UDC 517.947.2

ON THE BASIC EXISTENCE THEOREMS OF SPACE ELASTICITY THEORY

V. D. KUPRADZE

1. At the present time three proofs are known of the existence theorems of elasticity theory, based on the theory of singular integral equations [1, 2, 3]. Common to all of them is the use of the theory of regularization, which relies on the concept of the symbol of a bounded singular operator. In [1] the completion of the proof is achieved within the framework of Fredholm theory; in the works [2] and [3] it is achieved with the aid of theorems on the index of a singular operator.

In [4] it was first observed that, for the basic problems of elasticity theory, the regularizer can be constructed explicitly, without applying the general scheme based on the concept of the symbol. For this purpose [4] uses the theory of the spatial analogue of the Cauchy-type integral [5].

As we shall see, the same results can be obtained by starting from elementary facts of elasticity theory itself.

Thus, for the proof of the basic existence theorems of the spatial theory of elasticity, it proves sufficient to use the simplest properties of the singular integral in combination with the classical theory of integral equations. This circumstance, at least from the methodological point of view, deserves attention: as is known, until now proofs of the existence theorems, because of their excessive complexity, were not presented even in those books on elasticity theory in which Fredholm theory is systematically used. Now the situation is changing. In this connection it seems advisable to give a new exposition of the question, although repetitions cannot thereby be avoided (cf. [1], pp. 141—161).

2. Regular solutions of the system of equations of elasticity theory

\[ \mu \Delta u + (\lambda+\mu)\operatorname{grad}\operatorname{div} u = 0 \tag{1} \]

can be represented [1] in the form

\[ \delta(P)u(P)=\frac{1}{4\pi}\iint_S \left[\Gamma(Q,P)Tu(Q)-\Gamma_I(Q,P)u(Q)\right]\,dS_Q, \tag{2} \]

where the positive normal is directed outward and \(u(P)\) is the displacement vector at the point \(P(x_1,x_2,x_3)\); \(\Gamma(Q,P)\) is the matrix of fundamental solutions

\[ \Gamma(Q,P)=\left\|\Gamma_i^{(k)}\right\|_{i,k=1}^{3}, \]

\[ \Gamma_i^{(k)}(Q,P)= \left( m\,\frac{\partial r}{\partial x_i}\frac{\partial r}{\partial x_k} +n\,\delta_{ik} \right)\frac{1}{r(Q,P)}, \tag{3} \]

\[ m=\frac{\lambda+2\mu}{2\mu(\lambda+2\mu)}, \qquad n=\frac{\lambda+3\mu}{2\mu(\lambda+2\mu)}; \]

\(\delta_{ik}\) is the Kronecker symbol; \(Q\) is the point \((\xi_1,\xi_2,\xi_3)\); \(Tu(Q)\) is the stress vector on the surface element passing through the point \(Q\) with normal \(\nu(Q)=(\nu_1,\nu_2,\nu_3)\),

\[ Tu=2\mu\,\frac{\partial u}{\partial \nu(Q)}+\lambda \nu \operatorname{div} u+\mu(\nu\times \operatorname{rot} u), \tag{4} \]

\(\Gamma_1(Q,P)\) is the matrix of fundamental solutions of the first kind (solutions generated by the action of the operator \(T\) on the matrix \(\Gamma(Q,P)\)),

\[ \Gamma_1(Q,P)=\|T_k\Gamma^{(i)}\|_{i,k=1}^3, \]

\[ \begin{aligned} T_k\Gamma^{(i)}(Q,P) &=\frac{\mu}{\lambda+2\mu}\left[ \delta_{ik}\frac{\partial}{\partial \nu(Q)}\frac{1}{r(Q,P)} +\nu_i(Q)\frac{\partial}{\partial \xi_k}\frac{1}{r} -\nu_k(Q)\frac{\partial}{\partial \xi_i}\frac{1}{r} \right] \\ &\quad+\frac{3(\lambda+\mu)}{\lambda+2\mu} \frac{\partial r}{\partial \xi_i} \frac{\partial r}{\partial \xi_k} \frac{\partial}{\partial \nu(Q)} \frac{1}{r(Q,P)}, \end{aligned} \tag{5} \]

\(S\) is a closed Lyapunov surface bounding the domain \(B_i\), the domain of regularity of the solution; \(B_e\) is the complement of \(B_i\cup S\) in the whole space,

\[ \delta(P)= \begin{cases} E, & P\in B_i,\\[2mm] \dfrac{1}{2}E, & P\in S,\\[2mm] 0, & P\in B_e, \end{cases} \qquad (E\text{ is the unit vector}). \]

The structure of the right-hand side of equality (2) indicates the form of the simple- and double-layer potentials with undetermined densities, by means of which the solutions of the boundary-value problems may be expressed, if they exist.

Seeking the solution of the first problem (the displacements are prescribed on the boundary) in the form of a double-layer potential

\[ u(P)=\frac{1}{2\pi}\iint_S \Gamma_1(Q,P)\varphi(Q)\,dS_Q \]

and the solution of the second problem (the stresses are prescribed on the boundary) in the form of a simple-layer potential

\[ u(P)=\frac{1}{2\pi}\iint_S \Gamma(Q,P)\psi(Q)\,dS_Q, \]

we arrive at the adjoint systems of singular integral equations [1]

\[ \varphi(Q_0)-\varkappa \iint_S K(Q_0,Q)\varphi(Q)\,dS_Q=f(Q_0), \tag{6_1} \]

\[ \psi(Q_0)-\varkappa \iint_S K^*(Q,Q_0)\psi(Q)\,dS_Q=f(Q_0), \tag{6_2} \]

where

\[ \varkappa=\pm \frac{1}{2\pi}\,\frac{\mu}{\lambda+2\mu}, \qquad Q_0=(\xi_1^0,\xi_2^0,\xi_3^0)\in S, \]

\[ K(Q_0,Q)=\frac{1}{|\varkappa|}\left\|T_k\Gamma^{(i)}\right\|^3_{i,k=1} = M(Q,Q_0)+\frac{1}{\mu}N(Q,Q_0), \tag{7} \]

\[ M(Q,Q_0)=\frac{1}{r^3(Q,Q_0)}\times \]

