GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 2, 1995, 123-140

BASIC BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY FOR ANISOTROPIC BODIES

WITH CUTS. I

R. DUDUCHAVA, D. NATROSHVILI, AND E. SHARGORODSKY

Abstract. The three-dimensional problems of the mathematical theory of thermoelasticity are considered for homogeneous anisotropic bodies with cuts. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems of statics and pseudo-oscillations are proved in the Besov (𝔹_{p,q}^{s}) and Bessel-potential (ℍ_{p,q}^{s}) spaces by means of the classical potential methods and the theory of pseudodifferential equations on manifolds with boundary. Using the embedding theorems, it is proved that the solutions of the considered problems are Hölder continuous. It is shown that the displacement vector and the temperature distribution function are C^{α}-regular with any exponent α < 1/2.

This paper consists of two parts. In this part all the principal results are formulated. The forthcoming second part will deal with the auxiliary results and proofs.

Introduction

Three-dimensional crack problems evoke much interest in engineering applications. In this paper we investigate the three-dimensional boundary value problems (BVPs) of thermoelasticity in certain function spaces when the anisotropic elastic body under consideration contains any number of nonintersecting cuts in the form of two-dimensional smooth surfaces with smooth boundaries.

For domains bounded by smooth closed manifolds of the class C^{2+γ} the basic BVPs were completely investigated using the potential method by V. D. Kupradze and his collaborators [1] in the isotropic case and by D. Natroshvili [2] in the anisotropic case.

1991 Mathematics Subject Classification. 35C15, 35S15, 73M25. Key words and phrases. Thermoelasticity, anisotropic bodies, cuts, potentials, pseu-

dodifferential equations, boundary integral equations..

123

1072-947X/95/0300-0123$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Analogous three-dimensional crack problems of classical elasticity for isotropic bodies were treated in the Bessel-potential spaces ℍ_{2}^{s} by M. Costabel and E. Stephan in [3]. The BVPs for homogeneous anisotropic bodies were studied in the Besov (𝔹_{p,q}^{s}) and Bessel-potential (ℍ_{p}^{s}) spaces by R. Duduchava and collaborators in [4] who established more precise results on the regularity of solutions (C^{α}-regularity with α < 1/2). Each of the quoted papers contains an ample bibliography to the above-mentioned problems.

To illustrate our approach we consider two basic BVPs for an infinite domain. All the results obtained here remain valid (with minor modifications) for a bounded domain with interior cuts, i.e., when the cut surface does not touch the domain boundary.

This paper consists of two parts. The first part contains three sections. In the first section we formulate the problems and introduce the spaces

of functions and distributions needed for proving the unique solvability of the problems and, further on, the regularity properties of solutions.

In the second section we show the mapping properties of single- and double-layer potentials of thermoelasticity (both on the surface and from the surface to the space; see Theorems 2 and 4) and derive the integral representations of regular solutions.

The third section contains the formulations of the main theorems of the paper, concerning the existence and uniqueness of solutions of the problems discussed in the first section, the regularity of such solutions, and the explicit solvability properties of the corresponding boundary integral equations (see Theorems 5–9).

The proofs of Theorems 7 and 8 will be given in the forthcoming second part of the paper after recalling some auxiliary results.

§ 1. Formulation of the Problems

Let Ω^{+} be a bounded domain in ℝ^{3} with the smooth boundary ∂Ω^{+} = Σ and S be the connected part of Σ with the smooth boundary curve ∂S = ℓ ≠ ∅. Then S is a two-dimensional surface of an arbitrary configuration with the boundary ℓ. It is assumed that Ω^{−} := ℝ^{3}\\overline{Ω}^{+}, where \overline{Ω}^{+} = Ω^{+}∪Σ, ℝ_{S}^{3} = ℝ\\overline{S} and \overline{S} = S ∪ ℓ.

Let ℝ_{S}^{3} be filled with some homogeneous anisotropic elastic material having density ρ, elastic coefficients

c_{kjpq} = c_{pqkj} = c_{jkpq},  (1.1)

heat conductivity coefficients

λ_{ij} = λ_{ji}  (1.2)

and thermal capacity c_{0}.

BASIC BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY 125

In what follows ℝ_{S}^{3} is treated as an infinite elastic body with a cut along the surface \overline{S}. For simplicity Σ, S and ℓ will further be assumed to be C^{∞}-regular.

By u = (u_{1}, u_{2}, u_{3}) and u_{4} we denote the displacement vector field and the temperature field, respectively. The components of the thermal stress vector calculated on a surface element with the unit normal vector n = (n_{1}, n_{2}, n_{3}) have the form

[P(D_{x}, n)U]_{k} := [T(D_{x}, n)u]_{k} − β_{kj}n_{j}u_{4}, k = 1, 2, 3,

where U = (u_{1}, u_{2}, u_{3}, u_{4}), [T(D_{x}, n)u]_{k} := c_{kjpq}n_{j}D_{q}u_{p} are the components of the classical stress vector, D_{x} := (D_{1}, D_{2}, D_{3}), D_{p} = ∂/∂x_{p}, and the constants β_{ij} = β_{ji} are expressed in terms of the thermal and the elastic constants (cf. [5]). Here and in what follows, summation from 1 to 3 over repeated indices is meant.

The strain tensor components e_{kj} are defined by the formulas

e_{kj} = \frac{1}{2}(D_{j}u_{k} + D_{k}u_{j}), k, j = 1, 2, 3,

while the stress tensor components are related to e_{kj} as follows (Hooke’s law):

τ_{kj} = c_{kjpq}e_{pq} = c_{kjpq}D_{p}u_{q}.

Potential energy in classical elasticity reads

2W = e_{kj}τ_{kj} = c_{kjpq}e_{kj}e_{pq}.

