# Potential theory

Originally, studies related to the properties of forces which follow the law of gravitation. In the statement of this law given by I. Newton (1687) (cf. Newton laws of mechanics) the only forces considered are the forces of mutual attraction acting upon two material particles of small size or two material points. These forces are directly proportional to the product of the masses of these particles and inversely proportional to the square of the distance between them. Thus, the first and the most important problem from the point of view of celestial mechanics and geodesy was to study the forces of attraction of a material point by a finite smooth material body — a spheroid and, in particular, an ellipsoid (since many celestial bodies have this shape). After first partial achievements by Newton and others, studies carried out by J.L. Lagrange (1773), A. Legendre (1784–1794) and P.S. Laplace (1782–1799) became of major importance. Lagrange has established that a field of gravitational forces, as it is called now, is a potential field and has introduced a function which was later called by G. Green (1828) a potential function and by C.F. Gauss (1840) — just a potential. At present, the achievements of this initial period are included in courses on classical celestial mechanics (see also [2]).

Already Gauss and his contemporaries discovered that the method of potentials (cf. Potentials, method of) can be applied not only to solve problems in the theory of gravitation but, in general, to solve a wide range of problems in mathematical physics, in particular in electrostatics and magnetism. In this connection, potentials became to be considered not only for the physically realistic problems concerning the mutual attraction of positive masses, but also for problems with "masses" of arbitrary sign, or charges. The principal boundary value problems were defined, such as the Dirichlet problem and the Neumann problem, the electrostatic problem of the static distribution of charges on conductors or the Robin problem, and the problem of sweeping-out mass (see Balayage method). To solve the above-mentioned problems in the case of domains with sufficiently smooth boundaries certain types of potentials turned out to be efficient, i.e. special classes of parameter-dependent integrals such as volume potentials of distributed mass, single- and double-layer potentials, logarithmic potentials, Green potentials, etc. Results obtained by A.M. Lyapunov and V.A. Steklov at the end of the 19th century were fundamental for the creation of strong methods of solution of the principal boundary value problems. Studies in potential theory concerning properties of different potentials have acquired an independent significance.

In the first half of the 20th century, a great stimulus for the generalization of the principal problems and the completion of the existing formulations in potential theory was made on the basis of the general notions of a Radon measure, a capacity and generalized functions. Modern potential theory is closely related in its development to the theory of analytic, harmonic and subharmonic functions and to probability theory.

Together with further profound studies of classical boundary value problems and inverse problems (see Potential theory, inverse problems in) the modern period in the development of potential theory is characterized by the application of methods and notions of topology and functional analysis, and the use of abstract axiomatic methods (see Potential theory, abstract).

## Principal classes of potentials and their properties.

Let $S$ be a smooth closed surface, i.e. an $( n - 1 )$-dimensional smooth manifold without boundary, in the $n$-dimensional Euclidean space $\mathbf R ^ {n}$, $n > 2$, which bounds a bounded domain $G = G ^ {+}$, $\partial G = S$. Let $G ^ {-} = \mathbf R ^ {n} \setminus ( G ^ {+} \cup S )$ be the exterior unbounded domain. Let

$$E ( x , y ) = E ( | x - y | ) = \ \left \{ \begin{array}{ll} \frac{1}{\omega _ {n} ( n - 2 ) } \frac{1}{| x - y | ^ {n- 2} } , & n \geq 3 , \\ \frac{1}{2 \pi } \mathop{\rm ln} \frac{1}{| x - y | } , & n = 2 , \\ \end{array} \right .$$

be a principal fundamental solution of the Laplace equation $\Delta u \equiv \sum _ {k= 1} ^ {n} \partial ^ {2} u / \partial x _ {k} ^ {2} = 0$ in $\mathbf R ^ {n}$, where

$$| x - y | = \left [ \sum _ { k= 1} ^ { n } ( x _ {k} - y _ {k} ) ^ {2} \right ] ^ {1/2}$$

is the distance between two points $x = ( x _ {1}, \dots, x _ {n} )$ and $y = ( y _ {1}, \dots, y _ {n} )$ in $\mathbf R ^ {n}$, $\omega _ {n} = 2 \pi ^ {n/2} / \Gamma ( n / 2 )$ is the area of the unit sphere in $\mathbf R ^ {n}$ and $\Gamma$ is the gamma-function. The following three integrals, which depend on $x$ as a parameter,

$$\tag{1 } \left . \begin{array}{c} Z ( x) = \int\limits _ { G } \rho ( y) E ( x , y ) dy , \\ V ( x) = \int\limits _ { S } \mu ( y) E ( x , y ) dS ( y) , \\ W ( x) = \int\limits _ { S } \nu ( y) \frac \partial {\partial n _ {y} } E ( x , y ) \ dS ( y) , \end{array} \right \}$$

