In
mathematics and
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, potential theory is the study of
harmonic functions.
The term "potential theory" was coined in 19th-century
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
when it was realized that two fundamental
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the
gravitational potential and
electrostatic potential, both of which satisfy
Poisson's equation—or in the vacuum,
Laplace's equation.
There is considerable overlap between potential theory and the theory of Poisson's equation to the extent that it is impossible to draw a distinction between these two fields. The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about the
singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation. This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other.
Modern potential theory is also intimately connected with probability and the theory of
Markov chains. In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an
electrical network
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sour ...
on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials. Even in the finite case, the analogue I-K of the Laplacian in potential theory has its own maximum principle, uniqueness principle, balance principle, and others.
Symmetry
A useful starting point and organizing principle in the study of harmonic functions is a consideration of the
symmetries of the Laplace equation. Although it is not a symmetry in the usual sense of the term, we can start with the observation that the Laplace equation is
linear. This means that the fundamental object of study in potential theory is a linear space of functions. This observation will prove especially important when we consider function space approaches to the subject in a later section.
As for symmetry in the usual sense of the term, we may start with the theorem that the symmetries of the
-dimensional Laplace equation are exactly the
conformal
Conformal may refer to:
* Conformal (software), in ASIC Software
* Conformal coating in electronics
* Conformal cooling channel, in injection or blow moulding
* Conformal field theory in physics, such as:
** Boundary conformal field theory ...
symmetries of the
-dimensional
Euclidean space. This fact has several implications. First of all, one can consider harmonic functions which transform under irreducible representations of the
conformal group or of its
subgroups (such as the group of rotations or translations). Proceeding in this fashion, one systematically obtains the solutions of the Laplace equation which arise from separation of variables such as
spherical harmonic solutions and
Fourier series. By taking linear superpositions of these solutions, one can produce large classes of harmonic functions which can be shown to be dense in the space of all harmonic functions under suitable topologies.
Second, one can use conformal symmetry to understand such classical tricks and techniques for generating harmonic functions as the
Kelvin transform and the
method of images.
Third, one can use conformal transforms to map harmonic functions in one
domain to harmonic functions in another domain. The most common instance of such a construction is to relate harmonic functions on a
disk to harmonic functions on a half-plane.
Fourth, one can use conformal symmetry to extend harmonic functions to harmonic functions on conformally flat
Riemannian manifolds. Perhaps the simplest such extension is to consider a harmonic function defined on the whole of R
n (with the possible exception of a
discrete set of singular points) as a harmonic function on the
-dimensional
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. More complicated situations can also happen. For instance, one can obtain a higher-dimensional analog of Riemann surface theory by expressing a multi-valued harmonic function as a single-valued function on a branched cover of R
n or one can regard harmonic functions which are invariant under a discrete subgroup of the conformal group as functions on a multiply connected manifold or
orbifold.
Two dimensions
From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for more than two dimensions, one can surmise that potential theory in two dimensions is different from potential theory in other dimensions. This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a
complex analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
, one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis (such as
Schwarz's theorem,
Morera's theorem
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
Morera's theorem states that a continuous, complex-valued function ''f'' d ...
, the
Weierstrass-Casorati theorem,
Laurent series, and the classification of
singularities as
removable,
poles and
essential singularities) generalize to results on harmonic functions in any dimension. By considering which theorems of complex analysis are special cases of theorems of potential theory in any dimension, one can obtain a feel for exactly what is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results.
Local behavior
An important topic in potential theory is the study of the local behavior of harmonic functions. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. There are results which describe the local structure of
level sets of harmonic functions. There is
Bôcher's theorem, which characterizes the behavior of
isolated singularities of positive harmonic functions. As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities.
Inequalities
A fruitful approach to the study of harmonic functions is the consideration of inequalities they satisfy. Perhaps the most basic such inequality, from which most other inequalities may be derived, is the
maximum principle. Another important result is
Liouville's theorem, which states the only bounded harmonic functions defined on the whole of R
n are, in fact, constant functions. In addition to these basic inequalities, one has
Harnack's inequality, which states that positive harmonic functions on bounded domains are roughly constant.
One important use of these inequalities is to prove
convergence of families of harmonic functions or sub-harmonic functions, see
Harnack's theorem. These convergence theorems are used to prove the
existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being.
Etymology
The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistenti ...
of harmonic functions with particular properties.
Spaces of harmonic functions
Since the Laplace equation is linear, the set of harmonic functions defined on a given domain is, in fact, a
vector space. By defining suitable
norms and/or
inner products, one can exhibit sets of harmonic functions which form
Hilbert or
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
s. In this fashion, one obtains such spaces as the
Hardy space,
Bloch space
In the mathematical field of complex analysis, the Bloch space, named after French mathematician André Bloch and denoted \mathcal or ℬ, is the space of holomorphic functions ''f'' defined on the open unit disc D in the complex plane, such that t ...
,
Bergman space In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that t ...
and
Sobolev space.
See also
*
Subharmonic function
*
Kellogg's theorem
References
*
*
*
S. Axler, P. Bourdon, W. Ramey (2001). ''Harmonic Function Theory'' (2nd edition). Springer-Verlag. .
*
O. D. Kellogg (1969). ''Foundations of Potential Theory''. Dover Publications. .
*L. L. Helms (1975). ''Introduction to potential theory''. R. E. Krieger .
*
J. L. Doob. ''Classical Potential Theory and Its Probabilistic Counterpart'', Springer-Verlag, Berlin Heidelberg New York, .
*
*
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