From the physical standpoint, potential energy is assumed to be a positive definite quadratic form with respect to the variables e_{kj} = e_{jk}:

2W ≥ δe_{kj}e_{kj}, δ = const > 0.  (1.3)

Combining the static and the pseudo-oscillation cases, we consider the following system of equations of thermoelasticity:

A(D_{x}, τ)U(x) = F (x, τ), x ∈ ℝ_{S},  (1.4)

where F = (F_{1}, …, F_{4}) is a given vector with a compact support, I_{m} = ∥δ_{kj}∥_{m×m} is the unit matrix,

A(D, τ)=∥\begin{matrix}C(D) − ρτ^{2}I_{3} & \begin{matrix}−β_{1j}D_{j} \\ −β_{2j}D_{j} \\ −β_{3j}D_{j} \end{matrix} \\ α_{1j}D_{j}, α_{2j}D_{j}, α_{3j}D_{j}, & Λ(D) − c_{0}τ \end{matrix}∥_{4×4} ,  (1.5)

C(D) := ∥C_{kp}(D)∥_{3×3}, C_{kp}(D) := c_{kjpq}D_{j}D_{q},
Λ(D) := λ_{pq}D_{p}D_{q}, α_{kj} = −τT_{0}β_{kj},  (1.6)

T_{0} = const > 0 is a temperature of the medium in the natural state (cf. [5]), τ = σ + iω; τ = 0 corresponds to the static case, while τ = σ + iω, σ ≥ σ_{0} > 0 corresponds to the pseudo-oscillation case (equation (1.4) is obtained from the dynamic equations of thermoelasticity upon applying the Laplace transform).

Treating S as a double-sided surface, we consider two basic BVPs for equation (1.4) with a Dirichlet type boundary condition

[U(x)]^{±} = φ^{±}(x, τ), x ∈ S,  (1.7)

and with a Neumann type boundary condition

[B(D_{x}, n(x))U(x)]^{±} = ψ^{±}(x, τ), x ∈ S,  (1.8)

where the symbol [·]^{±} denotes the limiting value on S of a function (vector) from Ω^{±}, n(x) is the unit normal at x ∈ S outward with respect to Ω^{+}, φ^{±} = (φ_{1}^{±}, …, φ_{4}^{±}) and ψ^{±} = (ψ_{1}^{±}, …, ψ_{4}^{±}) are the known vector-functions,

B(D_{x}, n(x)) = ∥\begin{matrix}T(D_{x}, n(x)) & \begin{matrix}−β_{1j}n_{j}(x) \\ −β_{2j}n_{j}(x) \\ −β_{3j}n_{j}(x) \end{matrix} \\ 0, 0, 0, & λ_{pq}n_{p}(x)D_{q} \end{matrix}∥_{4×4} ,  (1.9)

T(D_{x}, n(x)) := ∥T_{kp}(D_{x}, n(x))∥_{3×3}, T_{kp}(D_{x}, n(x)) := c_{kjpq}n_{j}(x)D_{q}.

In problems (1.7) and (1.8) it is required that

u_{k}(x) = \begin{cases}o(1) & for τ = 0, \\ O(|x|^{N}) & for Re τ > 0, \end{cases} k = 1, 2, 3, 4,  (1.10)

for a sufficiently large |x| and some positive number N.

These conditions imply (see [6], [7])

D^{α}u_{k}(x) = \begin{cases}O(|x|^{−1−|α|}) & for τ = 0, \\ O(|x|^{−ν}) & for Re τ > 0, \end{cases}  (1.11)

as |x| → ∞, where α is an arbitrary multi-index, |α| = α_{1} + α_{2} + α_{3}, and ν is an arbitrary positive number.

The symmetry properties of coefficients (1.1) and the positive definiteness of the energy quadratic form (1.3) imply that the operator C(D) defined by (1.6) is a formally self-adjoint strongly elliptic matrix differential operator (see [8])

\begin{matrix}Re (C(ξ)η, η) = (C(ξ)η, η) = c_{kjpq}ξ_{j}ξ_{q}η_{p}\overline{η}_{k} ≥ δ_{0}|ξ|^{2}|η|^{2}, \\ δ_{0} = const > 0, ξ ∈ ℝ^{3}, η ∈ ℂ^{3}, \end{matrix} (1.12)

where ℂ^{3} is the three-dimensional complex Euclidean space, the bar designates complex conjugation, and (a, b) = ab = a_{k}\overline{b}_{k} for a, b ∈ ℂ^{3}.

In contrast to C(D) the operator A(D, τ) is elliptic but not self-adjoint. Denote by A^{∗}(D, τ) the operator formally adjoint to A(D, τ). It is obvious that

A^{∗}(D, τ) = A^{T} (−D, \overline{τ}),  (1.13)

where the superscript T denotes the transposition operator.

Note that the quadratic form Λ(ξ) defined by (1.6) is also positive definite (see [5]),

Λ(ξ) = λ_{pq}ξ_{p}ξ_{q} ≥ δ_{1}|ξ|^{2}, ξ ∈ ℝ^{3}, δ_{1} = const > 0.  (1.14)

A function f : \overline{Ω}^{±} → ℝ^{1} is said to be regular in Ω^{±} if f ∈ C^{2}(Ω^{±}) ∩ C^{1}(\overline{Ω}^{±}). A vector v = (v_{1}, …, v_{m}) (matrix Ã = ∥ã_{kj}∥_{m×m}) is said to be regular in Ω^{±} if all its components (entries) are regular functions in Ω^{±}. In general, v ∈ P (Ã ∈ P) means that all components of v (all entries of Ã) belong to the space P.

Let C^{k+γ}(\overline{Ω}^{±}), where k ≥ 0 is an integer and 0 < γ < 1, denote the space of functions u defined on \overline{Ω}^{±} whose derivatives D^{α}u of order |α| = k are Hölder continuous with the exponent γ. The space C^{k+γ}(Σ) is defined similarly (cf., for example, [1]).