\[ \times \left\| \begin{array}{ccc} \nu_1(Q)(\xi_1-\xi_1^0)+\nu_2(\xi_2-\xi_2^0)+\nu_3(\xi_3-\xi_3^0), & \nu_2(\xi_1-\xi_1^0)-\nu_1(\xi_2-\xi_2^0), & \nu_3(\xi_1-\xi_1^0)-\nu_1(\xi_3-\xi_3^0) \\[4pt] \nu_1(\xi_2-\xi_2^0)-\nu_2(\xi_1-\xi_1^0), & \nu_1(\xi_1-\xi_1^0)+\nu_2(\xi_2-\xi_2^0)+\nu_3(\xi_3-\xi_3^0), & \nu_3(\xi_2-\xi_2^0)-\nu_2(\xi_3-\xi_3^0) \\[4pt] \nu_1(\xi_3-\xi_3^0)-\nu_3(\xi_1-\xi_1^0), & \nu_2(\xi_3-\xi_3^0)-\nu_3(\xi_2-\xi_2^0), & \nu_1(\xi_1-\xi_1^0)+\nu_2(\xi_2-\xi_2^0)+\nu_3(\xi_3-\xi_3^0) \end{array} \right\|, \tag{8} \]

\[ N(Q,Q_0)= \left\| 3(\lambda+\mu)\frac{\partial r}{\partial \xi_i}\frac{\partial r}{\partial \xi_k} +2\mu\delta_{ik} \right\| \frac{\partial}{\partial \nu(Q)}\frac{1}{r(Q,Q_0)} \]

\[ (i,k=1,2,3). \tag{9} \]

\(f(Q_0)\) is a vector prescribed on \(S\), which we shall assume to belong to the Hölder class, and \(K^*(Q_0,Q)\) is the matrix conjugate to \(K(Q,Q_0)\).

In operator form, equation \((6_1)\) is written as

\[ (E-\varkappa K)\varphi=f. \]

We shall show that the equation

\[ (E+\varkappa M)(E-\varkappa K)\varphi=(E+\varkappa M)f, \]

for

\[ \varkappa\ne \pm \frac{1}{2\pi}, \tag{10} \]

is a Fredholm equation.

1. The matrix \(M(Q,Q_0)\), the principal part of the kernel, already occurs in works [6, 7] as the kernel of the spatial analogue of the Cauchy integral from the theory of functions of a complex variable. Subsequently, in [5], the spatial analogue of an integral of Cauchy type with the same kernel \(M(Q,Q_0)\) was considered; its basic properties were established and some applications were given.

In particular, the properties of the matrix \(M(Q,Q_0)\) used for constructing the regularizer were proved in [5]. They can, however, also be obtained directly from the formulas of elasticity theory.

First of all, obviously,

\[ \frac{1}{4\pi}\iint_S M(Q,P)\,dS_Q = \begin{cases} E, & P\in B_i,\\[2pt] \dfrac{1}{2}E, & P\in S,\\[2pt] 0, & P\in B_e. \end{cases} \tag{11} \]

This follows from the fact that the diagonal elements of the matrix (8), equal to

\[ -\frac{\partial}{\partial \nu(Q)}\frac{1}{r(Q,P)}, \]

satisfy (11), while for the off-diagonal elements we have

\[ \iint_{S+\gamma(\eta)} \left( \nu_1(Q)\frac{\partial}{\partial \xi_2}\frac{1}{r} - \nu_2(Q)\frac{\partial}{\partial \xi_1}\frac{1}{r} \right)dS_Q = \iiint_{B_i-\gamma_\eta} \left[ \frac{\partial}{\partial \xi_1} \left( \frac{\partial}{\partial \xi_2}\frac{1}{r} \right) - \frac{\partial}{\partial \xi_2} \left( \frac{\partial}{\partial \xi_1}\frac{1}{r} \right) \right] \]

\[ -\frac{\partial}{\partial \xi_2}\left(\frac{\partial}{\partial \xi_1}\frac{1}{r}\right)\Biggr]\,dv=0, \]

where \(\gamma(\eta)\) is the sphere of radius \(\eta\) with center at the point \(P\in B_i\) (a hemisphere, if \(P\in S\)); \(\gamma_\eta\) is the ball bounded by the sphere \(\gamma(\eta)\).

Noting that

\[ \iint_{\gamma(\eta)} \left(\nu_1\frac{\partial}{\partial \xi_2}\frac{1}{r} -\nu_2\frac{\partial}{\partial \xi_1}\frac{1}{r}\right)\,dS=0, \]

we obtain

\[ \iint_S \left(\nu_1\frac{\partial}{\partial \xi_2}\frac{1}{r} -\nu_2\frac{\partial}{\partial \xi_1}\frac{1}{r}\right)\,dS=0, \]

which proves (11).

Let \(Q_1(\eta_1,\eta_2,\eta_3)\) be an arbitrary point fixed on \(S\). About \(Q_1\), as center, describe a hemisphere of radius \(\varepsilon\), lying in \(B_i\), and call its surface \(\sigma(\varepsilon)\). Let \(S_\varepsilon\) be the part of \(S\) lying outside the hemisphere \(\sigma(\varepsilon)\). We shall prove that

\[ 4\pi M(Q_1,P)= \iint_{S_\varepsilon+\sigma(\varepsilon)} M(Q,P)M(Q_1,Q)\,dS_Q,\qquad P\in B_i. \tag{12} \]

For this purpose we shall verify the validity of the equality

\[ \iint_{S_\varepsilon+\sigma(\varepsilon)+\gamma(\eta)} M(Q,P)M(Q_1,Q)\,dS_Q=0. \tag{13} \]

We shall show this, for example, for the first element \(\Pi_1^{(1)}\) in the product \(M(Q,P)M(Q_1,Q)\) (i.e. for the projection onto the \(x_1\)-axis of the first column-vector). To simplify the computations we shall assume the coordinate system (rectilinear, orthogonal) to be oriented so that the positive \(x_1\)-axis coincides with the direction of the outer normal at the point \(Q_1\); then

\[ \nu_1(Q_1)=1,\qquad \nu_2(Q_1)=\nu_3(Q_1)=0. \]

