begin{matrix}κ_{3}∥g; H^{− \frac{1}{2}} (S_{1})∥ & for & g ∈ H^{− \frac{1}{2}} (S_{1}), \\ κ_{3}∥g; H̃^{− \frac{1}{2}} (S_{1})∥ & for & g ∈ H̃^{− \frac{1}{2}} (S_{1}), \end{matrix}

where κ_{3} is a positive constant independent of φ and g.

5. Galerkin Approximation of the BVI

In this section we treat the problem of numerical approximation of a solution to the BVI (4.11) by Galerkin’s method.

Suppose that H̃(h)^{\frac{1}{2}}(S\\overline{S_{2}}) is a finite dimensional subspace of H̃^{\frac{1}{2}} (S\\overline{S_{2}}) and let

K̃_{h}(S) = \{ψ_{h} ∈ H̃(h)^{\frac{1}{2}}(S\\overline{S_{2}}) : ψ_{h} · n ≤ 0 on S_{3}\}  (5.1)

be a convex closed nonempty subset of H̃^{\frac{1}{2}} (S\\overline{S_{2}}). Clearly, K̃_{h}(S) ⊂ K̃(S).

An element φ_{h} ∈ K̃_{h}(S) is said to be an approximate solution of the BVI (4.11) if

⟨Mφ_{h}, ψ_{h} − φ_{h}⟩_{S} ≥ ∫_{S_{1}} g · (ψ_{h} − φ_{h}) dS ∀ψ_{h} ∈ K̃_{h}(S).  (5.2)

The existence and uniqueness theorems for the solution of the BVI (5.2) follow immediately from Theorem 4.3. Furthermore, we have

Theorem 5.1. Let φ_{h} ∈ K̃_{h}(S) be a solution of BVI (4.11) and φ_{h} ∈ K̃_{h}(S) be an approximate solution of the BVI (5.2).

Then the following abstract error estimate holds:

∥φ − φ_{h}; H^{\frac{1}{2}} (S)∥^{2} ≤ c^{∗} inf_{ψ_{h}∈_{K_{h}}(S)} \{∥φ − ψ_{h}; H^{\frac{1}{2}} (S)∥^{2}

+|⟨Mφ, ψ_{h} − φ⟩_{S} − ∫_{S_{1}} g · (ψ_{h} − φ)dS|}  (5.3)

with some positive constant c^{∗} independent of g, f, and φ_{h}.

BOUNDARY VARIATIONAL INEQUALITY APPROACH 489

Proof. Due to the coerciveness and boundedness of the operator M (see Lemma 4.2) we derive

∥φ − φ_{h}; H^{\frac{1}{2}} (S)∥^{2} ≤ \frac{1}{κ_{2}} ⟨M(φ − φ_{h}), φ − φ_{h}⟩_{S}

= \frac{1}{κ_{2}} \{⟨M(φ − φ_{h}), φ − ψ_{h}⟩_{S} + ⟨M(φ − φ_{h}), ψ_{h} − φ_{h}⟩_{S}\}

≤ \frac{κ_{1}}{κ_{2}} ∥φ − φ_{h}; H^{\frac{1}{2}} (S)∥ ∥φ − ψ_{h}; H^{\frac{1}{2}} (S)∥ + \frac{1}{κ_{2}} \{⟨Mφ, ψ_{h} − φ_{h}⟩_{S}

− ⟨Mφ_{h}, ψ_{h} − φ_{h}⟩_{S}\} ≤ \frac{1}{2} ∥φ − φ_{h}; H^{\frac{1}{2}} (S)∥^{2}

+\frac{κ_{1}^{2}}{2κ_{2}^{2}} ∥φ − ψ_{h}; H^{\frac{1}{2}} (S)∥^{2} + \frac{1}{κ_{2}} \{⟨Mφ, ψ_{h} − φ⟩_{S}

−⟨Mφ, φ_{h} − φ⟩_{S} − ⟨Mφ_{h}, ψ_{h} − φ_{h}⟩_{S}} , where ψ_{h} ∈ K̃_{h}(S), and κ_{1} and κ_{2} are as in Lemma 4.2.

By virtue of (4.11) and (5.2) we conclude that

∥φ − φ_{h}; H^{\frac{1}{2}} (S)∥^{2} ≤ \frac{κ_{1}^{2}}{κ_{2}^{2}} ∥φ − ψ_{h}; H^{\frac{1}{2}} (S)∥^{2}

+\frac{2}{κ_{2}} \{⟨Mφ, ψ_{h} − φ⟩_{S} − ∫_{S_{1}} g · (φ_{h} − φ)dS − ∫_{S_{1}} g · (ψ_{h} − φ_{h})dS\}

= \frac{κ_{1}^{2}}{κ_{2}^{2}} ∥φ − ψ_{h}; H^{\frac{1}{2}} (S)∥^{2} + \frac{2}{κ_{2}} \{⟨Mφ, ψ_{h} − φ⟩_{S} − ∫_{S_{1}} g · (ψ_{h} − φ) dS\}

for all ψ_{h} ∈ K̃_{h}(S), whence (5.3) follows with c^{∗} =max\{\frac{2}{κ_{2}} , \frac{κ_{1}^{2}}{κ_{2}^{2}} \}.

Remark 5.2. Note that |⟨Mφ, ψ_{h} − φ⟩_{S}| ≤ κ_{1} ∥φ; H^{\frac{1}{2}} (S)∥ ∥ψ_{h} − φ; H^{\frac{1}{2}} (S)∥,

|∫_{S_{1}} g · (ψ_{h} − φ) dS| ≤ ∥g; L_{2}(S_{1})∥ ∥ψ_{h} − φ; L_{2}(S_{1})∥,

where κ_{1} is independent of φ and ψ_{h}. Therefore (5.3) implies the inequality

∥φ − φ_{h}; H^{\frac{1}{2}} (S)∥^{2} ≤ c^{∗∗} inf_{ψ_{h}∈_{K_{h}}(S)} \{∥φ − ψ_{h}; H^{\frac{1}{2}} (S)∥^{2}

+ ∥φ; H^{\frac{1}{2}} (S)∥ ∥ψ_{h} − φ; H^{\frac{1}{2}} (S)∥ + ∥g; L_{2}(S_{1})∥ ∥ψ_{h} − φ; L_{2}(S_{1})∥\}

with c^{∗∗} = max \{\frac{2}{κ_{2}} , \frac{κ_{1}^{2}}{κ_{2}^{2}} , \frac{2κ_{1}}{κ_{2}} \}.

Remark 5.3. Observe that the expression under modulus in the second term in the right-hand side of (5.3) is, actually, supported on the sub-manifold \overline{S_{3}} since Mφ = g on S_{1}, and ψ_{h} and φ_{h} vanish on S_{2}.

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(Received 11.04.2001)

Authors’ addresses:

A. Gachechiladze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia D. Natroshvili Department of Mathematics Georgian Technical University 77, Kostava St., Tbilisi 380075 Georgia E-mail: natrosh@hotmail.com