Assume that U = (u_{1}, …, u_{4}) and V = (v_{1}, …, v_{4}) are the regular vectors in Ω^{±} satisfying the conditions AU, A^{∗}V ∈ 𝕃_{1}(Ω^{±}) and (1.11). Then we have (see [2])

∫_{Ω^{±}} (AU, V) dx = ± ∫_{∂Ω^{±}}([BU]^{±}, [V]^{±}) dS −∫_{Ω^{±}}E(U, V) dx,  (1.15)

∫_{Ω^{±}} {(AU, V) − (U, A^{∗}V)}dx = ± ∫_{∂Ω^{±}} {([BU]^{±}, [V]^{±})dS −

− ([U]^{±}, [\overline{Q}V]^{±})\}dS,  (1.16)

∫_{Ω^{±}} \{(AU)_{k}\overline{U}_{k} + \frac{1}{\overline{τ}T_{0}} (\overline{AU})_{4}U_{4}\}dx = − ∫_{Ω^{±}} \{c_{ijkl}D_{l}u_{k}D_{j}\overline{u}_{i} +

+ ρτ^{2}u_{k}\overline{u}_{k} + \frac{1}{\overline{τ}T_{0}} λ_{ij}D_{j}u_{4}D_{i}\overline{u}_{4} + \frac{c_{0}}{T_{0}} u_{4}\overline{u}_{4}\}dx ±

± ∫_{∂Ω^{±}} \{[BU]_{k}^{±}[U_{k}]^{±} + \frac{1}{\overline{τ}T_{0}} [U_{4}]^{±}[\frac{∂}{∂ν} \overline{U}_{4}]^{±}\}dS,  (1.17)

where

E(U, V) = c_{kjpq}D_{q}u_{p}D_{j}\overline{v}_{k} + ρτ^{2}u_{k}\overline{v}_{k} − β_{kp}u_{4}D_{q}\overline{v}_{k} +

+ λ_{pq}D_{q}u_{4}D_{p}\overline{v}_{4} + c_{0}τu_{4}\overline{v}_{4} + τT_{0}β_{pq}D_{p}u_{p}\overline{v}_{4},  (1.18)

\frac{∂}{∂ν(x)} u_{4} = λ_{pq}n_{p}(x)D_{q}u_{4}(x),

Q(D_{x}, n(x)) := ∥\begin{matrix}T(D_{x}, n(x)) & \begin{matrix}T_{0}τβ_{1j}n_{j}(x) \\ T_{0}τβ_{2j}n_{j}(x) \\ T_{0}τβ_{3j}n_{j}(x) \end{matrix} \\ 0, 0, 0, & λ_{pq}n_{p}(x)D_{q} \end{matrix}∥_{4×4} .

In what follows the BVPs (1.7) and (1.8) will be investigated in different functional spaces.

To formulate these problems in exact terms we need the Sobolev 𝕎_{p}^{k}(ℝ^{3}), 𝕎_{p}^{k}(Ω), 𝕎_{p}^{k}(Σ), Sobolev–Slobodecky 𝕎_{p}^{s}(ℝ^{3}), 𝕎_{p}^{s}(Ω), 𝕎_{p}^{s}(Σ), the Besselpotential ℍ_{p}^{s}(ℝ^{3}), ℍ_{p}^{s}(Ω), ℍ_{p}^{s}(Σ), and the Besov 𝔹_{p,q}^{s}(ℝ^{3}), 𝔹_{p,q}^{s}(Ω), 𝔹_{p,q}^{s}(Σ) spaces (k = 0, 1, 2, …, −∞ < s < ∞, 1 < p < ∞, 1 ≤ q ≤ ∞). For the definitions of these spaces see [9].

Let 𝕏(ℝ^{3}) be one of the above-mentioned function spaces. For an arbitrary unbounded domain Ω^{−} ⊂ ℝ^{3} (with a smooth boundary ∂Ω^{−}) we denote by 𝕏_{loc}(Ω^{−}) the subset of distributions φ ∈ D′(Ω^{−}) with

φ|_{Ω_{R}^{−}} ∈ 𝕏(Ω_{R}^{−}), Ω_{R}^{−} = {x ∈ Ω : |x| < R}, ∀R > 0,

and by 𝕏_{comp}(Ω^{−}) the set of functions φ ∈ 𝕏(Ω^{−}) with compact supports.

𝕏_{loc}(ℝ_{S}^{3}) denotes the subset of distributions φ ∈ D′(ℝ_{S}^{3}) satisfying the conditions

φ|_{Ω^{+}} ∈ 𝕏(Ω^{+}), φ|_{Ω_{R}} ∈ 𝕏(Ω_{R}), Ω_{R} = \{x ∈ ℝ^{3}\\overline{Ω}^{+}, |x| ≤ R\},

for an arbitrary R > 0 and any bounded domain Ω^{+} with S ⊂ ∂Ω^{+}; here φ|_{Ω} is the restriction to Ω.

In particular, 𝕎_{p.loc}^{1}(ℝ_{S}^{3}) denotes the Sobolev space of functions φ on ℝ_{S}^{3} which are p-integrable on Ω\S for each compact domain Ω ⊂ ℝ^{3} together with their generalized derivatives of order 1

∥φ; 𝕎_{p}^{1}(Ω\S)∥ = { ∫_{Ω\S} (|φ(x)|^{p} + |∇φ(x)|^{p})dx}^{1/p} < ∞,

∇φ := (D_{1}φ, D_{2}φ, D_{3}φ).

For the open surface S ⊂ Σ the spaces ℍ_{p}^{t}(S), ℍ̃_{p}^{t}(S) are defined as

ℍ_{p}^{t}(S) = {r_{S}f : f ∈ ℍ_{p}^{t}(Σ)},
ℍ̃_{p}^{t}(S) = \{f ∈ ℍ_{p}^{t}(Σ) : supp f ⊂ \overline{S}\} ⊂ ℍ_{p}^{t}(Σ),

where r_{S}f = f|_{S} is the restriction.

The spaces 𝔹_{p,q}^{t}(S) and 𝔹̃_{p,q}^{t}(S) are defined similarly. Note that ℍ_{2}^{s} = 𝕎_{2}^{s} = 𝔹_{2,2}^{s}, 𝕎_{p}^{t} = 𝔹_{p,p}^{t}, and ℍ_{p}^{k} = 𝕎_{p}^{k} hold for any −∞ < s < ∞, for any positive and non-integer t, and for any non-negative integer k = 0, 1, 2, …, respectively.

In contrast to closed surfaces, even for infinitely smooth S, ℓ, φ^{±}, and ψ^{±} the solutions of problems {(1.4), F = 0, (1.7)} and {(1.4), F = 0, (1.8)} have in general no C^{α}-smoothness with α > 1/2 in the vicinity of ℓ = ∂S but are infinitely differentiable elsewhere.