where $n _ {y}$ is the direction of the exterior (with respect to $G ^ {+}$) normal to $S$ at a point $y \in S$, are called the volume potential, the single-layer potential and the double-layer potential, respectively. The functions $\rho ( y)$, $\mu ( y)$ and $\nu ( y)$ are called the densities of the corresponding potentials; hereafter they are assumed to be absolutely integrable over $G$ or $S$, respectively. For $n = 3$ (and sometimes for $n \geq 3$) the integrals (1) are called the Newton volume potential and the Newton single- and double-layer potentials; for $n = 2$ they are called logarithmic mass, single-layer or double-layer potentials, respectively. Let $\rho$ be of class $C ^ {1} ( G \cup S )$. Then the volume potential (cf. Newton potential) and its first derivatives are continuous everywhere on $\mathbf R ^ {n}$; moreover, they can be calculated by differentiation under the integral sign, i.e. $Z \in C ^ {1} ( \mathbf R ^ {n} )$. Further,

$$\lim\limits _ {| x | \rightarrow \infty } \frac{Z ( x) }{E ( x , 0 ) } = \ M ,\ M = \int\limits _ { G } \rho ( y) dy .$$

The second derivatives are continuous everywhere outside $S$, but they have a discontinuity when passing across the surface $S$; moreover, in $G ^ {+}$ they satisfy the Poisson equation $- \Delta Z = \rho ( x)$, $x \in G ^ {+}$, and in $G ^ {-}$ — the Laplace equation $\Delta Z = 0$, $x \in G ^ {-}$. The above-mentioned properties characterize a volume potential.

If $G _ {1}$ is a bounded domain in $\mathbf R ^ {n}$ with boundary $S _ {1} = \partial G _ {1}$ of class $C ^ {1}$, then Gauss' formula for a volume potential is valid:

$$\int\limits _ {S _ {1} } \frac{\partial Z }{\partial n _ {x} } \ d S _ {1} ( x) = - \int\limits _ {G \cap G _ {1} } \rho ( y) dy .$$

Let $\mu \in C ^ {1} ( S)$. The single-layer potential (cf. Simple-layer potential) $V ( x)$ is a harmonic function when $x \notin S$; moreover,

$$\lim\limits _ {| x| \rightarrow \infty } \frac{V ( x) }{E ( x , 0 ) } = \ M ,\ M = \int\limits _ { S } \mu ( y) dS ( y) ;$$

in particular, $\lim\limits _ {| x| \rightarrow \infty } V ( x) = 0$ for $n \geq 3$, but $\lim\limits _ {| x| \rightarrow \infty } V ( x) = 0$ when $n = 2$ if and only if $\int _ {S} \mu ( y) dS ( y) = 0$. A single-layer potential is continuous everywhere on $\mathbf R ^ {n}$, $V \in C ( \mathbf R ^ {n} )$, moreover, $V ( x)$ and its tangential derivatives are continuous when passing across the surface $S$. The normal derivative of a single-layer potential has a discontinuity when passing across the surface $S$:

$$\left ( \frac{\partial V }{\partial n _ {x} } \right ) ^ {+} = \frac{1}{2} \mu ( x) + \frac{\partial V ( x) }{\partial n _ {x} } ,$$

$$\left ( \frac{\partial V }{\partial n _ {x} } \right ) ^ {-} = - \frac{1}{2} \mu ( x) + \frac{\partial V ( x) }{\partial n _ {x} } ,\ x \in S ,$$

where $( \partial V / \partial n _ {x} ) ^ {+}$ and $( \partial V / \partial n _ {x} ) ^ {-}$ are the limit values of the normal derivative from $G ^ {+}$ and $G ^ {-}$, respectively, i.e.

$$\left ( \frac{\partial V }{\partial n _ {x} } \right ) ^ {+} = \lim\limits _ {\begin{array}{c} x ^ \prime \rightarrow x \\ x ^ \prime \in G ^ {+} \end{array} } \ \frac{\partial V ( x ^ \prime ) }{\partial n _ {x} } ,$$

$$\left ( \frac{\partial V }{\partial n _ {x} } \right ) ^ {-} = \lim\limits _ {\begin{array}{c} x ^ \prime \rightarrow x \\ x ^ \prime \in G ^ {-} \end{array} } \ \frac{\partial V ( x ^ \prime ) }{\partial n _ {x} } .$$

$\partial V ( x) / \partial n _ {x}$ denotes the so-called direct value of the normal derivative of a single-layer potential calculated over the surface $S$, i.e.

$$\frac{\partial V ( x) }{\partial n _ {x} } = \int\limits _ { S } \mu ( y) \frac \partial {\partial n _ {x} } E ( x , y ) dS ( y) ,\ x \in S .$$

It is a continuous function of the points $x \in S$, and the kernel $\partial E ( x , y ) / \partial n _ {x}$ has a weak singularity on $S$,

$$\left | \frac \partial {\partial n _ {x} } E ( x , y ) \right | \leq \ \frac{\textrm{ const } }{| x - y | ^ {n- 2} } ,\ \ x , y \in S .$$

These properties characterize a single-layer potential.

