mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...

s. 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 r ...

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 p ...

s of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the

gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric po ...

and

electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...

, both of which satisfy

Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...

—or in the vacuum,

Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...

. 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 chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

s. 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 sources, c ...

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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

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 Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...

. 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

n

-dimensional Laplace equation are exactly the conformal symmetries of the

n

-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. This fact has several implications. First of all, one can consider harmonic functions which transform under irreducible representations of the

conformal group In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Se ...

or of its

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s (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 In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...

solutions and

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

. 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 The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and ...

and the

method of images The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which the domain of the sought function is extended by the addition of its mirror image with respect to a symmetry hyperplane. As a resul ...

. Third, one can use conformal transforms to map harmonic functions in one

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

to harmonic functions in another domain. The most common instance of such a construction is to relate harmonic functions on a

disk Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...

to harmonic functions on a half-plane. Fourth, one can use conformal symmetry to extend harmonic functions to harmonic functions on conformally flat

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

s. Perhaps the simplest such extension is to consider a harmonic function defined on the whole of Rⁿ (with the possible exception of a

discrete set ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...

of singular points) as a harmonic function on the

n

-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 Rⁿ 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 In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

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 an ...

, 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 In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) of ''n'' ...

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'' defined ...

, the Weierstrass-Casorati theorem,

Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...

, and the classification of singularities as removable,

poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...

and

essential singularities In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...

) 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 set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...

s of harmonic functions. There is

Bôcher's theorem In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher. Bôcher's theorem in complex analysis In complex analysis, the theorem states that the finite zeros of the derivative r'(z) of a non ...

, 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 In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...

. Another important result is Liouville's theorem, which states the only bounded harmonic functions defined on the whole of Rⁿ are, in fact, constant functions. In addition to these basic inequalities, one has

Harnack's inequality In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions ...

, which states that positive harmonic functions on bounded domains are roughly constant. One important use of these inequalities is to prove

convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Wei ...

of families of harmonic functions or sub-harmonic functions, see

Harnack's theorem In the mathematical field of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of sequences of harmonic functions. Given a sequence of harmonic function ...

. 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 ontology, ontological Property (philosophy), property of being. Etymology The term ''existence'' comes from Old French ''existence'', from Medieval ...

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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

. By defining suitable

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

s and/or

inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

s, one can exhibit sets of harmonic functions which form

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

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 vector ...

s. In this fashion, one obtains such spaces as the

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . ...

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 In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...

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 Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory. The theory of martingales was developed by Doob. Early life and education Doob was born in Cincinnati, Ohio, ...