We have

\[ \Pi_1^{(1)}=(M^{(1)}(Q,P))^*M_1^{(1)}(Q_1,Q) = \frac{1}{r^3r_1^3} \{\nu_1(Q)[(\xi_1-x_1)(\eta_1-\xi_1)- \]

\[ -(\xi_2-x_2)(\eta_2-\xi_2)-(\xi_3-x_3)(\eta_3-\xi_3)] +\nu_2(Q)[(\xi_2-x_2)(\eta_1-\xi_1)+ \]

\[ +(\xi_1-x_1)(\eta_2-\xi_2)] +\nu_3(Q)[(\xi_3-x_3)(\eta_1-\xi_1)+(\xi_1-x_1)(\eta_3-\xi_3)]\}^{1)} \]

\[ \bigl(r\equiv r(Q,P),\qquad r_1\equiv r(Q_1,Q)\bigr). \]

Denote

\[ A_1=\frac{1}{r^3r_1^3} [(\xi_1-x_1)(\eta_1-\xi_1)-(\xi_2-x_2)(\eta_2-\xi_2)-(\xi_3-x_3)(\eta_3-\xi_3)], \]

\[ A_2=\frac{1}{r^3r_1^3} [(\xi_2-x_2)(\eta_1-\xi_1)+(\xi_1-x_1)(\eta_2-\xi_2)]. \]

\(^{1)}\) \((M^{(1)}(Q,P))^*\) is the first row of the matrix \(M(Q,P)\).

\[ A_3=\frac{1}{r_1^3 r_1^3}\left[(\xi_3-x_3)(\eta_1-\xi_1)+(\xi_1-x_1)(\eta_3-\xi_3)\right]. \]

Obviously, for the vector \(A(A_1,A_2,A_3)\) in the domain from which the points \(P\) and \(Q_1\) have been excluded, the equality

\[ \operatorname{div}_Q A=0 \]

holds and, consequently,

\[ \iint_{S_\varepsilon+\sigma(\varepsilon)+\gamma(\eta)} \left(M^{(1)}(Q,P)\right)*M^{(1)}(Q_1,Q)\,dS_Q=0. \]

Repeating this reasoning for the other elements, we arrive at (13). But

\[ \iint_{S_\varepsilon+\sigma(\varepsilon)+\gamma(\eta)} M(Q,P)M(Q_1,Q)\,dS_Q = \iint_{S_\varepsilon+\sigma(\varepsilon)} M(Q,P)M(Q_1,Q)\,dS_Q+ \]

\[ +\iint_{\gamma(\eta)} \{M(Q,P)[M(Q_1,Q)-M(Q_1,P)]+M(Q,P)M(Q_1,P)\}\,dS_Q=0, \]

and, passing to the limit as \(\eta\to0\), we shall have

\[ \iint_{S_\varepsilon+\sigma(\varepsilon)} M(Q,P)M(Q_1,Q)\,dS_Q = -\lim_{\eta\to0}\iint_{\gamma(\eta)} M(Q,P)M(Q_1,P)\,dS_Q \]

and, in accordance with (11)\(^1\), we obtain equality (12); from this equality (15) follows easily. Without dwelling on this, let us show that the same result can be obtained directly from formula (2). Consider it in the domain bounded by the surface \(S_\varepsilon+\sigma(\varepsilon;\tau)\), where \(\sigma(\varepsilon;\tau)\) lies in \(B_i\) and depends on the parameter \(\tau\) in such a way that

\[ \lim_{\tau\to0}\sigma(\varepsilon;\tau)=S-S_\varepsilon. \]

Taking the point \(P\equiv Q_0\) lying on \(S_\varepsilon+\sigma(\varepsilon;\tau)\), from (2) we shall have

\[ 2\pi E u(Q_0)= \iint_{S_\varepsilon+\sigma(\varepsilon;\tau)} \left[\Gamma(Q,Q_0)Tu(Q)-\Gamma_{\mathrm I}(Q,Q_0)u(Q)\right]\,dS_Q. \]

Put \(\lambda+\mu=0\). Then \(u(Q)\) will be a solution of the Laplace equation, and if we take into account that

\[ \Gamma_{\mathrm I}(Q,Q_0)=\left\|T_k\Gamma^{(i)}\right\|_{i,k=1}^3 = \frac{\mu}{\lambda+2\mu}M(Q,Q_0)+\frac{1}{\lambda+2\mu}N(Q,Q_0), \]

then, on the basis of (3), (4), (8), and (9), we may write:

\[ 2\pi u(Q_0)=\frac{1}{\mu} \iint_{S_\varepsilon+\sigma(\varepsilon;\tau)} \left\{ E\frac{1}{r(Q,Q_0)} \left[ 2\mu\frac{\partial u}{\partial \nu(Q)} +\mu(\nu\times\operatorname{rot}u)-\mu\nu\,\operatorname{div}u \right]\right. \]

\[ \left. -\left[ M(Q,Q_0)+2E\frac{\partial}{\partial \nu(Q)}\frac{1}{r(Q,Q_0)} \right]u(Q) \right\}\,dS_Q \]

\(^1\) It should be borne in mind that in the preceding formula the normal is directed inward to \(\gamma(\eta)\).

and since

\[ \iint_{S_\varepsilon+\sigma(\varepsilon;\tau)} E\left( \frac{1}{r(Q,Q_0)}\frac{\partial u}{\partial \nu} -u(Q)\frac{\partial}{\partial \nu(Q)}\frac{1}{r(Q,Q_0)} \right)dS_Q=2\pi u(Q_0), \]

we finally have

\[ \begin{aligned} -2\pi E u(Q_0) &= \iint_{S_\varepsilon+\sigma(\varepsilon;\tau)} E\,\frac{\nu\times \operatorname{rot}u-\nu\,\operatorname{div}u}{r(Q,Q_0)}\,dS_Q \\ &\quad- \iint_{S_\varepsilon+\sigma(\varepsilon;\tau)} M(Q,Q_0)u(Q)\,dS_Q . \end{aligned} \tag{14} \]