Hence we seek solutions of the BVPs (1.7) and (1.8) from the Sobolev space 𝕎_{p,loc}^{1}(ℝ_{S}^{3}) provided that

φ^{±} ∈ 𝔹_{p,p}^{1−1/p}(S), φ^{0} = φ^{+} − φ^{−} ∈ 𝔹̃_{p,p}^{1−1/p}(S)  (1.19)

for the Dirichlet type problem (1.7) and

ψ^{±} ∈ 𝔹_{p,p}^{−1/p}(S), ψ^{0} = ψ^{+} − ψ^{−} ∈ 𝔹̃_{p,p}^{−1/p}(S)  (1.20)

for the Neumann type problem (1.8).

Further the problems {(1.4), F = 0, (1.7), (1.10), (1.19)} and {(1.4), F = 0, (1.8), (1.10), (1.20)} will be referred to as Problem D and Problem N, respectively. Note that property (1.11) holds for solutions of Problems D and N as well.

For s > 1/p by the trace theorem (see [9], Theorem 3.3.3) we have

\begin{matrix}g^{±} ∈ 𝔹_{p,p}^{s−1/p}(S) if g ∈ ℍ_{p,loc}^{s}(ℝ_{S}^{3}) ∪ 𝕎_{p,loc}^{s}(ℝ_{S}^{3}), \\ g^{±} ∈ 𝔹_{p,q}^{s−1/p}(S) if g ∈ 𝔹_{p,q,loc}^{s}(ℝ_{S}^{3}). \end{matrix}  (1.21)

Therefore (1.7) and (1.19) are compatible and correctly defined if U ∈ 𝕎_{p,loc}^{1}(ℝ_{S}^{3}).

As to (1.8), (1.20), we should give some additional explanation, since D_{j}U_{k} ∈ 𝕃_{p,loc}(ℝ_{S}^{3}) = 𝕎_{p,loc}^{0}(ℝ_{S}^{3}) and they have no traces on S in general.

We can make condition (1.8) meaningful for U ∈ 𝕎_{p,loc}^{1}(ℝ_{S}^{3}) with AU ∈ 𝕃_{p,loc}(ℝ_{S}^{3}) using equality (1.15). Indeed, it can be rewritten as

⟨[B(D_{x}, n(x))U]^{+}, V^{+}⟩_{Σ} =
∫_{Ω^{+}} (A(D)U, V )dx + ∫_{Ω^{+}} E(U, V) dx,  (1.22)

for U ∈ 𝕎_{p}^{1}(Ω^{+}), AU ∈ 𝕃_{p}(Ω^{+}), ∀V ∈ 𝕎_{p′}^{1}(Ω^{+});

⟨[B(D_{x}, n(x))U]^{−}, V^{−}⟩_{Σ} =−∫_{Ω^{−}}(A(D)U, V )dx− ∫_{Ω^{−}}E(U, V)dx, (1.23)

for U ∈𝕎_{p,loc}^{1}(Ω^{−}), AU ∈𝕃_{p}(Ω^{−}), ∀V ∈𝕎_{p′,comp}^{1}(Ω^{−}). Here p′ = p/(p − 1)

and ⟨·, ·⟩_{Σ} defines the duality between 𝔹_{p,p}^{−1/p}(Σ) and 𝔹_{p′,p′}^{1/p}(Σ) given by

⟨f, g⟩_{Σ} = ∫_{Σ} fg dS

for the smooth functions f and g.

Relations (1.22) and (1.23) define [B(D, n(x))U]^{±} ∈ 𝔹_{p,p}^{−1/p}(S) correctly, since by virtue of (1.21) their right-hand side expressions exist for any V ∈ 𝕎_{p′}^{1}(Ω^{+}) and V ∈ 𝕎_{p′,comp}^{1}(Ω^{−}), respectively, and V^{±} ∈ 𝔹_{p′,p′}^{1/p}(Σ).

§ 2. Properties of Fundamental Solutions and Potentials

By A(ξ, τ) denote the symbol matrix of the operator A(D_{x}, τ) (see (1.5)). Obviously,

A(ξ, τ) = A(−iξ, τ)

and the matrix distribution

Φ(x, τ) := (2π)^{−3} ∫_{ℝ^{3}} e^{−iξτ} A^{−1}(ξ, τ) dξ  (2.1)

represents the fundamental matrix of the operator A(D_{x}, τ), i.e.,

A(D_{x}, τ)Φ(x, τ) = δ(x)I_{4}

where δ(·) is the Dirac distribution. The fundamental matrix of the formally adjoint operator reads

Φ^{∗}(x, τ) = Φ^{T} (−x, τ).  (2.2)

The entries of these matrices are of the class C^{∞}(ℝ^{3}\{0}) and for Re τ > 0 they, together with all their derivatives, decay faster than any negative power of |x| at infinity. For τ = 0 we have

D^{α}Φ_{kj}(x, 0) = O(|x|^{−1−|α|}) as |x| → ∞

(see [2]).

Near the origin the main singular parts of the matrices Φ(x, τ) and Φ^{∗}(x, τ) coincide and have the form (see [2], [10])

Φ(x) := ∥\begin{matrix}Γ(x) & 0 0 0 \\ 0 0 0 & γ(x) \end{matrix}∥_{4×4} ,  (2.3)

where Γ(·) is the fundamental matrix of the classical operator C(D) while γ(·) is the fundamental function of the operator Λ(D)

\begin{aligned}Γ(x) & = (2π)^{−3} ∫_{ℝ^{3}} e^{ixξ}C^{−1}(ξ) dξ = (8π^{2}|x|)^{−1} ∫_{0}^{2π}C^{−1}(aη̃) dφ, \\ C(ξ) & = ∥c_{kjpq}(−iξ_{j})(−iξ_{q})∥_{3×3}, \\ γ(x) & = (2π)^{−3}∫_{ℝ^{3}} e^{ixξ}Λ^{−1}(−iξ) dξ =−\{4π|L|^{1/2}(L^{−1}x, x)^{1/2}\}^{−1}, \\ Λ(ξ) & = λ_{pq}ξ_{q}ξ_{p} \end{aligned} (2.4)

(see [11]) with η̃ = (cos φ, sin φ, 0), L = ∥λ_{pq}∥_{3×3}, |L| = det L; here a = ∥a_{jk}∥_{3×3} is an orthogonal matrix with the property a^{T} x = (0, 0, |x|).

