# Singular integral equation

An equation containing the unknown function under the integral sign of an improper integral in the sense of Cauchy (cf. Cauchy integral). Depending on the dimension of the manifold over which the integrals are taken, one distinguishes one-dimensional and multi-dimensional singular integral equations. In comparison with the theory of Fredholm equations (cf. Fredholm equation), the theory of singular integral equations is more complex. For example, the theories of one-dimensional and multi-dimensional singular integral equations, both in the formulation of definitive results and in the methods used to establish them, differ significantly from one another. In the one-dimensional case, the theory is more fully developed, and its results are formulated more simply than the corresponding results in the multi-dimensional case. In what follows, main attention will be given to the one-dimensional case.

An important class of one-dimensional singular integral equations are those with a Cauchy kernel:

$$\tag{1 } a ( t) \phi ( t) + \frac{b ( t) }{\pi i } \int\limits _ \Gamma \frac{\phi ( \tau ) }{\tau - t } \ d \tau + \int\limits _ \Gamma k ( t, \tau ) \phi ( \tau ) \ d \tau = f ( t),$$

$$t \in \Gamma ,$$

where $a$, $b$, $k$, $f$ are known functions, $k$ is a Fredholm kernel (see Integral operator), $\phi$ is the desired function, $\Gamma$ is a planar curve, and the improper integral is to be understood as a Cauchy principal value, i.e.

$$\int\limits _ \Gamma \frac{\phi ( \tau ) }{\tau - t } \ d \tau = \ \lim\limits _ {\epsilon \rightarrow 0 } \ \int\limits _ {\Gamma _ \epsilon } \frac{\phi ( \tau ) }{\tau - t } \ d \tau ,\ t \in \Gamma ,$$

where $\Gamma _ \epsilon = \Gamma \setminus l _ \epsilon$, $l _ \epsilon$ being the arc $t ^ \prime tt ^ {\prime\prime}$ on $\Gamma$ such that $tt ^ \prime$ and $tt ^ {\prime\prime}$ are both of length $\epsilon$.

The operator $K$ defined by the left-hand side of (1) is called a singular operator (or sometimes a general singular operator):

$$\tag{2 } K = aI + bS + V,$$

where $I$ is the identity operator, $S$ is a singular integral operator (sometimes called a singular integral operator with Cauchy kernel), i.e.

$$( S \phi ) ( t) = \ { \frac{1}{\pi i } } \int\limits _ \Gamma \frac{\phi ( \tau ) }{\tau - t } \ d \tau ,\ \ t \in \Gamma ,$$

and $V$ is the integral operator with kernel $k ( t, \tau)$.

The operator $K _ {0} = aI + bS$ is called the characteristic part of the singular operator $K$, or the characteristic singular operator, and the equation

$$\tag{3 } a ( t) \phi ( t) + \frac{b ( t) }{\pi i } \int\limits _ \Gamma \frac{\phi ( t) }{\tau - t } \ d \tau = f ( t),\ \ t \in \Gamma ,$$

is called a characteristic singular integral equation, the functions $a$ and $b$ being the coefficients of the corresponding operator or equation.

The equation

$$a ( t) \psi ( t) - { \frac{1}{\pi i } } \int\limits _ \Gamma \frac{b ( \tau ) \psi ( \tau ) }{\tau - t } \ d \tau +$$

$$+ \int\limits _ \Gamma k ( \tau , t) \psi ( \tau ) d \tau = g ( t),\ t \in \Gamma ,$$

is called the adjoint of equation (1), and the operator $K ^ \prime = aI + SbI + V ^ \prime$ ($V ^ \prime$ being the integral operator with kernel $k ( \tau , t)$) is called the adjoint of $K$. In particular, $K _ {0} ^ \prime = aI + SbI$ is the adjoint of $K _ {0}$.

The operators $K$, $K _ {0}$, $K ^ \prime$, $K _ {0} ^ \prime$, or their corresponding equations, are said to be of normal type if the functions

$$A = a + b,\ \ B = a - b$$

do not vanish anywhere on $\Gamma$. In this case one also says that the coefficients of the operator or equation satisfy the normality condition.