Each column of the matrix \(M(Q_1,Q)\), as a function of \(Q\), is a harmonic vector in the domain under consideration; therefore one may set \(u(Q)=M^{(i)}(Q_1,Q)\) \((i=1,2,3)\). Let, for example, \(i=1\). It is easy to see that the first integral on the right in (14) then vanishes. Indeed, obviously,

\[ M^{(1)}(Q_1,Q) = \frac{1}{r^3(Q_1,Q)} (\eta_1-\xi_1,\eta_2-\xi_2,\eta_3-\xi_3), \]

and for the projection of the vector
\([\nu(Q)\times \operatorname{rot}_Q M^{(1)}(Q_1,Q)-\nu(Q)\operatorname{div}_Q M^{(1)}(Q_1,Q)]\)
onto the \(x_1\)-axis, as is easy to see, we have

\[ \begin{aligned} &\nu_1(Q)\left( \frac{\partial}{\partial \xi_1}\frac{\eta_1-\xi_1}{r^3(Q_1,Q)} + \frac{\partial}{\partial \xi_2}\frac{\eta_2-\xi_2}{r^3(Q_1,Q)} + \frac{\partial}{\partial \xi_3}\frac{\eta_3-\xi_3}{r^3(Q_1,Q)} \right) \\ &\quad -\nu_2(Q)\left( \frac{\partial}{\partial \xi_2}\frac{\eta_1-\xi_1}{r^3(Q_1,Q)} - \frac{\partial}{\partial \xi_1}\frac{\eta_2-\xi_2}{r^3(Q_1,Q)} \right) \\ &\quad -\nu_3(Q)\left( \frac{\partial}{\partial \xi_3}\frac{\eta_1-\xi_1}{r^3(Q_1,Q)} - \frac{\partial}{\partial \xi_1}\frac{\eta_3-\xi_3}{r^3(Q_1,Q)} \right) =0 . \end{aligned} \]

Repeating this argument for the other elements, we obtain from (14)

\[ \iint_{S_\varepsilon+\sigma(\varepsilon;\tau)} M(Q,Q_0)M(Q_1,Q)\,dS_Q -2\pi M(Q_1,Q_0)=0 . \tag{14^{1}} \]

On the other hand,

\[ \begin{aligned} \iint_{\sigma(\varepsilon;\tau)} M(Q,Q_0)M(Q_1,Q)\,dS_Q &= \iint_{\sigma(\varepsilon;\tau)} [M(Q,Q_0)-M(Q_1,Q_0)]\times \\ &\quad\times M(Q_1,Q)\,dS_Q +M(Q_1,Q_0) \iint_{\sigma(\varepsilon;\tau)} [M(Q_1,Q)+M(Q,Q_1)]\,dS_Q \\ &\quad -M(Q_1,Q_0) \iint_{(S-S_\varepsilon)+\sigma(\varepsilon;\tau)} M(Q,Q_1)\,dS_Q +M(Q_1,Q_0) \iint_{(S-S_\varepsilon)} M(Q,Q_1)\,dS_Q \\ &= \iint_{\sigma(\varepsilon;\tau)} [M(Q,Q_0)-M(Q_1,Q_0)]M(Q_1,Q)\,dS_Q \\ &\quad +M(Q_1,Q_0) \iint_{\sigma(\varepsilon;\tau)} [M(Q_1,Q)+M(Q,Q_1)]\,dS_Q +2\pi M(Q_1,Q_0)+ \end{aligned} \]

\[ + M(Q_1,Q_0)\iint_{(S-S_\varepsilon)} M(Q,Q_1)\,dS_Q . \]

Let us pass to the limit in this equality first as \(\tau \to 0\). Obviously, \(\sigma(\varepsilon,\tau)\) then turns into \((S-S_\varepsilon)\), and all the integrals converge absolutely except the last one, which exists in the sense of the principal value. Finally passing to the limit as \(\varepsilon \to 0\), we ultimately find

\[ \lim_{\varepsilon\to 0}\left\{\lim_{\tau\to 0}\iint_{\sigma(\varepsilon;\tau)} M(Q,Q_0)M(Q_1,Q)\,dS_Q\right\}=2\pi M(Q_1,Q) \]

and, consequently, from \((14^1)\) we have

\[ \iint_S M(Q,Q_0)M(Q_1,Q)\,dS_Q=0. \tag{15} \]

Consider the integral

\[ \frac{1}{2\pi}\iint_S M(Q_1,Q)\varphi(Q_1)\,dS_{Q_1},\qquad Q\in S, \]

where \(\varphi(Q_1)\) is a vector defined on \(S\), satisfying the Hölder condition. After carrying out a number of the identical transformations indicated below, in which the properties (11) and (15) are used, we shall have

\[ \frac{1}{4\pi^2}\iint_S M(Q,Q_0)\,dS_Q \iint_S M(Q_1,Q)\varphi(Q_1)\,dS_{Q_1}= \]

\[ =\frac{1}{4\pi^2}\iint_S M(Q,Q_0)\,dS_Q \iint_S M(Q_1,Q)\,[\varphi(Q_1)-\varphi(Q_0)]\,dS_{Q_1}+ \]

\[ +\frac{1}{4\pi^2}\iint_S M(Q,Q_0)\,dS_Q \iint_S M(Q_1,Q)\,dS_{Q_1}\,\varphi(Q_0)=\varphi(Q_0)+ \]

\[ +\frac{1}{4\pi^2}\iint_S dS_{Q_1}\iint_S M(Q,Q_0)M(Q_1,Q)[\varphi(Q_1)-\varphi(Q_0)]\,dS_Q=\varphi(Q_0). \]

Thus

\[ \iint_S M(Q,Q_0)\,dS_Q \iint_S M(Q_1,Q)\varphi(Q_1)\,dS_{Q_1} =4\pi^2 E\varphi(Q_0). \tag{16} \]

1. Returning to what was said at the end of item 2, we shall have, by virtue of (7),

\[ (E+\varkappa M)\left(E-\varkappa M-\frac{\varkappa}{\mu}N\right)\varphi =(E+\varkappa M)f \]

and, by virtue of (16),

\[ E(1-4\pi^2\varkappa^2)\varphi+\varkappa F_1\varphi=\Phi_1, \tag{17} \]

where

\[ F_1=-\frac{1}{\mu}EN-\frac{\varkappa}{\mu}MN \tag{18\(_1\)} \]

is a completely continuous operator, and

\[ \Phi_1 = (E + \varkappa M) f. \tag{18_2} \]

We shall assume, in accordance with (10), that

\[ \varkappa \ne \pm \frac{1}{2\pi}, \]

which is obviously equivalent to the inequality

\[ \lambda + \mu \ne 0. \tag{19} \]