Let $\nu \in C ^ {1} ( S)$. The double-layer potential $W ( x)$ is a harmonic function for $x \notin S$; moreover,

$$\lim\limits _ {| x| \rightarrow \infty } \omega _ {n} | x | ^ {n- 1} W ( x) = \ M ,\ M = \int\limits _ { S } \nu ( y) dS ( y) .$$

When passing across the surface $S$ the double-layer potential has a discontinuity (whence its name):

$$W ^ {+} ( x) = - \frac{1}{2} \nu + W ( x) ,\ W ^ {-} ( x) = \frac{1}{2} \nu ( x) + W ( x) ,\ x \in S ,$$

where $W ^ {+} ( x)$ and $W ^ {-} ( x)$ are the limit values of the double-layer potential from $G ^ {+}$ and $G ^ {-}$, respectively, that is,

$$W ^ {+} ( x) = \lim\limits _ {\begin{array}{c} x ^ \prime \rightarrow x \\ x ^ \prime \in G ^ {+} \end{array} } \ W ( x ^ \prime ) ,\ W ^ {-} ( x) = \lim\limits _ {\begin{array}{c} x ^ \prime \rightarrow x \\ x ^ \prime \in G ^ {-} \end{array} } W ( x ^ \prime ) .$$

$W ( x)$ when $x \in S$ denotes the so-called direct value of the double-layer potential calculated over the surface $S$, that is,

$$W ( x) = \int\limits _ { S } \nu ( y) \frac \partial {\partial n _ {y} } E ( x , y ) dS ( y) ,\ x \in S .$$

It is a continuous function of the points $x \in S$, and the kernel $\partial E ( x , y ) / \partial n _ {y}$ has a weak singularity on $S$,

$$\left | \frac \partial {\partial n _ {y} } E ( x , y ) \right | \leq \ \frac{\textrm{ const } }{| x - y | ^ {n- 2} } ,\ \ x , y \in S .$$

The tangential derivatives of a double-layer potential also have a discontinuity when passing across the surface $S$, but the normal derivative $\partial W ( x) / \partial n _ {x}$ retains its value when passing across $S$:

$$\left ( \frac{\partial W }{\partial n _ {x} } \right ) ^ {+} = \ \left ( \frac{\partial W }{\partial n _ {x} } \right ) ^ {-} ,\ \ x \in S .$$

These properties characterize a double-layer potential.

In the case of a constant density $\nu = 1$ Gauss' formula for a double-layer potential holds:

$$- \int\limits _ { S } \frac \partial {\partial n _ {x} } E ( x , y ) \ d S ( y) = q ( x) = \left \{ \begin{array}{ll} 1 , &x \in G ^ {+} , \\ \frac{1}{2} , &x \in S , \\ 0, &x \in G ^ {-} . \\ \end{array} \right .$$

The integral at the left-hand side of this equality is interpreted (when divided by $\omega _ {n} ( n- 2)$) as the solid angle at which the surface $S$ is seen from the point $x$.

Below, certain properties of potentials under weaker restrictions on the densities and the surface $S$ are given.

If $\rho \in L _ {1} ( G)$, then $Z ( x)$ is a harmonic function for $x \in G ^ {-}$ and $Z ( x)$ is summable on $G ^ {+}$. If $\rho \in L _ {p}$, $1 \leq p \leq n / 2$, then $Z \in L _ {q} ( \mathbf R ^ {n} )$, $1 / p + 1 / q = 1$, $1 < q < n p / ( n - 2 p )$; if $\rho \in L _ {p} ( G)$, $p > n / 2$, then $Z \in C ( \mathbf R ^ {n} )$. If $\rho \in L _ {p} ( G)$, $1 \leq p \leq n$, then $Z \in W _ {q} ^ {1} ( \mathbf R ^ {n} )$, $1 < q < n p / ( n - p )$; if $\rho \in L _ {p} ( G)$, $p > n$, then $Z \in C ^ {1} ( \mathbf R ^ {n} )$. If $\rho \in L _ {2} ( G)$, then the generalized second derivatives of $Z ( x)$ exist, they are also of class $L _ {2} ( G)$ and are expressed by singular integrals:

$$\frac{\partial ^ {2} Z }{\partial x _ {i} \partial x _ {j} } = - \frac{1}{n} \delta _ {ij} \rho ( x) + \int\limits _ { G } \rho ( y) \frac{\partial ^ {2} }{\partial x _ {i} \partial x _ {j} } E ( x , y ) dy ,$$

$$i , j = 1, \dots, n ,$$

where $\delta _ {ij} = 1$ for $i = j$, $\delta _ {ij} = 0$ for $i \neq j$; if $\rho \in L _ {p} ( G)$, $1 < p < \infty$, then all generalized derivatives $\partial ^ {2} Z / \partial x _ {i} \partial x _ {j}$ also exist and belong to $L _ {p} ( \mathbf R ^ {n} )$. If $\rho \in L _ {p} ( G)$, $1 \leq p < + \infty$, then $Z ( x)$ is a generalized solution of the Poisson equation $- \Delta Z = \rho ( x)$, $x \in G$. If $\rho \in C ^ {( 0 , \alpha ) } ( G)$ and $S \in C ^ {( 1 , \alpha ) }$, $0 < \alpha < 1$, then $Z \in C ^ {( 2 , \alpha ) }$ in $G ^ {+}$ or $G ^ {-}$. If $\rho \in C ^ {( l , \alpha ) } ( G)$ and $S \in C ^ {( k + 1 , \alpha ) }$, $0 < \alpha < 1$, $l , k$ integers, $0 \leq l \leq k$, then $Z \in C ^ {( l + 2 , \alpha ) } ( G ^ {+} )$.

Let $S \in C ^ {( 1 , \alpha ) }$, $0 < \alpha < 1$, let $\overline{D}\;$ be a closed bounded domain such that $G ^ {+} \cup S \subset D \subset \overline{D}\; \subset \mathbf R ^ {n}$. If $\mu \in L _ {p} ( S)$, $p = 1 , 2$, then $V \in L _ {p} ( \overline{D}\; )$, $V \in L _ {p} ( S)$, $\partial V / \partial x _ {i} \in L _ {p} ( \overline{D}\; )$, $p = 1 , 2$; $i = 1, \dots, n$. If the density is bounded and summable, then

$$V \in C ^ {( 0 , \lambda ) } \ \ \textrm{ for all } \ \lambda \in ( 0 , 1 ) .$$

If $\mu \in C ^ {( 0 , \alpha ) } ( S)$, $0 < \alpha < 1$, then $V \in C ^ {( 1 , \alpha ) }$ in $G ^ {+}$ or $G ^ {-}$. If $\nu \in C ^ {( 0 , \alpha ) } ( S)$, then $W \in C ^ {( 0 , \alpha ) }$ in $G ^ {+}$ or $G ^ {-}$.

If $\mu \in C ^ {( l , \alpha ) } ( S)$ and $S \in C ^ {( k + 1 , \alpha ) }$, $0 < \alpha < 1$, $l , k$ integers, $0 \leq l \leq k$, then $V \in C ^ {( l + 1 , \alpha ) }$ in $G ^ {+}$ or $G ^ {-}$. If $\nu \in C ^ {( l , \alpha ) } ( S)$ and $S \in C ^ {( k + 1 , \alpha ) }$, $0 < \alpha < 1$, $l , k$ integers, $0 \leq l \leq k + 1$, then $W \in C ^ {( l , \alpha ) }$ in $G ^ {+}$ or $G ^ {-}$.

For potentials and their derivatives extended by continuity on $S$ the above-described properties of smoothness are also valid under the corresponding smoothness conditions on the density and the surface $S$.

## Representation of functions and solution of the principal boundary value problems in potential theory using potentials.

Let $\Phi ( x)$ be a function of class $C ^ {2} ( G \cup S )$ and let $S$ be a smooth surface of class $C ^ {2}$. Then the following integral identity (Green formula) holds:

$$\tag{2 } - \int\limits _ { G } \Delta \Phi ( y) E ( x , y ) \ d y +$$

$$+ \int\limits _ { S } \left ( \frac{\partial \Phi ( y) }{\partial n _ {y} } E ( x , y ) - \Phi ( y) \frac{\partial E ( x , y ) }{\partial n _ {y} } \right ) d S ( y) = q ( x) \Phi ( x) .$$

In particular, in $G$ the function $\Phi ( x)$ can be represented as the sum of a volume potential and single- and double-layer potentials, with respective densities

$$\rho ( y) = - \Delta \Phi ( y) ,\ \ \mu ( y) = \frac{\partial \Phi ( y) }{\partial n _ {y} } ,\ \ \nu ( y) = - \Phi ( y) .$$