Let $H _ \alpha ( \Gamma )$, $0 < \alpha \leq 1$, be the class of functions $\{ f \}$ defined on $\Gamma$ and satisfying the condition

$$| f ( t _ {1} ) - f ( t _ {2} ) | \leq \textrm{ const } | t _ {1} - t _ {2} | ^ \alpha ,$$

for all $t _ {1} , t _ {2} \in \Gamma$. If $f$ belongs to $H _ \alpha ( \Gamma )$ for some admissible value $\alpha$ and knowledge of the numerical value of $\alpha$ is not required, then one writes $f \in H ( \Gamma )$, or even $f \in H$ if it is clear from the context which contour $\Gamma$ is meant.

The set $H$ is called a Hölder class of functions, and if $f \in H$ one says that $f$ satisfies a Hölder condition or that $f$ is an $H$-function.

Let $G$ be a complex-valued continuous function that does not vanish on an oriented closed simple smooth contour $\Gamma$, and let

$$\tag{4 } \kappa = \ { \frac{1}{2 \pi } } [ \mathop{\rm arg} G ( t)] _ \Gamma ,$$

where $[ \cdot ] _ \Gamma$ denotes the increment of the function between brackets after a single circuit of $\Gamma$ in the positive direction. The integer $\kappa$ is called the index of the function $G$, $\kappa = \mathop{\rm ind} G$.

## Solution of the characteristic singular integral equation and its adjoint.

Let $\Gamma$ be a simple, closed, oriented, smooth contour on which the positive direction is chosen in such a way that it bounds a finite domain on the left, let the coordinate origin lie in this domain, let $a, b, f \in H ( \Gamma )$, and let $a$ and $b$ satisfy the normality condition. Further, let $\kappa$ be defined by (4), with

$$\tag{5 } G = \frac{a - b }{a + b } .$$

Then the following assertions hold.

1) If $\kappa \geq 0$, then the equation (3) is solvable in $H ( \Gamma )$ for any right-hand side $f \in H ( \Gamma )$, and all its $H$-solutions are given by the formula (see [1], [2])

$$\tag{6 } \phi ( f ) = \ a _ {*} ( t) f ( t) - \frac{b _ {*} ( t) \omega ( t) }{\pi i } \int\limits _ \Gamma \frac{f ( \tau ) }{\omega ( \tau ) ( \tau - t) } \ d \tau +$$

$$+ b _ {*} ( t) \omega ( t) p _ {\kappa - 1 } ( t),$$

where

$$a _ {*} = \ { \frac{a}{a ^ {2} - b ^ {2} } } ,\ \ b _ {*} = \ { \frac{b}{a ^ {2} - b ^ {2} } } ,$$

$$\omega ( t) = t ^ {- {\kappa / 2 } } \sqrt {a ^ {2} ( t) - b ^ {2} ( t) } \mathop{\rm exp} \left [ { \frac{1}{2 \pi i } } \int\limits _ \Gamma \frac{ \mathop{\rm ln} [ \tau ^ {- \kappa } G ( \tau )] }{\tau - t } d \tau \right ] ,$$

and $p _ {\kappa - 1 }$ is an arbitrary polynomial of degree $\kappa - 1$ ($p _ {- 1} = 0$). If $\kappa < 0$, then equation (3) is solvable in $H ( \Gamma )$ if and only if $f$ satisfies the condition

$$\int\limits _ \Gamma \frac{t ^ {k} }{\omega ( t) } f ( t) dt = 0,\ \ k = 0, \dots, - \kappa - 1.$$

When these conditions hold, (3) has a unique $H$-solution, given by the formula (6) with $p _ {\kappa - 1 } = 0$.