Under this condition, (17) is a system of Fredholm equations on \(S\). Denote

\[ (1 - 4\pi^2\varkappa^2)^{-1} F_1 = F,\qquad (1 - 4\pi^2\varkappa^2)^{-1} \Phi_1 = \Phi \]

and write equation (17) in the form

\[ \varphi + \varkappa F\varphi = \Phi. \tag{20} \]

As a Fredholm equation, (20), after a finite number of iterations, is reduced to an equation with a continuous kernel. If \(\varkappa\) is distinct from the poles of the resolvent of this equation, then, having found the (unique) solution of the equation

\[ (E + \varkappa^{s+1} F^{(s)})\psi = \Phi, \]

(where \(s\) is the number of the last iteration and \(F^{(s)}\) is the \(s\)-th iterated operator), we shall have for the solution of equation (20)

\[ \varphi = \psi + \sum_{k=1}^{s} \varkappa^k F^{(k-1)} \psi \qquad (F^{(0)} \equiv F). \]

Hence it is clear that, for the indicated \(\varkappa\), the solution is represented in the following form:

\[ \varphi(Q_0) = (1 - 4\pi^2\varkappa^2)^{-1} f(Q_0) + \varkappa \iint_S N_1(Q,Q_0;\varkappa) f(Q)\, dS_Q, \tag{21} \]

where

\[ N_1(Q,Q_0;\varkappa) = (1 - 4\pi^2\varkappa^2)^{-1} M(Q,Q_0) + O\{r^{-2+\varepsilon}(Q,Q_0)\},\qquad \varepsilon > 0. \tag{22} \]

Using (16) and proceeding as usual [1], from (21), (22), and \((6_1)\) we obtain the equation for the “resolvent”

\[ N_1(Q,Q_0;\varkappa) - (1 - 4\pi^2\varkappa^2)^{-1} K(Q_0,Q) \]

\[ {} - \varkappa \iint_S N_1(Q,Q_0;\varkappa) K(Q_0,Q)\, dS_Q = 0. \tag{23} \]

It follows from what has been said that if \((6_1)\) has a solution in the Hölder class, then it will be a solution of equation (20) and, for \(\varkappa\) distinct from the values excluded above, will be represented in the form (21), where \(N_1\) satisfies (23).

Let us now show that equation \((6_1)\), for all \(\varkappa\) except for a certain countable set of values, indeed has a solution in the class \(H\) (the Hölder class). We shall seek this solution in the form

\[ \varphi(Q_0) = \sigma(Q_0) - \varkappa \iint_S M^*(Q,Q_0)\sigma(Q)\, dS_Q, \tag{24} \]

where \(M^*(Q,Q_0)\) is the matrix transposed with respect to \(M(Q,Q_0)\). Taking into account that

\[ M^*(Q,Q_0)=-M(Q,Q_0)+2E\,\frac{\partial}{\partial \nu(Q)}\,\frac{1}{r(Q,Q_0)}, \]

and, substituting (24) into \((6_1)\), we obtain, by virtue of (16),

\[ E(1-4\pi^2\varkappa^2)\sigma(Q_0)-\varkappa \iint\limits_S F_2(Q,Q_0;\varkappa)\sigma(Q)\,dS_Q=f(Q_0), \tag{25} \]

where \(F_2(Q,Q_0;\varkappa)\), as is easily verified, is a “completely continuous kernel.”

Suppose that \(\varkappa\) does not coincide with the poles of the resolvent of this equation; then [1], having found from (25) the solution (the unique one) and substituting it into (24), we shall have

\[ \varphi(Q_0)=(1-4\pi^2\varkappa^2)^{-1}f(Q_0)+\varkappa \iint\limits_S N_2(Q,Q_0;\varkappa)f(Q)\,dS_Q, \tag{26} \]

where

\[ N_2(Q,Q_0;\varkappa)-(1-4\pi^2\varkappa^2)^{-1}K(Q_0,Q)- \]

\[ -\varkappa \iint\limits_S K(Q_1,Q_0)N_2(Q_1,Q;\varkappa)\,dS_{Q_1}=0. \tag{27} \]

Combining the results obtained, we arrive at the conclusion:

For \(\varkappa\) different from a certain countable set of values, equation \((6_1)\) has a unique solution, represented simultaneously in the forms (21) and (26); consequently,

\[ N_1(Q,Q_0;\varkappa)\equiv N_2(Q,Q_0;\varkappa) \tag{28} \]

identically with respect to \(Q\) and \(Q_0\) and for \(\varkappa\) not coinciding with the exceptional ones. As an identity of meromorphic functions, (28) indicates that \(N_1\) and \(N_2\) have common poles, and consequently (28) is valid for all \(\varkappa\) different from the poles. By virtue of (28), the functional equations (23) and (27) take the following form:

\[ N(Q,Q_0;\varkappa)-(1-4\pi^2\varkappa^2)^{-1}K(Q,Q_0)- \]

\[ -\varkappa \iint\limits_S N(Q,Q_1;\varkappa)K(Q_1,Q_0)\,dS_{Q_1}=0, \tag{29} \]

\[ N(Q,Q_0;\varkappa)-(1-4\pi^2\varkappa^2)^{-1}K(Q,Q_0)- \]

\[ -\varkappa \iint\limits_S K(Q,Q_1)N(Q_1,Q_0;\varkappa)\,dS_{Q_1}=0. \tag{30} \]

We have obtained Fredholm’s first theorem:

In the \(\varkappa\)-plane, with the exceptional pair of points

\[ \varkappa=\pm \frac{1}{2\pi} \]

excluded, there exists a meromorphic matrix \(N(Q,Q_0;\varkappa)\) such that, for all \(\varkappa\) not coinciding with the poles of this matrix, equation \((6_1)\) has a unique solution expressed by the equality

\[ \varphi(Q_0)=(1-4\pi^2\varkappa^2)^{-1}f(Q_0)+\varkappa \iint\limits_S N(Q_0,Q;\varkappa)f(Q)\,dS_Q, \]

and the resolvent \(N(Q,Q_0;\varkappa)\) satisfies the functional equations (29) and (30).

1. The classical Fredholm theory, as is known, is based on the functional equations of the resolvent. From these equations all the fundamental propositions of the theory are derived. Equations (29), (30) are analogues of the named equations and from them, as we shall see below, the fundamental theorems for systems of singular equations \((6_1), (6_2)\) are obtained analogously to the classical case. For one equation (with a special kernel) this was first shown by Giraud.

The term denoted by \(O\{r^{-2+\varepsilon}(Q,Q_0)\}\) on the right-hand side of (22), under closer consideration [1], turns out to be a continuous function of the points \(Q\) and \(Q_0\). This is a consequence of the fact that this term contains only expressions of the form

\[ \iint_S \Omega(Q_1,Q_0;\varkappa)\,M(Q_1,Q)\,dS_{Q_1},\qquad Q_0,Q\in S, \]

(where \(\Omega(Q_1,Q_0;\varkappa)\) is a matrix continuous in Hölder’s sense) which, by virtue of well-known theorems of Giraud [1], belong to the class \(H\).

Moreover, the term under consideration is evidently a meromorphic function of \(\varkappa\).

These remarks will be used to study the expansion of the resolvent \(N(Q,Q_0;\varkappa)\) in powers of \(\varkappa\) in a neighborhood of the pole \(\varkappa=\varkappa_0\). Let this expansion have the form

\[ N(Q,Q_0;\varkappa)= \sum_{\alpha=0}^{\infty} A_{(\alpha)}(Q,Q_0)(\varkappa-\varkappa_0)^\alpha + \sum_{\beta=1}^{p} B_{(\beta)}(Q,Q_0)(\varkappa-\varkappa_0)^{-\beta}, \tag{31} \]

where \(A_{(\alpha)}(Q,Q_0)\), \(B_{(\beta)}(Q,Q_0)\) are matrices. It is clear that negative powers of \((\varkappa-\varkappa_0)\) could arise only from the second term in (22), which is a meromorphic function of \(\varkappa\) (whereas the first term is evidently holomorphic with respect to \(\varkappa\)). But the second term, as has just been noted, is continuous with respect to \(Q,Q_0\); consequently, the matrices \(B_{(\beta)}(Q,Q_0)\), \(\beta=1,2,\ldots,p\), are continuous. In an analogous way one can verify that the matrices \(A_{(\alpha)}(Q,Q_0)\), \(\alpha=1,2,3,\ldots\), at the point \(Q\equiv Q_0\) have a pole of the second order (we shall use this remark in the proof of the third Fredholm theorem).

Substituting (31) into (29) and (30) and comparing the coefficients of \((\varkappa-\varkappa_0)^{-p}\), we obtain

\[ B_{(p)}(Q,Q_0)-\varkappa_0\iint_S B_{(p)}(Q,Q_1)\,K(Q_1,Q_0)\,dS_{Q_1}=0, \tag{32} \]

\[ B_{(p)}(Q,Q_0)-\varkappa_0\iint_S K(Q,Q_1)\,B_{(p)}(Q_1,Q_0)\,dS_{Q_1}=0. \tag{33} \]

From equations (29), (30) it follows (for details see [1], pp. 147–150)

\[ N(Q,Q_0;\varkappa)-N(Q,Q_0;\nu) + \frac{4\pi^2\nu(\varkappa-\nu)}{1-4\pi^2\nu^2}\,N(Q,Q_0;\varkappa) + \]

\[ + \frac{4\pi^2\varkappa(\varkappa-\nu)}{1-4\pi^2\varkappa^2}\,N(Q,Q_0;\nu) -(\varkappa-\nu)\times \]

\[ \times \iint_S N(Q,Q_1;\varkappa)\,N(Q_1,Q_0;\nu)\,dS_{Q_1}=0. \tag{34} \]

Denote

\[ \sum_{\alpha=0}^{\infty} A_{(\alpha)}(Q,Q_0)(\varkappa-\varkappa_0)^\alpha = A(Q,Q_0;\varkappa), \]

\[ \sum_{\beta=1}^{p} B_{(\beta)}(Q,Q_0)(\varkappa-\varkappa_0)^{-\beta} = B(Q,Q_0;\varkappa) \tag{35} \]

and let

\[ u=\varkappa-\varkappa_0,\qquad v=\nu-\varkappa_0. \]

Substitute into (34) the value of \(N\) from (31) and compare the coefficients of the terms containing the variables \(u\) and \(v\) in positive powers; then we obtain

\[ \iint_S A(Q,Q_1;\varkappa)\,A(Q_1,Q_0;\nu)\,dS_{Q_1} = \frac{1}{\varkappa-\nu}\,[A(Q,Q_0;\varkappa)-A(Q,Q_0;\nu)] + \]

\[ +\frac{4\pi^2\nu}{1-4\pi^2\nu^2}\,A(Q,Q_0;\varkappa) + \frac{4\pi^2\varkappa}{1-4\pi^2\varkappa^2}\,A(Q,Q_0;\nu). \tag{36} \]

For \(\nu=0\), (36) becomes (29); consequently, \(A(Q,Q_0;\varkappa)\) is the resolvent of the kernel \(A(Q,Q_0;0)\).

Continuing the comparison of coefficients, when considering nonnegative powers of the variable \(u\), we obtain

\[ \iint_S A(Q,Q_1;\varkappa)\,B(Q_1,Q_0;\nu)\,dS_{Q_1} = \frac{4\pi^2\varkappa}{1-4\pi^2\varkappa^2}\,B(Q,Q_0;\nu), \tag{37} \]

comparison of the remaining terms gives

\[ \iint_S B(Q,Q_1;\varkappa)\,A(Q_1,Q_0;\nu)\,dS_{Q_1} = \frac{4\pi^2\nu}{1-4\pi^2\nu^2}\,B(Q,Q_0;\varkappa). \tag{38} \]

Finally, putting \(\varkappa=0\) in (37) and \(\nu=0\) in (38), we obtain

\[ \iint_S A(Q,Q_1;0)\,B(Q_1,Q_0;\nu)\,dS_{Q_1} = \]

\[ = \iint_S B(Q,Q_1;\varkappa)\,A(Q_1,Q_0;0)\,dS_{Q_1} =0 \tag{39} \]

and, in particular, for \(\varkappa=\nu=0\),

\[ \iint_S A(Q,Q_1;0)\,B(Q_1,Q_0;0)\,dS_{Q_1} = \]

\[ = \iint_S B(Q,Q_1;0)\,A(Q_1,Q_0;0)\,dS_{Q_1} =0. \tag{40} \]

1. Equality (33), in which \(Q_0\) is regarded as a parameter, is a homogeneous system corresponding to the nonhomogeneous system (6) for \(\varkappa=\varkappa_0\)

\[ \varphi(Q)-\varkappa_0\iint_S K(Q,Q_1)\,\varphi(Q_1)\,dS_{Q_1}=0. \tag{41} \]

Consequently, the homogeneous system (41) is satisfied by the vectors \(B_{(p)}^{(1)}(Q,Q_0)\), \(B_{(p)}^{(2)}\), \(B_{(p)}^{(3)}\) \(\bigl(B_{(p)}^{(k)}\) is the \(k\)-th column of the matrix \(B_{(p)}\bigr)\), which simultaneously cannot be identically zero; consequently, (41) has distinct

nonzero solutions. On the other hand, all solutions of (41) are at the same time solutions of a certain Fredholm equation and, consequently, the number of linearly independent solutions is finite. Denote them by \(\varphi_1(Q), \varphi_2(Q), \ldots, \varphi_r(Q)\). Any other solution, in particular \(\dot B_{(p)}^{(k)}(Q,Q_0)\), is expressed in the form

\[ \dot B_{(p)}^{(k)}(Q,Q_0)=\sum_{j=1}^{r}\psi_j^{(k)}(Q_0)\varphi_j(Q), \tag{42} \]

where \(\psi_j^{(k)}(Q_0)\) are continuous scalar functions. This follows from the fact that \(\dot B_{(p)}^{(k)}(Q,Q_0)\) are continuous and \(\varphi_j(Q)\) are linearly independent.

Equality (32) can be rewritten in the following form:

\[ \dot B_{(p)}^{*}(Q,Q_0)-\chi \iint_S K^*(Q_1,Q_0)\dot B_{(p)}^{*}(Q,Q_1)\,dS_{Q_1}=0. \]

Considering \(Q\) as a parameter, we have the homogeneous system corresponding to the inhomogeneous system \((6_2)\)

\[ \psi(Q_0)-\chi_0 \iint_S K^*(Q_1,Q_0)\psi(Q_1)\,dS_{Q_1}=0. \tag{43} \]

Each of the vectors \(\dot B_{(p)}^{(s)*}(Q,Q_0)\) \((s=1,2,3)\), for all \(Q\), satisfies system (43), and since at least one of these vectors is nonzero, while equation (43) can have only a finite number of linearly independent solutions

\[ \psi_1(Q_0),\quad \psi_2(Q_0),\ \ldots,\ \psi_{r_*}(Q_0), \]

it follows that

\[ \dot B_{(p)}^{(s)*}(Q,Q_0)=\sum_{j=1}^{r_*}\varphi_j^{(s)}(Q)\psi_j(Q_0)\quad (s=1,2,3), \tag{44} \]

where \(\varphi_j^{(s)}(Q)\) are scalar continuous functions of \(Q\).

Projecting (42) onto the axis \(x_s\) and taking into account that

\[ \dot B_{(p)s}^{(k)}(Q,Q_0)=\dot B_{(p)k}^{(s)*}(Q,Q_0), \]

we shall have

\[ \dot B_{(p)k}^{(s)*}(Q,Q_0)=\sum_{j=1}^{r}\psi_j^{(k)}(Q_0)\varphi_{js}(Q). \]

Introduce the vectors

\[ \omega_j=\bigl(\psi_j^{(1)},\ \psi_j^{(2)},\ \psi_j^{(3)}\bigr)\quad (j=1,2,\ldots,r). \]

Then, obviously,

\[ \dot B_{(p)}^{(s)*}(Q,Q_0)=\sum_{j=1}^{r}\varphi_{js}(Q)\omega_j(Q_0). \]

Substituting this expression into (43), we obtain

\[ \sum_{j=1}^{r}\varphi_{js}(Q) \left[ \omega_j(Q_0)-\chi_0\iint_S K^*(Q_1,Q_0)\omega_j(Q_1)\,dS_{Q_1} \right]=0, \]

and, taking into account the linear independence of the functions

\[ \varphi_{1s},\ \varphi_{2s},\ \ldots,\ \varphi_{rs}, \]

we conclude that all \(\omega_j\) \((j=1,2,\ldots,r)\) are solutions of (43), and hence

\[ r_* \geqslant r . \]

Similarly, projecting (44) onto the axis \(x_k\) \((k=1,2,3)\) and taking into account that

\[ B_{(p)k}^{(s)*}(Q,Q_0)=B_{(p)s}^{(k)}(Q,Q_0), \]

we shall have

\[ B_{(p)s}^{(k)}(Q,Q_0)=\sum_{j=1}^{r_*}\varphi_j^{(s)}(Q)\psi_{jk}(Q_0)\qquad (s=1,2,3). \]

Introduce the vectors

\[ \sigma_j=\bigl(\varphi_j^{(1)},\varphi_j^{(2)},\varphi_j^{(3)}\bigr)\qquad (j=1,2,\ldots,r_*), \]

then

\[ B_{(p)}^{(k)}(Q,Q_0)=\sum_{j=1}^{r_*}\psi_{jk}(Q_0)\sigma_j(Q). \]

Substituting the latter into equation (41), we obtain

\[ \sum_{j=1}^{r_*}\psi_{jk}(Q_0)\left[\sigma_j(Q)-\varkappa_0\iint_S K(Q,Q_1)\sigma_j(Q_1)\,dS_{Q_1}\right]=0, \]

whence, in the same way as above,

\[ r\geqslant r_*, \]

and finally

\[ r=r_*. \]

The second Fredholm theorem is proved.