For a function $u ( x)$ of class $C ^ {1} ( G \cup S )$ that is harmonic on $G$ the following identity holds:

$$\tag{3 } \int\limits _ { S } \left ( \frac{\partial u ( y) }{\partial n _ {y} } E ( x , y ) - u ( y) \frac{\partial E ( x , y ) }{\partial n _ {y} } \right ) \ d S ( y) = q ( x) u ( x) .$$

Hence, such a function $u ( x)$ can be represented in $G$ by the sum of single- and double-layer potentials with densities $\mu ( y) = \partial u ( y) / \partial n _ {y}$, $\nu ( y) = - u ( y)$, respectively. However, the densities in (3) cannot be arbitrarily given on $S$; they are connected by the integral relation obtained from (3) for $x \in G ^ {-}$.

A central place in potential theory is occupied by the Dirichlet and the Neumann boundary value problem (also called the first and the second boundary value problem (cf. also Dirichlet problem; Neumann problem)) for the domains $G ^ {+}$ (interior problems) and $G ^ {-}$ (exterior problems) which, under the assumption of sufficient smoothness, can be completely studied by reducing them to the integral equations of potential theory.

The interior Dirichlet problem: Find a function $u ( x)$ of class $C ( G ^ {+} \cup S )$, $S \in C ^ {( 1 , \alpha ) }$, $0 < \alpha < 1$, harmonic in $G ^ {+}$, which satisfies the boundary condition $u ( x) = \phi ^ {+} ( x)$, $x \in S$, where $\phi ^ {+} ( x)$ is a given continuous function on $S$. The solution to this problem always exists, is unique and can be obtained in the form of a double-layer potential

$$u ( x) = \int\limits _ { S } \nu ( y) \frac \partial {\partial n _ {y} } E ( x , y ) d S ( y)$$

with density $\nu$ which is obtained as the unique solution of the Fredholm integral equation of the second kind

$$- \frac{1}{2} \nu ( x) + \int\limits _ { S } \nu ( y) \frac \partial {\partial n _ {y} } E ( x , y ) \ d S ( y) = \phi ^ {+} ( x) ,\ x \in S .$$

The interior Neumann problem: Find a function $u ( x)$ of class $C ^ {1} ( G ^ {+} \cup S )$, $S \in C ^ {( 1 , \alpha ) }$, $0 < \alpha < 1$, harmonic in $G ^ {+}$, which satisfies the boundary condition $\partial u ( x) / \partial n _ {x} = \psi ^ {+} ( x)$, $x \in S$, where $\psi ^ {+} ( x)$ is a given continuous function on $S$. A solution to this problem exists if and only if the function $\psi ^ {+} ( x)$ satisfies the orthogonality condition

$$\tag{4 } \int\limits _ { S } \psi ^ {+} ( x) d S ( x) = 0 .$$

This solution is obtained up to an arbitrary additive constant $C$ in the form $u ( x) = V ( x) + C$, where

$$V ( x) = \int\limits _ { S } \mu ( y) E ( x , y ) d S ( y)$$

is a single-layer potential whose density $\mu$ is obtained from the following Fredholm integral equation of the second kind:

$$\tag{5 } \frac{1}{2} \mu ( x) + \int\limits _ { S } \mu ( y) \frac \partial {\partial n _ {x} } E ( x , y ) \ d S ( y) = \psi ^ {+} ( x) ,\ \ x \in S .$$

The continuous homogeneous equation has a non-trivial solution $\mu _ {0} ( x)$ and the inhomogeneous equation (5) is solvable under the condition (4); moreover, its general solution has the form $\mu ( x) + c \mu _ {0} ( x)$, where $c$ is an arbitrary constant.

The exterior Dirichlet problem: Find a function $u ( x)$ of class $C ( G ^ {-} \cup S )$, $S \in C ^ {( 1 , \alpha ) }$, $0 < \alpha < 1$, harmonic in $G ^ {-}$, $0 \in G ^ {+}$, which satisfies the boundary condition $u ( x) = \phi ^ {-} ( x)$, $x \in S$, where $\phi ^ {-} ( x)$ is a given continuous function on $S$. Here, $u ( x)$ is assumed to be regular at infinity, i.e.

$$\lim\limits _ {| x| \rightarrow \infty } | x | ^ {n- 2} u ( x) = \textrm{ const } .$$

