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
be a smooth closed surface, i.e. an
-dimensional
smooth manifold without boundary, in the
-dimensional
Euclidean space
,
,
which bounds a bounded domain
,
.
Let
be the exterior unbounded domain. Let
be a principal
fundamental solution
of the
Laplace equation
in
,
where
is the distance between two points
and
in
,
is the area of the unit sphere in
and
is the gamma-function. The following three integrals, which depend on
as a parameter,
where
is the direction of the exterior (with respect to
)
normal to
at a point
,
are called the
volume potential,
the
single-layer potential
and the
double-layer potential,
respectively. The functions
,
and
are called the
densities
of the corresponding potentials; hereafter they are
assumed to be absolutely integrable over
or
,
respectively. For
(and sometimes for
)
the integrals
(1)
are called the
Newton volume potential
and the
Newton single-
and
double-layer potentials;
for
they are called
logarithmic mass,
single-layer
or
double-layer potentials,
respectively. Let
be of class
.
Then the volume potential (cf.
Newton potential)
and its first derivatives are continuous everywhere on
;
moreover, they can be calculated by differentiation under the integral sign, i.e.
.
Further,
The second derivatives are continuous everywhere outside
,
but they have a discontinuity when passing across the surface
;
moreover, in
they satisfy the
Poisson equation
,
,
and in
— the
Laplace equation
,
.
The above-mentioned properties characterize a volume potential.
If
is a bounded domain in
with boundary
of class
,
then
Gauss' formula for a volume potential
is valid:
Let
.
The single-layer potential (cf.
Simple-layer potential)
is a harmonic function when
;
moreover,
in particular,
for
,
but
when
if and only if
.
A single-layer potential is continuous everywhere on
,
,
moreover,
and its tangential derivatives are continuous when passing across the surface
.
The normal derivative of a single-layer potential
has a discontinuity when passing across the surface
:
where
and
are the limit values of the normal derivative from
and
,
respectively, i.e.
denotes the so-called
direct value of the normal derivative of a single-layer potential
calculated over the surface
,
i.e.
It is a continuous function of the points
,
and the kernel
has a weak singularity on
,
These properties characterize a single-layer potential.
Let
.
The
double-layer potential
is a harmonic function for
;
moreover,
When passing across the surface
the double-layer potential has a discontinuity (whence its name):
where
and
are the limit values of the double-layer potential from
and
,
respectively, that is,
when
denotes the so-called
direct value of the double-layer potential
calculated over the surface
,
that is,
It is a continuous function of the points
,
and the kernel
has a weak singularity on
,
The tangential derivatives of a double-layer potential also
have a discontinuity when passing across the surface
,
but the normal derivative
retains its value when passing across
:
These properties characterize a double-layer potential.
In the case of a constant density
Gauss' formula for a double-layer potential
holds:
The integral at the left-hand side of this equality is interpreted (when divided by
)
as the solid angle at which the surface
is seen from the point
.
Below, certain properties of potentials under weaker
restrictions on the densities and the surface
are given.
If
,
then
is a harmonic function for
and
is summable on
.
If
,
,
then
,
,
;
if
,
,
then
.
If
,
,
then
,
;
if
,
,
then
.
If
,
then the generalized second derivatives of
exist, they are also of class
and are expressed by singular integrals:
where
for
,
for
;
if
,
,
then all generalized derivatives
also exist and belong to
.
If
,
,
then
is a generalized solution of the Poisson equation
,
.
If
and
,
,
then
in
or
.
If
and
,
,
integers,
,
then
.
Let
,
,
let
be a closed bounded domain such that
.
If
,
,
then
,
,
,
;
.
If the density is bounded and summable, then
If
,
,
then
in
or
.
If
,
then
in
or
.
If
and
,
,
integers,
,
then
in
or
.
If
and
,
,
integers,
,
then
in
or
.
For potentials and their derivatives extended by continuity on
the above-described properties of smoothness are also valid under the
corresponding smoothness conditions on the density and the surface
.
Representation of functions and solution of the principal boundary value problems in potential theory using potentials.
Let
be a function of class
and let
be a smooth surface of class
.
Then the following integral identity
(Green formula)
holds:
In particular, in
the function
can be represented as the sum of a volume
potential and single- and double-layer potentials, with respective densities
For a function
of class
that is harmonic on
the following identity holds:
Hence, such a function
can be represented in
by the sum of single- and double-layer potentials with densities
,
,
respectively. However, the densities in
(3)
cannot be arbitrarily given on
;
they are connected by the integral relation obtained from
(3)
for
.
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
(interior problems)
and
(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
of class
,
,
,
harmonic in
,
which satisfies the boundary condition
,
,
where
is a given continuous function on
.
The solution to this problem always exists, is unique
and can be obtained in the form of a double-layer potential
with density
which is obtained as the unique solution of
the Fredholm integral equation of the second kind
The
interior Neumann problem:
Find a function
of class
,
,
,
harmonic in
,
which satisfies the boundary condition
,
,
where
is a given continuous function on
.
A solution to this problem exists if and only if the function
satisfies the orthogonality condition
This solution is obtained up to an arbitrary additive constant
in the form
,
where
is a single-layer potential whose density
is obtained from the following Fredholm integral equation of the second kind:
The continuous homogeneous equation has a non-trivial solution
and the inhomogeneous equation
(5)
is solvable under the
condition
(4);
moreover, its general solution has the form
,
where
is an arbitrary constant.
The
exterior Dirichlet problem:
Find a function
of class
,
,
,
harmonic in
,
,
which satisfies the boundary condition
,
,
where
is a given continuous function on
.
Here,
is assumed to be regular at infinity, i.e.
The solution of this problem always exists, is unique and can be obtained in the form
where
is a constant and
is a double-layer potential whose density
is a solution of the following Fredholm integral equation of the second kind:
The corresponding homogeneous equation has the non-trivial solution
.
Under an adequate choice of the constant
,
the solution of the inhomogeneous equation
(6)
takes the form
where
is an arbitrary constant and
is a particular solution of
(6).
The constant
is chosen in the form
where the density
must satisfy the condition
This density
is a non-trivial solution of the equation
(5)
of the interior Neumann problem with data
,
,
satisfying the normalization condition
which is equivalent to
(7)
for
.
The single-layer potential
with density
is called an
equilibrium potential
or
Robin potential.
The density
provides a solution to the
Robin problem
or the electrostatic problem on the distribution of charges on the conductor
generating a constant equilibrium potential in
.
A certain complexity in solving the exterior Dirichlet problem is
due to the fact that, in general, the harmonic function
that is regular at infinity decreases slower than the double-layer potential as
and, thus, in the general case
cannot be represented only by one double-layer potential.
The
exterior Neumann problem:
Find a function
of class
,
,
,
harmonic in
,
,
which satisfies the boundary condition
,
,
where
is a given continuous function on
;
in addition,
is assumed to be regular at infinity. For
the solution of this problem always exists and is unique; for
a solution exists if and only if the following condition holds:
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
whose density is a solution of the following
Fredholm integral equation of the second kind:
For
the solution of this equation always exists and is unique. For
the corresponding homogeneous equation has a non-trivial solution
.
Thus, the inhomogeneous equation
(9)
with the
solvability condition
(8)
has a unique solution
such that
and its general solution is of the form
,
where
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
where
is a harmonic function in
that is continuous with respect to
on
and that satisfies, for each
,
the boundary condition
,
.
The solution of the (interior) Dirichlet problem
of class
for the Poisson equation
,
,
with the boundary condition
,
,
can be represented in the form
The integrals
which depend on the parameter
,
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
,
,
with boundary condition
,
,
is reduced to the following Fredholm integral equation
of the second kind with a self-adjoint kernel:
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
be a positive
Borel measure
on
with compact support
.
The
potential of the measure,
exists everywhere in
in the sense of a mapping
for
and
for
(i.e. here the value
is also allowed), is a
superharmonic function
everywhere in
and is harmonic outside the support
.
For a measure
of arbitrary sign with compact support the potential
is defined on the basis of the canonical decomposition
,
,
,
as
.
At the points
where both potentials
and
assume the value
,
this potential is not defined. If the measure
is concentrated on a smooth surface
,
then the double-layer potential of the measure
is determined similarly to
(10):
The potential
(10)
is finite,
everywhere on
except at the points of a
polar set,
which is characterized as a set of outer
capacity
zero. If
everywhere except on a set of outer capacity zero, then
.
If the measure
,
,
is concentrated on a set of capacity zero, then
.
The following maximum principle is valid:
i.e. the least upper bound of
equals the least upper bound of the restriction of
to
.
If this restriction is continuous (in the general case, including
)
at a point
,
then the potential
is continuous at that point
in
.
The potentials of a measure,
,
can be reduced to potentials of densities
(1)
if and only if the measure
is absolutely continuous with respect to the Lebesgue measure in
or on
,
respectively (see
[3]–
[6]).
If
is a
generalized function,
or distribution, on
,
then its
potential
is defined as the convolution
,
which is also a generalized function. For instance, if
is a generalized function with compact support, then the Poisson equation
is valid on
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
with sufficiently smooth boundary
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
whose boundary
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
,
,
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
prescribed on the boundary
of an arbitrary bounded domain
is
resolutive,
i.e. the generalized Perron–Wiener solution
for this function exists and, moreover, is unique. In general,
in
1939
M. Brelot
has shown that a finite measurable function
on
is resolutive if and only if
is integrable with respect to
harmonic measure
on
.
The generalized solution
does not take the prescribed values
at all boundary points. A point
is called a
regular point
if for any finite continuous function
on
the generalized solution
takes the value
,
that is,
All other points
are called
irregular points;
they include the isolated points of the boundary when
and the Lebesgue spine for
.
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
.
For the Dirichlet problem one can construct a
generalized Green function
,
which can be defined, e.g., for an arbitrary fixed point
in the following way:
The generalized Green function preserves some properties of the
classical Green function, for example, the symmetry property
,
but
,
,
if and only if
is a regular point of the boundary
(see
[4],
[6]).
The studies of potentials with other kernels, different from
,
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]).