In the investigation of equations \((6_1)\), \((6_2)\), an important role is played by the property of biorthonormality of the set of eigenvectors of equations (41) and (42). The proof of this property is based on the theorem on the simplicity of the poles of the resolvent [1]. For \(p=1\) we have

\[ B_{(1)}(Q,Q_0)=-\varkappa_0 B(Q,Q_0;0) \]

and, according to (33),

\[ B(Q,Q_0;0)-\varkappa_0\iint_S K(Q,Q_1)B(Q_1,Q_0;0)\,dS_{Q_1}=0. \]

Moreover, from (29) for \(\varkappa=0\) it follows that

\[ N(Q,Q_0;0)=K(Q,Q_0) \]

or

\[ A(Q,Q_0;0)+B(Q,Q_0;0)=K(Q,Q_0). \tag{45} \]

Substituting the value \(B(Q,Q_0;0)\) into the preceding integral equality and using (40), we obtain

\[ B(Q,Q_0;0)-\varkappa_0\iint_S B(Q,Q_1;0)B(Q_1,Q_0;0)\,dS_{Q_1}=0 \tag{45'} \]

or

\[ B_{(1)}(Q,Q_0)+\iint_S B_{(1)}(Q,Q_1)B_{(1)}(Q_1,Q_0)\,dS_{Q_1}=0 \]

and, finally,

\[ B_{(1)s}^{(k)}(Q,Q_0)+\iint_S \sum_{i=1}^{3} B_{(1)s}^{(i)}(Q,Q_1)B_{(1)i}^{(k)}(Q_1,Q_0)\,dS_{Q_1}=0. \]

Substituting here the bilinear expressions, in accordance with (42), and comparing coefficients, we obtain the desired biorthonormality formulas

\[ \iint\limits_S \varphi_i(Q)\,\psi_j(Q)\,dS = \begin{cases} 1, & i=j,\\ 0, & i\ne j \end{cases} \qquad (i,\ j=1,\ 2,\ 3,\ \ldots,\ r). \]

7. Let us turn to the proof of Fredholm’s third theorem. Let \(\omega(Q)\) and \(\sigma(Q)\) be solutions of the equations

\[ \omega(Q)-\chi_0\iint\limits_S B(Q,Q_1;\,0)\,\omega(Q_1)\,dS_{Q_1}=f(Q), \tag{46_1} \]

\[ \sigma(Q)-\chi_0\iint\limits_S A(Q,Q_1;\,0)\,\sigma(Q_1)\,dS_{Q_1}=f(Q), \tag{46_2} \]

then

\[ \varphi(Q)=\omega(Q)+\sigma(Q)-f(Q) \tag{47} \]

will be a solution of equation \((6_1)\) for \(\chi=\chi_0\). This is verified as follows: according to (45), equation \((6_1)\) takes the form

\[ \varphi(Q_0)-\chi_0\iint\limits_S [A(Q_0,Q;\,0)+B(Q_0,Q;\,0)]\,\varphi(Q)\,dS_Q=f(Q_0). \]

Substitute here the value from (47), replace in \((46_1)\) and \((46_2)\) \(Q_0\) by \(Q\) and \(Q\) by \(Q_1\), multiply the first on the left by \(A(Q_0,Q;\,0)\,dS_Q\), the second by \(B(Q_0,Q;\,0)\,dS_Q\), integrate over \(S\), and use the orthogonality formulas (40); then one is easily convinced of the validity of what is being proved.

Thus the question reduces to solving equations \((46_1)\) and \((46_2)\); but \((46_2)\) is an equation with singular kernel \(A(Q,Q_1;\,0)\) (see p. 716), and its resolvent is \(A(Q,Q_1;\,\chi)\) (see p. 717), therefore \(\chi=\chi_0\) is not a characteristic number for equation \((46_2)\), and its solution is found by Fredholm’s first theorem, already proved in item 4.

As for the vector \(\omega(Q_0)\), it can be constructed only in the case when \(f(Q_0)\) satisfies certain additional conditions. Indeed, equation \((46_1)\) is a Fredholm equation with continuous kernel \(B(Q,Q_1;\,\chi)\) (see p. 716); consequently, \(\chi=\chi_0\), as a pole of the resolvent, is a characteristic number of this equation, and for its solvability it is sufficient that the conditions

\[ \iint\limits_S f(Q)\,\tau_j(Q)\,dS_Q=0, \tag{48} \]

be fulfilled, where \(\tau_j(Q)\) \((j=1,\ 2,\ \ldots,\ l)\) are \(l\) linearly independent solutions of the adjoint homogeneous equation

\[ \tau(Q)-\chi_0\iint\limits_S B^*(Q_1,Q;\,0)\,\tau(Q_1)\,dS_{Q_1}=0. \tag{49} \]

We shall show that \(l=r\) and that the system of vectors \(\tau_j(Q)\) coincides with the complete system of fundamental solutions of the system of equations (43) adjoint to the given one. For this purpose let us turn to the equation adjoint to equation (45), in which \(Q_0\) and \(Q\) are interchanged:

\[ B^*(Q_0,Q;\,0)-\chi_0\iint\limits_S B^*(Q_1,Q;\,0)B^*(Q_0,Q_1;\,0)\,dS_{Q_1}=0. \]

Considering \(Q_0\) as a parameter and comparing with (49), we find

\[ \tau(Q)=B^{(k)*}(Q_0,Q;0)\quad (k=1,2,3). \]

But (see p. 719)

\[ B^*(Q_0,Q;0)=\frac{1}{\chi_0}\,B^*_{(1)}(Q_0,Q) \]

and, according to (44),

\[ B^{(k)*}_{(1)}(Q_0,Q)=\sum_{j=1}^{r}\varphi_j^{(k)}(Q_0)\psi_j(Q). \]

Consequently,

\[ \tau(Q)=-\frac{1}{\chi_0}\sum_{j=1}^{r}\varphi_j^{(k)}(Q_0)\psi_j(Q). \]

Thus, an arbitrary solution of equation (49) is represented as a linear combination of the linearly independent vectors \(\{\psi_j(Q)\}\) \((j=1,2,\ldots,r)\). This means that \(\{\psi_j(Q)\}_{j=1}^{3}\) constitute a complete system of fundamental solutions of equation (49); in this case the conditions (48) become the following:

\[ \iint_S f(Q)\psi_j(Q)\,dS=0\quad (j=1,2,\ldots,r), \tag{50} \]

which proves the sufficiency of conditions (50) for the solvability of \((6_1)\). The proof of necessity is elementary.

Fredholm’s third theorem is proved.

References

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2. Mikhlin S. G. Multidimensional Singular Integrals and Integral Equations. Moscow, Fizmatgiz, 1962.
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4. Mazya V. G., Sapozhnikova V. D. A note on the regularization of the singular system of isotropic elasticity theory. Vestnik of Leningrad University, No. 7, issue 2, 1964.
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Received by the editors
February 28, 1967

Tbilisi State University