It is evident that the equalities

Φ(tx) = t^{−1}Φ(x), Φ(x) = Φ^{T} (x) = Φ(−x)  (2.5)

hold for any positive t > 0. Near the origin Φ_{kj}(x, τ) has the asymptotics

D^{α}[Φ_{kj}(x, τ) − Φ_{kj}(x)] = \begin{cases}O(ln |x|), & if α = 0, \\ O(|x|^{−|α|}), & if |α| > 0. \end{cases}  (2.6)

The properties of generalized potentials corresponding to these matrices in the case of closed surfaces were studied in [2], [10]. Due to these results from now on we shall assume without loss of generality that F = 0 in (1.4), as the particular solution of (1.4) can be written explicitly using the generalized Newtonian potentials (see, for example, [12]).

On account of (1.16) we obtain the following integral representation of a regular vector:

∫_{Ω^{ω}} Φ(x − y, τ)A(D_{y}, τ)U(y) dy ± ∫_{∂Ω^{±}} {[Q(D_{y}, n(y))Φ^{T} (x − y, τ)]^{T} ×

×[U(y)]^{±} − Φ(x − y, τ)[B(D_{y}, n(y))U(y)]^{±}}d_{y}S =
= \begin{cases}U(x), & x ∈ Ω^{±}, \\ 0, & x ∈ Ω^{∓}. \end{cases}  (2.7)

We introduce the generalized single- and double-layer potentials

P_{Σ}^{1}g(x) := ∫_{Σ} Φ(x − y, τ)g(y) d_{y}S,  (2.8)

P_{Σ}^{2}g(x) := ∫_{Σ} [Q(D_{y}, n(y))Φ^{T} (x − y, τ)]^{T} g(y) d_{y}S,  (2.9)
x ∈ ℝ_{Σ}^{3} = ℝ^{3}\Σ,

which are the solutions of the homogeneous equation (1.4), i.e., for F = 0. The same notation will be used for the direct values of P_{Σ}^{j}g(x), x ∈ Σ (j = 1, 2). Note in this connection that for x ∈ Σ integral (2.9) exists only in the sense of the Cauchy principal value, while (2.8) exists as a usual improper integral (see [2]). In a similar manner we define

(P_{Σ}^{3}g)(x) := ∫_{Σ} B(D_{x}, n(x))Φ(x − y, τ)g(y) d_{y}S  (2.10)

for x ∈ ℝ_{Σ}^{3} and x ∈ Σ; here n(x) is the C_{0}^{∞}-extension of the exterior unit normal vector from Σ onto ℝ^{3}.

Lemma 1. The equalities

[P_{Σ}^{2}g]^{±}(x) = ±\frac{1}{2}g(x) + P_{Σ}^{2}g(x), k ≥ 0,  (2.11)
[B(D_{x}, n(x))P_{Σ}^{1}g]^{±}(x) = ∓\frac{1}{2}g + P_{Σ}^{3}g(x), k ≥ 0,  (2.12)
[P_{Σ}^{1}g]^{+}(x) = [P_{Σ}^{1}g]^{−}(x) = P_{Σ}^{1}g(x), k ≥ 0,  (2.13)
[B(D_{x}, n(x))P_{Σ}^{2}g]^{+}(x) = [B(D_{x}, n(x))P_{Σ}^{2}g]^{−}(x) ≡
= P_{Σ}^{4}g(x), k ≥ 1,  (2.14)

are fulfilled for any g ∈ C^{k+γ}(Σ), 0 < γ < 1 and x ∈ Σ.

The operators

P_{Σ}^{j} : C^{k+γ}(Σ) → C^{k+2−j+γ}(\overline{Ω}^{±}), j = 1, 2,  (2.15)
P_{Σ}^{j} : C^{k+γ}(Σ) → C^{k+2−j+γ}(Σ), j = 1, 2,  (2.16)
P_{Σ}^{3} : C^{k+γ}(Σ) → C^{k+γ}(Σ),  (2.17)
P_{Σ}^{4} : C^{k+1+γ}(Σ) → C^{k+γ}(Σ)  (2.18)

are bounded for 0 < γ < 1 and any integer k ≥ 0.

We retain notation of type (2.8)–(2.10) and (2.14) for the potentials and the corresponding operators when the closed surface Σ is replaced by the open surface S. The potentials P^{j} possess essentially different properties and require an especially careful approach.

Theorem 2. Let s ∈ ℝ, 1 < p < ∞, 1 ≤ q ≤ ∞. P_{Σ}^{1}, P_{Σ}^{2}, P_{Σ}^{3} and P_{Σ}^{4} are the pseudodifferential operators of orders −1, 0, 0 and 1, respectively.

The following operators are bounded (cf. (2.15)–(2.18)):

P_{Σ}^{1} : ℍ_{p}^{s}(Σ) → ℍ_{p}^{s+1}(Σ),

P_{Σ}^{1} : 𝔹_{p,q}^{s}(Σ) → 𝔹_{p,q}^{s+1}(Σ), (2.19)

P_{Σ}^{2}, P_{Σ}^{3} : ℍ_{p}^{s}(Σ) → ℍ_{p}^{s}(Σ),

P_{Σ}^{2}, P_{Σ}^{3} : 𝔹_{p,q}^{s}(Σ) → 𝔹_{p,q}^{s}(Σ), (2.20)

P_{Σ}^{4} : ℍ_{p}^{s+1}(Σ) → ℍ_{p}^{s}(Σ),

P_{Σ}^{4} : 𝔹_{p,q}^{s+1}(Σ) → 𝔹_{p,q}^{s}(Σ). (2.21)

Proof. The first claim of the theorem is proved in [13], while the other follows from the well-known properties of the boundedness of pseudodifferential operators (see, for example, [14]).