2) The singular integral equation adjoint to (3),

$$\tag{7 } a ( t) \psi ( t) - { \frac{1}{\pi i } } \int\limits _ \Gamma \frac{b ( \tau ) \psi ( \tau ) }{\tau - t } \ d \tau = g ( t),\ \ t \in \Gamma ,$$

is solvable in $H$ for any $g \in H ( \Gamma )$ if $\kappa \leq 0$, and all its $H$-solutions are given by the formula

$$\tag{8 } \psi ( t) = a _ {*} ( t) g ( t) +$$

$$+ { \frac{1}{\pi i \omega ( t) } } \int\limits _ \Gamma \frac{\omega ( \tau ) b _ {*} ( \tau ) g ( \tau ) }{\tau - t } d \tau + \frac{p _ {- \kappa - 1 } }{\omega ( t) } .$$

But if $\kappa > 0$, equation (7) is solvable if and only if $g$ satisfies the $\kappa$ conditions:

$$\int\limits _ \Gamma t ^ {k} b ( t) \omega ( t) g ( t) dt = 0,\ \ k = 0, \dots, \kappa - 1,$$

and if these conditions hold, the solution is given by (8) with $p _ {- \kappa - 1 } = 0$.

Noether's theorems. Let $\nu$ and $\nu ^ \prime$ be the numbers of linearly independent solutions of the homogeneous equations $K _ {0} \phi = 0$ and $K _ {0} ^ \prime \psi = 0$, respectively. Then the difference $\nu - \nu ^ \prime$ is called the index of the operator $K _ {0}$ or of the equation $K _ {0} \phi = 0$:

$$\mathop{\rm ind} K = \nu - \nu ^ \prime .$$

### Theorem 1.

The homogeneous singular integral equations $K _ {0} \phi = 0$ and $K _ {0} ^ \prime \psi = 0$ have a finite number of linearly independent solutions.

### Theorem 2.

Necessary and sufficient conditions for the solvability of the non-homogeneous equation (3) are:

$$\int\limits _ \Gamma f ( t) \psi _ {j} ( t) \ dt = 0,\ \ j = 1, \dots, \nu ^ \prime ,$$

where $\psi _ {1}, \dots, \psi _ {\nu ^ \prime }$ is a complete set of linearly independent solutions of the adjoint homogeneous equation $K _ {0} ^ \prime \psi = 0$.

### Theorem 3.

The index of $K _ {0}$ (cf. Index of an operator) is equal to the index of the function $G$ defined by equation (5), i.e.

$$\tag{9 } \mathop{\rm ind} K _ {0} = \ { \frac{1}{2 \pi } } \left [ \mathop{\rm arg} \frac{a - b }{a + b } \right ] _ \Gamma .$$

These theorems remain valid in the case of the general singular integral equation (1), that is, in these theorems $K _ {0}$, $K _ {0} ^ \prime$ can be replaced by $K$, $K ^ \prime$, respectively. It is only necessary to bear in mind that, in the case of general singular integral equations, $\nu$ and $\nu ^ \prime$ are both non-zero in general, in contrast to the case of characteristic singular integral equations, when one of them must be zero.

Theorems 1–3 are called after F. Noether, who first proved them [9] in the case of a one-dimensional singular integral equation with Hilbert kernel:

$$\tag{10 } a ( s) \phi ( s) + \frac{b ( s) }{2 \pi } \int\limits _ {- \pi } ^ \pi \phi ( t) \mathop{\rm cot} \ { \frac{t - s }{2} } dt +$$

$$+ \int\limits _ {- \pi } ^ \pi k ( s, t) \phi ( t) dt = f ( s),\ - \pi \leq s \leq \pi .$$

These theorems are analogous to the Fredholm theorems (see Fredholm equation) and differ from them only in that the numbers of linearly independent solutions of the homogeneous equation and its adjoint are in general distinct, that is, whereas the index of a Fredholm equation is always equal to zero, a singular integral equation can have non-zero index.

Like the Noether theorems, the formulas (6) and (8) remain valid in the case when $\Gamma = \cup \Gamma _ {k}$ consists of a finite number of smooth mutually-disjoint closed contours. In this case the symbol $[ \cdot ] _ \Gamma$ in (4) denotes the sum of the increments of the function between brackets after a circuit of each of the contours $\Gamma _ {k}$.