The solution of this problem always exists, is unique and can be obtained in the form

$$u ( x) = W ( x) + \frac{A}{| x | ^ {n- 2} } ,$$

where $A$ is a constant and

$$W ( x) = \int\limits _ { S } \nu ( y) \frac \partial {\partial n _ {y} } E ( x , y ) d S ( y)$$

is a double-layer potential whose density $\nu$ is a solution of the following Fredholm integral equation of the second kind:

$$\tag{6 } \frac{1}{2} \nu ( x) + \int\limits _ { S } \nu ( y) \frac \partial {\partial n _ {y} } E ( x , y ) d S ( y) = \phi ^ {-} ( x) - \frac{A}{| x | ^ {n- 2} } ,$$

$$x \in S .$$

The corresponding homogeneous equation has the non-trivial solution $\widetilde \nu _ {0} = 1$. Under an adequate choice of the constant $A$, the solution of the inhomogeneous equation (6) takes the form

$$\nu ( y) = \nu ^ {-} ( y) + C ,$$

where $C$ is an arbitrary constant and $\nu ^ {-} ( y)$ is a particular solution of (6). The constant $A$ is chosen in the form

$$A = - \int\limits _ { S } \phi ^ {-} ( x) \nu _ {0} ( x) d S ( x) ,$$

where the density $\nu _ {0}$ must satisfy the condition

$$\tag{7 } \int\limits _ { S } \nu _ {0} ( y) \frac{1}{| y | ^ {n- 2} } d S ( y) = 1 .$$

This density $\nu _ {0}$ is a non-trivial solution of the equation (5) of the interior Neumann problem with data $\psi ^ {+} ( x) = 0$, $x \in S$, satisfying the normalization condition

$$V _ {0} ( x) \equiv \int\limits _ { S } \nu _ {0} ( y) E ( x , y ) \ d S ( y) = 1 ,\ x \in G ^ {+} \cup S ,$$

which is equivalent to (7) for $n \geq 3$. The single-layer potential $V _ {0} ( x)$ with density $\nu _ {0} ( x)$ is called an equilibrium potential or Robin potential. The density $\nu _ {0} ( x)$ provides a solution to the Robin problem or the electrostatic problem on the distribution of charges on the conductor $S$ generating a constant equilibrium potential in $G ^ {+}$. A certain complexity in solving the exterior Dirichlet problem is due to the fact that, in general, the harmonic function $u ( x)$ that is regular at infinity decreases slower than the double-layer potential as $| x | \rightarrow \infty$ and, thus, in the general case $u ( x)$ cannot be represented only by one double-layer potential.

The exterior Neumann problem: Find a function $u ( x)$ of class $C ^ {1} ( G ^ {-} \cup S )$, $S \in C ^ {( 1 , \alpha ) }$, $0 < \alpha < 1$, harmonic in $G ^ {-}$, $0 \in G ^ {+}$, which satisfies the boundary condition $\partial u ( x) / \partial n _ {x} = \psi ^ {-} ( x)$, $x \in S$, where $\psi ^ {-} ( x)$ is a given continuous function on $S$; in addition, $u ( x)$ is assumed to be regular at infinity. For $n \geq 3$ the solution of this problem always exists and is unique; for $n = 2$ a solution exists if and only if the following condition holds:

$$\tag{8 } \int\limits _ { S } \psi ^ {-} ( x) d S ( x) = 0 ,$$

Moreover, this solution is defined up to an arbitrary additive constant. This solution of the exterior Neumann problem can be represented in the form of a single-layer potential

$$u ( x) = \int\limits _ { S } \mu ( y) E ( x , y ) d S ( y)$$

whose density is a solution of the following Fredholm integral equation of the second kind:

$$\tag{9 } - \frac{1}{2} \mu ( x) + \int\limits _ { S } \mu ( y) \frac \partial {\partial n _ {x} } E ( x , y ) d S ( y) = \psi ^ {-} ( x) ,\ \ x \in S .$$

For $n \geq 3$ the solution of this equation always exists and is unique. For $n = 2$ the corresponding homogeneous equation has a non-trivial solution $\mu _ {0} ( x)$. Thus, the inhomogeneous equation (9) with the solvability condition (8) has a unique solution $\widetilde \mu ( x)$ such that

$$\int\limits _ { S } \widetilde \mu ( x) d S ( x) = 0 ,$$

and its general solution is of the form $\mu ( x) = \widetilde \mu ( x) + c \mu _ {0}$, where $c$ is an arbitrary constant.