Let Σ_{0} be an m-dimensional C^{∞}-smooth compact manifold without boundary and embedded in ℝ^{n} (n ≥ m). Consider the distribution v×δ_{Σ_{0}} ∈ 𝕊′(ℝ^{n}) defined by the formula

⟨v × δ_{Σ_{0}}, φ⟩ := ⟨v, φ|_{Σ_{0}}⟩, φ ∈ 𝕊(ℝ^{n}),  (2.22)

for any v ∈ ℍ_{p}^{s}(Σ_{0}) (v ∈ 𝔹_{p,q}^{s}(Σ_{0}), 1 < p < ∞, 1 ≤ q ≤ ∞).

The above definition is correct, since the restriction φ|_{Σ_{0}} ∈ C^{∞}(Σ_{0}).

Lemma 3. Let v ∈ 𝔹_{p,p}^{s}(Σ_{0}), (v ∈ 𝔹_{p,q}^{s}(Σ_{0})), 1 < p < ∞, 1 ≤ q ≤ ∞,

s < 0. Then v×δ_{Σ_{0}} ∈ ℍ_{p}^{s− \frac{n−m}{p′}} (ℝ^{n}) (v×δ_{Σ_{0}} ∈ 𝔹_{p,q}^{s− \frac{n−m}{p′}} (ℝ^{n})), p′ = p/(p−1).

Proof. Applying the trace theorem (see [9], Theorem 3.3.3), we conclude

that any function ψ from ℍ_{p′}^{−s+ \frac{n−m}{p′}} (ℝ^{n}) (from 𝔹_{p′,q′}^{−s+ \frac{n−m}{p′}} (ℝ^{n})) has the trace ψ|_{Σ_{0}} ∈ 𝔹_{p′,p′}^{−s} (ψ|_{Σ_{0}} ∈ 𝔹_{p′,q′}^{−s}(Σ_{0})). Hence, by virtue of definition (2.22), v ×

δ_{Σ_{0}} represents a bounded functional on ℍ_{p′}^{−s+ \frac{n−m}{p′}} (ℝ^{n}) (on 𝔹_{p′,q′}^{−s+ \frac{n−m}{p′}} (ℝ^{n}), 1 < q < ∞) and by the duality property (see [9], Theorem 2.11.1) we get the proof for 1 < q < ∞.

For q = 1, ∞ the proof is accomplished by interpolation (see [9], Theorem 3.3.6).

Theorem 4. Let s ∈ ℝ, 1 < p < ∞, 1 ≤ q ≤ ∞, j = 1, 2. Then the operators

P_{Σ}^{j} : 𝔹_{p,p}^{s}(Σ) → ℍ_{p}^{s+2−j+1/p}(Ω^{+}) ∩ C^{∞}(Ω^{+}),
P_{Σ}^{j} : 𝔹_{p,q}^{s}(Σ) → 𝔹_{p,q}^{s+2−j+1/p}(Ω^{+}) ∩ C^{∞}(Ω^{+}),

(2.23)

P_{Σ}^{j} : 𝔹_{p,p}^{s}(Σ) → ℍ_{p,loc}^{s+2−j+1/p}(Ω^{−}) ∩ C^{∞}(Ω^{−}),
P_{Σ}^{j} : 𝔹_{p,q}^{s}(Σ) → 𝔹_{p,q,loc}^{s+2−j+1/p}(Ω^{−}) ∩ C^{∞}(Ω^{−})

(2.24)

are bounded and for these extended operators formulas (2.11)–(2.14) remain valid in the corresponding spaces.

Representation (2.7) holds for U ∈ 𝕎_{p,loc}^{1}(Ω^{±}) if, in addition, A(D_{x}, τ)U = 0 in Ω^{±}.

Proof. Let us first consider P_{Σ}^{1} and s < 0. Assume that φ ∈ C_{0}^{∞}(ℝ^{3}), φ(ξ) = 1 for |ξ| ≤ 1. If g ∈ 𝔹_{p,p}^{s}(Σ) (g ∈ 𝔹_{p,q}^{s}(Σ)), we have

P_{Σ}^{1}g = Φ ∗ (g × δ_{Σ}) = −F^{−1}A^{−1}F(g × δ_{Σ}) =
= −F^{−1}A^{−1}(ξ)[1 − φ(ξ)]F(g × δ_{Σ}) −
− F^{−1}A^{−1}(ξ)φ(ξ)F(g × δ_{Σ}) ≡ P_{Σ,1}^{1}g + P_{Σ,2}^{1}g,  (2.25)

where F (F^{−1}) is the direct (inverse) Fourier transform.

From Lemma 3 it follows that g×δ_{Σ} ∈ ℍ_{p}^{s−1/p′}(ℝ^{3}) (g×δ_{Σ} ∈ 𝔹_{p,q}^{s−1/p′}(ℝ^{3})). Applying the theorem on the boundedness of pseudodifferential operators (see [14]), we obtain

P_{Σ,1}^{1}g ∈ ℍ_{p}^{s+1+1/p}(ℝ^{3}) (P_{Σ,1}^{1}g ∈ 𝔹_{p,q}^{s+1+1/p}(ℝ^{3})).  (2.26)

For the second summand in (2.25) we have

P_{Σ,2}^{1}g = −F^{−1}A^{−1}(ξ)F[F^{−1}(φ(ξ)F(g × δ_{Σ}))] ≡
≡ −F^{−1}A^{−1}Ff = Φ ∗ f with f = F^{−1}φF(g × δ_{Σ}).  (2.27)

Since the pseudodifferential operator F^{−1}φF is of order −∞, we have f ∈ C^{∞}(ℝ^{3}) and therefore (see (2.27))

A(D_{x})P_{Σ,2}^{1}g = A(D_{x})(Φ ∗ f) = (A(D_{x})Φ) ∗ f = δ ∗ f = f. (2.28)

Thus P_{Σ,2}^{1}g is the solution of an elliptic system with an infinitely smooth right-hand side. Therefore P_{Σ,2}^{1}g ∈ C^{∞}(ℝ^{3}) (see, for example, [15], Chapter I, Corollary 4.1 or [16], Chapter III, Theorem 1.4).