The case when $\Gamma$ is a finite union of smooth mutually-disjoint open contours requires special consideration. If $\phi$ is an $H$-function inside any closed part of every $\Gamma _ {k}$ not containing the end points, and if close to either end $c$ it can be written in the form $\phi ( t) = \phi _ {*} ( t) | t - c | ^ {- \alpha }$, $0 \leq \alpha = \textrm{ const } < 1$, where $\phi _ {*}$ is an $H$-function in a neighbourhood of $c$ containing $c$, then one says that $\phi$ belongs to the class $H ^ {*}$. If $a, b \in H$, $g, g \in H ^ {*}$ and in $H ^ {*}$ one looks for solutions of the equations (3), (7), then one can define $\kappa$ and $\phi$ in such a way that (6) and (8) remain valid. Furthermore, if one defines in a corresponding way subclasses of $H ^ {*}$ in which one looks for solutions of a given singular integral equation and its adjoint, then the Noether theorems also remain valid (see [1]).

The above results can be extended in various ways. It can be shown (see [1]) that under certain conditions they also remain valid in the case of a piecewise-smooth contour $\Gamma$ (that is, when $\Gamma$ is the union of a finite number of smooth open arcs, which are mutually-disjoint except for their end points). Singular integral equations can also be studied in the Lebesgue function spaces $L _ {p} ( \Gamma )$ and $L _ {p} ( \Gamma , \rho )$, where $p > 1$ and $\rho$ is a certain weight (see [4][7]). [4][6] contain results which directly extend those stated above.

Let $\Gamma$ be a simple rectifiable contour with equation $t = t ( s)$, $0 \leq s \leq \gamma$, where $s$ is the arc-length on $\Gamma$ starting from some fixed point and $\gamma$ is the length of $\Gamma$. One says that a function $f$ defined on $\Gamma$ is almost-everywhere finite, measurable, integrable, etc., if the function $f ( t ( s))$ has the corresponding property on the interval $[ 0, \gamma ]$. The Lebesgue integral of $f$ on $\Gamma$ is defined by

$$\int\limits _ \Gamma f ( t) dt = \ \int\limits _ { 0 } ^ \gamma f ( t ( s)) t ^ \prime ( s) ds.$$

Let $L _ {p} ( \Gamma )$ denote the set of measurable functions on $\Gamma$ such that $| f | ^ {p}$ is integrable on $\Gamma$. The function class $L _ {p} ( \Gamma )$, $p \geq 1$, becomes a Banach space if one introduces the norm by

$$\| f \| = \ \left ( \int\limits _ \Gamma | f | ^ {p} ds \right ) ^ {1/p} .$$

If in equations (3), (7) the equalities hold almost-everywhere, with continuous coefficients $a$ and $b$ satisfying the normality condition and $f, g \in L _ {p} ( \Gamma )$, $p > 1$, then 1) and 2) remain valid upon replacing $H$ by $L _ {p} ( \Gamma )$, $p > 1$. Furthermore, if the solutions of $K \phi = f$, where $K$ has the form (2), are sought in $L _ {p} ( \Gamma )$, $p > 1$, and the solutions of its homogeneous adjoint $K ^ \prime \psi = 0$ are sought in $L _ {p ^ \prime } ( \Gamma )$, where $p ^ \prime = p/( p - 1)$, then the Noether theorems also remain valid and $V$ may be any completely-continuous operator on $L _ {p} ( \Gamma )$.

When $\Gamma$ is a finite union of open contours, or if $\Gamma$ is closed but the coefficients of the singular integral equation are not continuous, then solutions of the equations can often be found in weighted function spaces $L _ {p} ( \Gamma , \rho )$, $p > 1$ ($f \in L _ {p} ( \Gamma , \rho ) \iff \rho f \in L _ {p} ( \Gamma )$). Under specific conditions on the weight function $\rho$, results analogous to the above are valid.

## The regularization problem.

One of the basic problems in the theory of singular integral equations is the regularization problem, that is, the problem of reducing a singular integral equation to a Fredholm equation.