Boundary value problems in potential theory can also be solved using a Green function. For instance, for the (interior) Dirichlet problem the Green function has the form

$$G ( x , y ) = E ( x , y ) + g ( x , y ) ,\ \ x \in G ^ {+} \cup S ,\ y \in G ^ {+} ,$$

where $g ( x , y )$ is a harmonic function in $G ^ {+}$ that is continuous with respect to $x$ on $G ^ {+} \cup S$ and that satisfies, for each $y \in G ^ {+}$, the boundary condition $g ( x , y ) = 0$, $x \in S$. The solution of the (interior) Dirichlet problem $u ( x)$ of class $C ^ {2} ( G ^ {+} ) \cap C ( G ^ {+} \cup S )$ for the Poisson equation $- \Delta u ( x) = f ( x)$, $x \in G ^ {-}$, with the boundary condition $u ( x) = \phi ^ {+} ( x)$, $x \in S$, can be represented in the form

$$u ( x) = \int\limits _ {G ^ {+} } f ( y) G ( x , y ) d y + \int\limits _ { S } \phi ^ {+} ( y) \frac \partial {\partial n _ {y} } G ( x , y ) d S ( y) ,\ x \in G ^ {+} .$$

The integrals

$$\int\limits _ { G } \rho ( y) G ( x , y ) \ d y ,\ \int\limits _ { S } \nu ( y) \frac \partial {\partial n _ {y} } G ( x , y ) d S ( y) ,$$

which depend on the parameter $x$, are called the Green volume potential (of the Dirichlet problem) and the Green double-layer potential, respectively. Their properties are similar to the properties of the potentials (1).

Green functions allow one to reduce eigen value problems to integral equations. For instance, the Dirichlet problem $- \Delta \mu = \lambda u ( x)$, $x \in G ^ {+}$, with boundary condition $u ( x) = 0$, $x \in S$, is reduced to the following Fredholm integral equation of the second kind with a self-adjoint kernel:

$$u ( x) - \lambda \int\limits _ {G ^ {+} } u ( y) G ( x , y ) \ d y = 0 ,\ x \in G ^ {+} .$$

## Further generalization of some fundamental concepts in potential theory.

Simultaneously with profound studies on the properties of the potentials (1), defined by densities of a more or less general form, and of their applications, the concept of potential itself has undergone a deep generalization related to the concepts of a Radon measure and a Radon integral. This process started in the 1920s.

Let $\lambda \geq 0$ be a positive Borel measure on $\mathbf R ^ {n}$ with compact support $\supp \lambda$. The potential of the measure,

$$\tag{10 } E \lambda ( x) = \int\limits E ( x , y ) d \lambda ( y) ,$$

exists everywhere in $\mathbf R ^ {n}$ in the sense of a mapping $E \lambda : \mathbf R ^ {n} \rightarrow [ 0 , \infty ]$ for $n \geq 3$ and $E \lambda : \mathbf R ^ {2} \rightarrow ( - \infty , \infty ]$ for $n = 2$ (i.e. here the value $+ \infty$ is also allowed), is a superharmonic function everywhere in $\mathbf R ^ {n}$ and is harmonic outside the support $\supp \lambda$. For a measure $\lambda$ of arbitrary sign with compact support the potential $E \lambda$ is defined on the basis of the canonical decomposition $\lambda = \lambda ^ {+} - \lambda ^ {-}$, $\lambda ^ {+} \geq 0$, $\lambda ^ {-} \geq 0$, as $E \lambda = E \lambda ^ {+} - E \lambda ^ {-}$. At the points $x \in \mathbf R ^ {n}$ where both potentials $E \lambda ^ {+} ( x)$ and $E \lambda ^ {-} ( x)$ assume the value $+ \infty$, this potential is not defined. If the measure $\lambda \geq 0$ is concentrated on a smooth surface $S$, then the double-layer potential of the measure $\lambda$ is determined similarly to (10):

$$\frac{\partial E }{\partial n _ {y} } \lambda ( x) = \ \int\limits \frac \partial {\partial n _ {y} } E ( x , y ) d \lambda ( y) .$$

The potential (10) is finite, $E \lambda < + \infty$ everywhere on $\mathbf R ^ {n}$ except at the points of a polar set, which is characterized as a set of outer capacity zero. If $E \lambda ( x) = 0$ everywhere except on a set of outer capacity zero, then $\lambda = 0$. If the measure $\lambda \geq 0$, $\lambda \neq 0$, is concentrated on a set of capacity zero, then $\sup E \lambda = + \infty$. The following maximum principle is valid:

$$E \lambda ( x) \leq \sup \{ {E \lambda ( y) } : { y \in \supp \lambda } \} ,$$

i.e. the least upper bound of $E \lambda ( x)$ equals the least upper bound of the restriction of $E \lambda$ to $\supp \lambda$. If this restriction is continuous (in the general case, including $+ \infty$) at a point $x _ {0} \in \supp \lambda$, then the potential $E \lambda ( x)$ is continuous at that point $x _ {0}$ in $\mathbf R ^ {n}$. The potentials of a measure, $E \lambda$, can be reduced to potentials of densities (1) if and only if the measure $\lambda$ is absolutely continuous with respect to the Lebesgue measure in $G$ or on $S$, respectively (see [3][6]).

If $T$ is a generalized function, or distribution, on $\mathbf R ^ {n}$, then its potential is defined as the convolution $E * T$, which is also a generalized function. For instance, if $T$ is a generalized function with compact support, then the Poisson equation $\Delta ( E T ) = - T$ is valid on $\mathbf R ^ {n}$ in the sense of the theory of generalized functions. Potentials of measures can be considered as a particular case of potentials of distributions. For potentials of distributions see [3], [4], [9].

For domains $G = G ^ {+}$ with sufficiently smooth boundary $S$ the method of potentials provides an efficient solution to the Dirichlet problem. One of the principal directions of development in potential theory consists in finding methods to prove the existence and uniqueness of a solution to the Dirichlet problem for wider classes of domains (see Balayage method; Dirichlet principle; Perron method; Schwarz alternating method). However, in 1910 S. Zaremba noted that for a plane domain $G$ whose boundary $\partial G = S$ has isolated points the Dirichlet problem is not always solvable in the above classical formulation; in addition, in 1912 H. Lebesgue has shown that it is not always solvable also for spatial domains homeomorphic to a closed sphere but with a sufficiently sharp edge at the boundary entering inside the domain (a so-called Lebesgue spine, see Irregular boundary point), i.e. there exist continuous functions $\phi ^ {+} ( x)$, $x \in \partial G$, for which the Dirichlet problem cannot be solved in any way.

Thus, the generalized Perron–Wiener solution to the Dirichlet problem for the Laplace equation obtained in the course of development of the Perron method is of great importance. As has been shown by N. Wiener (1924), in this case any finite continuous function $\phi = \phi ^ {+}$ prescribed on the boundary $S = \partial G$ of an arbitrary bounded domain $G \subset \mathbf R ^ {n}$ is resolutive, i.e. the generalized Perron–Wiener solution $H _ \phi ( x)$ for this function exists and, moreover, is unique. In general, in 1939 M. Brelot has shown that a finite measurable function $\phi$ on $S$ is resolutive if and only if $\phi$ is integrable with respect to harmonic measure on $S$.

The generalized solution $H _ \phi$ does not take the prescribed values $\phi$ at all boundary points. A point $x _ {0} \in S$ is called a regular point if for any finite continuous function $\phi$ on $S$ the generalized solution $H _ \phi ( x)$ takes the value $\phi ( x _ {0} )$, that is,

$$\lim\limits _ {x \rightarrow x _ {0} } H _ \phi ( x) = \ \phi ( x _ {0} ) ,\ x \in G .$$

All other points $x _ {0} \in S$ are called irregular points; they include the isolated points of the boundary when $n \geq 2$ and the Lebesgue spine for $n \geq 3$. It turned out (the Kellogg–Evans theorem, 1933) that the set of irregular points has outer capacity zero, i.e. this set is in some sense thin. The set of regular points is dense in $S$.

For the Dirichlet problem one can construct a generalized Green function $G$, which can be defined, e.g., for an arbitrary fixed point $y \in G$ in the following way:

$$G ( x , y ) = E ( x , y ) - H _ {E ( x , y ) } ( x) ,\ \ x \in G .$$

The generalized Green function preserves some properties of the classical Green function, for example, the symmetry property $G ( x , y ) = G ( y , x )$, but $\lim\limits _ {x \rightarrow x _ {0} } G ( x , y ) = 0$, $x \in G$, if and only if $x _ {0}$ is a regular point of the boundary $S$ (see [4], [6]).

The studies of potentials with other kernels, different from $E ( x , y )$, and the study of the Dirichlet problem for compacta and the stability of the Dirichlet problem are of great importance (see [6], [4]). Their application to the solution of boundary value problems is intensively developed (see Bessel potential; Non-linear potential; Riesz potential; and also [3], [11]).

#### References

 [1] N.M. [N.M. Gyunter] Günter, "Potential theory and its application to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian) [2] L.N. Sretenskii, "Theory of the Newton potential" , Moscow-Leningrad (1946) (In Russian) [3] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) [4] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) [5] O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) (Re-issue: Springer, 1967) [6] M.V. Keldysh, "On the resolutivity and the stability of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–231 (In Russian) [7] A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian) [8] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian) [9] V.S. Vladimirov, "Equations of mathematical physics" , M. Dekker (1971) (Translated from Russian) [10] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) [11] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) [12] S.G. Mikhlin, "Linear partial differential equations" , Moscow (1977) (In Russian) [13] A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (Translated from Russian)