For any g ∈ 𝔹_{p,p}^{s}(Σ) (g ∈ 𝔹_{p,q}^{s}(Σ) and any compact domain Ω ⊂ ℝ^{3} we obtain P_{Σ}^{1}g ∈ ℍ_{p}^{s+1+1/p}(Ω) (P_{Σ}^{1}g ∈ 𝔹_{p,q}^{s+1+1/p}(Ω)).

It should be noted that the convergence g_{n} ⟶^{𝔹_{p,q}^{s}(Σ)} g as n → ∞ implies

the convergence P_{Σ}^{1}g_{n} ⟶^{D′(Ω)} P_{Σ}^{1}g (see [17, §7]). Therefore the graph of the operator P_{Σ}^{1} : 𝔹_{p,p}^{s}(Σ) → ℍ_{p}^{s+1+1/p}(Ω) (P_{Σ}^{1} : 𝔹_{p,p}^{s}(Σ) → 𝔹_{p,q}^{s+1+1/p}(Ω)) is closed. It remains for us to apply the closed graph theorem (see, for example, [18, Theorem 2.15]).

Let us proceed to the case s ≥ 0. Assume that s = m+1/p′, m = 0, 1, . . .. For a function g ∈ 𝔹_{p,p}^{s}(Σ) we choose a sequence {g_{n}}_{n∈ℕ} ⊂ 𝔹_{p,p}^{s}(Σ) ∩ C^{1+ε}(Σ), ε > 0, such that lim_{n→∞} ∥(g_{n} − g)|𝔹_{p,p}^{s}(Σ)∥ = 0.

By Lemma 1 we have P_{Σ}^{1}g_{n} ∈ C^{2+ε}(\overline{Ω}) ⊂ 𝕎_{p}^{2}(Ω) representing the solution of the boundary value problem

A(D_{x})U(x) = 0, x ∈ Ω,  (2.29)
U|_{Σ} = P_{Σ}^{1}g_{n}|_{Σ},  (2.30)

where

P_{Σ}^{1}g_{n}|_{Σ} ∈ C^{2+ε}(Σ).

By Theorem 2 (see (2.19))

∥P_{Σ}^{1}g_{n}|𝔹_{p,p}^{s+1}(Σ)∥ ≤ C∥g_{n}|𝔹_{p,p}^{s}(Σ)∥,  (2.31)

where C = const is independent of g_{n}. Using the a priori estimates (see [19], [20] or [21], Chapter V), we obtain P_{Σ}^{1}g_{n} ∈ 𝕎_{p}^{m+2}(Ω) = ℍ_{p}^{s+1+1/p}(Ω) and

∥P_{Σ}^{1}g_{n}|ℍ_{p}^{s+1+1/p}(Ω)∥ ≤ C_{1}(∥P_{Σ}^{1}g_{n}|_{Σ}|𝔹_{p,p}^{s+1}(Σ)∥ + ∥P_{Σ}^{1}g_{n}|𝕃_{p}(Ω)∥).

The results proved above for s < 0 and the embedding theorem yield

∥P_{Σ}^{1}g_{n}|𝕃_{p}(Ω)∥ ≤ C_{2}∥g_{n}|𝔹_{p,q}^{s}(Σ)∥,

since g_{n} ∈ 𝔹_{p,q}^{−ε}(Σ) implies P_{Σ}^{1}g_{n} ∈ 𝔹_{p,q}^{−ε+1+1/p}(Ω) ⊂ 𝕃_{p}(Ω).

Taking into account (2.31), we arrive at

∥P_{Σ}^{1}g_{n}|ℍ_{p}^{s+1+1/p}(Ω)∥ ≤ C_{2}∥g_{n}|𝔹_{p,p}^{s}(Σ)∥.  (2.32)

A similar inequality holds for g_{n}−g_{m}. Therefore {P_{Σ}^{1}g_{n}}_{n∈ℕ} represents a fundamental sequence in ℍ_{p}^{s+1+1/p}(Ω). From the proven part of the theorem and the above embedding theorem we obtain

lim_{n→∞} ∥(P_{Σ}^{1}g − P_{Σ}^{1}g_{n})|𝕃_{p}(Ω)∥ = 0.

Therefore P_{Σ}^{1}g ∈ ℍ_{p}^{s+1+1/p}(Ω) and

∥P_{Σ}^{1}g|ℍ_{p}^{s+1+1/p}(Ω)∥ ≤ C_{2}∥g|𝔹_{p,p}^{s}(Σ)∥.  (2.33)

For P_{Σ}^{1} the proof is completed using interpolation (see [9]). For the operator P_{Σ}^{2} the proof is similar, the only difference being that for s ≥ 0 we should begin with s = \frac{1}{p′} + 1 instead of s = \frac{1}{p′} and apply (2.20).

§ 3. Main Theorems

The next five theorems can be regarded as the main results of both parts of this work. Two of them (Theorems 7 and 8) will be proved in §5.

Theorem 5. Let φ^{+} and φ^{−} be given vector-functions satisfying (1.19). Then U ∈ 𝕎_{p,loc}^{1}(ℝ_{S}^{3}) is a solution of Problem D if and only if

U(x) = (P_{S}^{2}φ^{0})(x) − (P_{S}^{1}φ)(x), x ∈ ℝ_{S}^{3},  (3.1)

where φ^{0} = φ^{+} − φ^{−} ∈ 𝔹̃_{p,p}^{1/p′}(S), while φ ∈ 𝔹̃_{p,p}^{−1/p}(S) solves the system of pseudodifferential equation

P_{S}^{1}φ = f on S,  (3.2)

with

f = P_{S}^{2}φ^{0} − \frac{1}{2}(φ^{+} + φ^{−}).

Proof. By Theorem 4 and formula (2.7) we obtain the following representation of an arbitrary solution U of the homogeneous equation (1.4), F = 0,

± \{(P_{Σ}^{2}U^{±})(x) − (P_{Σ}^{1}(BU)^{±})(x)\} = \begin{cases}U(x), & x ∈ Ω^{±}, \\ 0, & x ∈ Ω^{∓}, \end{cases}  (3.3)

which, upon taking the differences, yields (3.1) with φ^{0} = U^{+} − U^{−} and φ = (BU)^{+} − (BU)^{−}, if we take into account that

(U)^{+} − (U)^{−} = 0, (BU)^{+} − (BU)^{−} = 0 on Σ\\overline{S}.