Let $E$ and $E _ {1}$ be Banach spaces, which may coincide, and let $A: E \rightarrow E _ {1}$ be a bounded linear operator. A bounded operator $B$ is called a left regularizer of $A$ if $BA = I + V$, where $I$, $V$ are the identity and a completely-continuous operator on $E$, respectively. If the equations $A \phi = f$ and $BA \phi = Bf$ are equivalent for each $f \in E _ {1}$, then $B$ is called a left equivalent regularizer of $A$. A bounded operator $B$ is called a right regularizer of $A$ if $AB = I _ {1} + V _ {1}$, where $I _ {1}$, $V _ {1}$ are the identity and a completely-continuous operator on $E _ {1}$, respectively. If the equations $A \phi = f$ and $AB \psi = f$ are simultaneously solvable or unsolvable as $f$ ranges over $E _ {1}$, and in the case of solvability the relation $\phi = B \psi$ holds between their solutions, then $B$ is called a right equivalent regularizer of $A$. If $B$ is simultaneously a left and right regularizer of $A$, then it is called a two-sided regularizer, or simply a regularizer of $A$. One says that $A$ admits left, right, two-sided, equivalent, regularization if it has a left, right, two-sided, or equivalent regularizer, respectively.

Let $K$ be the operator defined by (2), where $\Gamma$ is a closed simple smooth contour, $a$ and $b$ are $H$-functions (or continuous functions) satisfying the normality condition and $V$ is a completely-continuous operator on $L _ {p} ( \Gamma )$, $p > 1$. Then $K$ has an uncountable set of regularizers on $L _ {p} ( \Gamma )$, e.g. one of which is the operator

$$M = \ { \frac{a}{a ^ {2} - b ^ {2} } } I - { \frac{b}{a ^ {2} - b ^ {2} } } S.$$

For $K$ to admit left equivalent regularization, it is necessary and sufficient that its index $\kappa$ is non-negative [7]. One can take $M$ to be an equivalent left regularizer. If $\kappa < 0$, then $K$ admits right equivalent regularization, which can be realized using $M$ (see [1]).

## Systems of singular integral equations.

If in (1) $a$, $b$ and $k$ are square matrices of order $n$, regarded as matrices of linear transformations of an unknown vector $\phi = ( \phi _ {1}, \dots, \phi _ {n} )$, and $f = ( f _ {1}, \dots, f _ {n} )$ is a known vector, then (1) is called a system of singular integral equations. It is said to be of normal type if the matrices $A = a + b$ and $B = a - b$ are non-singular on $\Gamma$, that is, $\mathop{\rm det} A \neq 0$ and $\mathop{\rm det} B \neq 0$ for all $t \in \Gamma$.

The Noether theorems remain valid for a system of singular integral equations in the class $H$ (see [1], [3]), and can be extended to the case of Lebesgue function spaces (see [4], [5]). In contrast to the case of a single equation, a characteristic system of singular integral equations cannot, in general, be solved by quadratures, although there is a formula similar to (9) for the index (see [1]):

$$\mathop{\rm ind} K = \ { \frac{1}{2 \pi } } [ \mathop{\rm arg} \mathop{\rm det} \ A ^ {- 1} B] _ \Gamma .$$

For a system of singular integral equations, regularization problems (see [3]) are similar to those for a single equation.

There have been several investigations of both one singular integral equation and a system of such equations when the normality condition is violated (see [11] and the bibliography contained therein).

## Multi-dimensional singular integral equations.

These are equations of the form

$$\tag{11 } a ( t) \phi ( t) + \int\limits _ \Gamma \frac{g ( t, \theta ) }{r ^ {m} } \phi ( \tau ) d \tau + ( V \phi ) ( t) = \ f ( t),\ \ t \in \Gamma ,$$

where $\Gamma$ is a domain in the Euclidean space $E _ {m}$, $m > 1$. $\Gamma$ may be bounded or unbounded, and can, in particular cases, coincide with $E _ {m}$; $t$ and $\tau$ are points of $E _ {m}$, $r = | t - \tau |$, $\theta = ( \tau - t)/r$, $d \tau$ is the volume element in $E _ {m}$, and $V$ is a completely-continuous operator on the Banach function space in which the solution is sought. Further, $a$ and $g$ are given functions and the improper singular integral is understood in the principle value sense, that is,

$$\tag{12 } \int\limits _ \Gamma \phi ( \tau ) \frac{g ( t, \theta ) }{r ^ {m} } \ d \tau = \ \lim\limits _ {\epsilon \rightarrow 0 } \ \int\limits _ {\Gamma \setminus \{ r < \epsilon \} } \phi ( \tau ) \frac{g ( t, \theta ) }{r ^ {m} } \ d \tau .$$

Here $t$ is called the pole, $g ( t, \theta )$ the characteristic and $\phi$ the density of the singular integral (12). As a rule, the limit in (12) does not exist when the following condition is violated:

$$\tag{13 } \int\limits _ \sigma g ( t, \theta ) d \sigma = 0,$$

where $\sigma$ is the unit sphere with centre at the origin. Thus it is assumed that (13) always holds.

In the theory of multi-dimensional singular integral equations, an important role is played by the notion of a symbol (cf. Symbol of an operator). It is defined in terms of the functions $a$ and $g$, and from a given symbol the original singular operator can be recovered up to a completely-continuous term. Composition of singular operators corresponds to multiplication of their symbols. It has been shown [7] that under certain restrictions (11) admits a regularization in the space $L _ {p}$, $p > 1$, if and only if the absolute value of its symbol has a positive lower bound, and in this case the Fredholm theorems hold.

## Historical survey.

The study of one-dimensional singular integral equations originated in the works of D. Hilbert and H. Poincaré at almost the same time as the formulation of the theory of Fredholm equations (cf. Fredholm equation). A special case, a singular integral equation with Cauchy kernel, was considered much earlier in the doctoral thesis of Yu.V. Sokhotskii, published in St. Petersburg in 1873; however, this research remained unnoticed.

Basic results on the formulation of a general theory of the equations (1) and (10) were obtained at the beginning of the 1920s by Noether [9] and T. Carleman [10]. Noether first introduced the concept of an index and proved the theorems 1–3 above by applying the method of left regularization. This method was first described (in various special cases) by Poincaré and Hilbert, but its general form is due to Noether. A crucial point in the realization of the above method involves the application of a permutation (composition) formula for repeated singular integrals in the Cauchy principal value sense (the Poincaré–Bertrand formula). For certain special classes of equations (3), Carleman had the basic idea behind a method for reducing this equation to the following boundary value problem in the theory of analytic functions (the linear conjugacy problem, see [1] and Riemann–Hilbert problem (analytic functions)):

$$\Phi ^ {+} ( t) = G ( t) \Phi ^ {-} ( t) + g ( t),\ \ t \in \Gamma ,$$

and found a way of constructing an explicit solution. Carleman and I.N. Vekua found a method of regularizing equation (1) involving a solution of the characteristic equation (3).

The great significance, both theoretical and practical, of singular integral equations became especially apparent towards the end of the 1930s in connection with the solution of certain very important problems in the mechanics of a solid medium (the theory of elasticity, hydro- and aeromechanics, and others) and theoretical physics. The theory of one-dimensional singular integral equations was significantly advanced in the 1940s and reached a final form (in a definite sense) in the works of Soviet mathematicians. A presentation of this theory in Hölder classes of functions is to be found in a monograph of one of its creators, N.I. Muskhelishvili (see [1]). This monograph also stimulated scientific investigations in certain other directions, for example in the theory of singular integral equations not satisfying the Hausdorff normality condition, singular integral equations with non-diagonal singularities (with displacements), Wiener–Hopf equations, multi-dimensional singular integral equations, etc.

The earliest studies of multi-dimensional singular integral equations were carried out in 1928 by F. Tricomi, who established a permutation formula for two-dimensional singular integrals and applied it to the solution of a class of singular integral equations. In this direction, the fundamental work was done in 1934 by G. Giraud, who proved the validity of the Fredholm theorems for certain classes of multi-dimensional singular integral equations on Lyapunov manifolds.

#### References

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