Applying Theorem 4, from (3.1) it follows that

U^{±} = ±\frac{1}{2}φ^{0} + P_{S}^{2}φ^{0} − P_{S}^{1}φ = φ^{±}

and, upon taking the sum U^{+} + U^{−}, we obtain equation (3.2).

Theorem 6. Let 1 < p < ∞ and ψ^{±} ∈ 𝔹_{p,p}^{−1/p}(S), ψ^{0} = ψ^{+} − ψ^{−} ∈ 𝔹̃_{p,p}^{−1/p}(S) be given functions. Then U ∈ 𝕎_{p,loc}^{1}(ℝ_{S}^{3}) is a solution of Problem N if and only if

U(x) = (P_{S}^{2}ψ)(x) − (P_{S}^{1}ψ^{0})(x), x ∈ ℝ_{S}^{3},  (3.4)

where ψ ∈ 𝔹̃_{p,p}^{1/p}(S) solves the pseudodifferential equation

P_{S}^{4}ψ = g,  (3.5)

where

g = \frac{1}{2}(ψ^{+} + ψ^{−}) + P_{S}^{3}ψ^{0}.

Proof. The theorem is proved similarly to Theorem 5. We would like only to note that if equation (3.5) has a solution, then ψ and the boundary values U^{±} are related by the equality

ψ = U^{+} − U^{−} ∈ 𝔹̃_{p,p}^{1/p}(S).

Theorem 7. (i) The operators

P_{S}^{1} : 𝔹̃_{p,q}^{ν}(S) → 𝔹_{p,q}^{ν+1}(S),  (3.6)
P_{S}^{1} : ℍ̃_{p}^{ν}(S) → ℍ_{p}^{ν+1}(S)  (3.7)

are bounded for any 1 < p < ∞, 1 ≤ q ≤ ∞, ν ∈ ℝ; (ii) (3.6) is a Fredholm operator if the condition

1/p − 3/2 < ν < 1/p − 1/2  (3.8)

is fulfilled; (iii) (3.7) is a Fredholm operator if and only if condition (3.8) is fulfilled; (iv) operators (3.6) and (3.7) are invertible for all ν satisfying (3.8).

Theorem 8. (i) The operators

P_{S}^{4} : 𝔹̃_{p,q}^{ν+1}(S) → 𝔹_{p,q}^{ν}(S),  (3.9)
P_{S}^{4} : ℍ̃_{p}^{ν+1}(S) → ℍ_{p}^{ν}(S)  (3.10)

are bounded for any 1 < p < ∞, 1 ≤ q ≤ ∞, ν ∈ ℝ; (ii) (3.9) is a Fredholm operator if (3.8) is fulfiled; (iii) (3.10) is a Fredholm operator if and only if (3.8) is fulfilled; (iv) operators (3.9) and (3.10) are invertible for all ν satisfying (3.8).

Let us assume that M is a smooth manifold with the boundary ∂M ≠ ∅ and introduce the notation H_{∞}^{s}(M) := ∩_{2<p<∞}ℍ_{p}^{s}(M) = ∩_{2<p<∞}𝔹_{p,q}^{s}(M),

H̃_{∞}^{s}(M) := ∩_{2<p<∞}ℍ̃_{p}^{s}(M) = ∩_{2<p<∞}𝔹̃_{p,q}^{s}(M). Obviously, H_{∞}^{s}(M) = H̃_{∞}^{s}(M)

if −1/2 ≤ s ≤ 0.

Theorem 9. Let φ^{±} ∈H_{∞}^{1/2}(S), φ^{+} − φ^{−} ∈H̃_{∞}^{1/2}(S) and ψ^{±} ∈H_{∞}^{−1/2}(S). Then the solutions of Problems D and N are real analytic vectors in ℝ_{S}^{3}, vanishing at infinity. For their restrictions to Ω^{±} we have the inclusions U|_{Ω^{±}} ∈ H_{∞}^{1/2+1/p}(Ω^{±}).

Therefore

U ∈ ⋂_{α<1/2} C^{α}(ℝ_{S}^{3}),

where

C^{α}(ℝ_{S}^{3}) := \{φ : φ ∈ C^{α}(\overline{Ω}^{±}), φ^{+}(x) = φ^{−}(x) if x ∈ Σ\S\}.

Proof. Assume that Theorems 7 and 8 are proved (see §5). Due to representation (3.1) and (3.4) U(x) is a real analytic vector satisfying condition (1.11) at infinity. For its traces we have U^{±} ∈ ∩_{1<p<∞}ℍ_{p}^{1/2}(S). If we now

take a sufficiently large p, the proof will follow from Theorems 2 and 4 and the well-known embedding

ℍ_{p}^{ν}(ℝ_{S}^{3}) ⊂ C^{ν−μ}(ℝ_{S}^{3}), μ > 3/p  (3.11)

(see [9]).

Theorem 9 implies that the traces U^{±} on both faces of the crack surface S belong to the Hölder space ∩_{α<1/2}C^{α}(S) and U^{+}(x) = U^{−}(x) for x ∈ ℓ = ∂S.

Remark 10. Operators (2.16) and (2.18) are invertible for any k = 0, 1, ... and 0 < γ < 1 if Re τ > 0 (see [2]).

Remark 11. Theorem 9 shows the advantage of considering equations (3.2) and (3.5) in the spaces 𝔹_{p,p}^{ν}(S) (or ℍ_{p}^{ν}(S)) with p ≠ 2, since if we stick to the case p = 2, we shall not be able to obtain the above results on smoothness for U|_{Ω^{±}} and U^{±}.

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(Received 01.10.1993)

Authors’ addresses:

R. Duduchava A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z. Rukhadze St., Tbilisi 380093 Republic of Georgia

D. Natroshvili Georgian Technical University Department of Mathematics (99) 77, M. Kostava St., Tbilisi 380075 Republic of Georgia

E. Shargorodsky Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia