Potential theory may be defined as the study of harmonic functions. In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...

Definition and comments

The term "potential theory" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from potentials which satisfied Laplace's equation. Hence, potential theory was the study of functions which could serve as potentials. Nowadays, we know that nature is more complicated: the equations which describe forces are systems of non-linear partial differential equations such as the Einstein equations and the Yang-Mills equations, and the Laplace equation is only valid as a limiting case. Nevertheless, the term "potential theory" has remained as a convenient term for describing the study of functions which satisfy the Laplace equation. It is also still the case that the Laplace equation is used in applications in several areas of physics like heat conduction and electrostatics. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Force (disambiguation). ... In physics, a potential may refer to the scalar potential or to the vector potential. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

Obviously, there is considerable overlap between potential theory and the theory of the Laplace equation. To the extent that it is possible 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. Of course, 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. In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

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. Sphere symmetry group o. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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. 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 or 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. In mathematics, a conformal map is a function which preserves angles. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... Spherical Harmonic is a fantasy novel by Catherine Asaro which tells the story of Pharaoh Dyhianna (Dehya) Selei, ruler of the Skolian Imperialate, after the Radiance War fought by the Imperialate and their enemy Eubian Concord. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

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. 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 superharmonic functions. ... The method of image charges (also known as the method of images and method of mirror charges) is a basic problem solving tool in electrostatics. ...

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. In mathematics, the domain of a function is the set of all input values to the function. ... In geometry, a disk is the region in a plane contained inside of a circle. ...

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 Rⁿ (with the possible exception of a discrete set of singular points) as a harmonic function on the n-dimensional sphere. More complicated situations can also happen. For instance, one can obtain a higher-dimensional analog of Riemann surface theory by expressing a multiply 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 Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In topology, a point x of a set S is called an isolated point, if there exists a neighbourhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an... A sphere is a symmetrical geometrical object. ... In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ...

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, 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 Schwartz's theorem, Morera's theorem, 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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In complex analysis, Moreras theorem states that if the integral of a continuous complex-valued function f of a complex variable along every simple closed curve within an open set is zero, that is, if for C any simple closed curve, then f is differentiable at every point in... The Weierstrass-Casorati theorem in complex analysis describes the remarkable behavior of holomorphic functions near essential singularities. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be defined without creating any problems. ... In complex analysis, an essential singularity of a function is a severe singularity near which the function exhibits extreme behavior. ...

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. In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ... In mathematics, Bôchers theorem, named after Maxime Bôcher, states that the finite zeros of the derivative of a nonconstant rational function that are not multiple zeros are also the positions of equilibrium in the field of force due to particles of positive mass at the zeros of... In complex analysis, a branch of mathematics, an isolated singularity is a singularity which contains no other singularities close to it. ...

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 Rⁿ are, in fact, constant functions. In addition to these basic inequalities, one has such inequalities as Cauchy's estimate, Harnack's inequality, and the Schwarz lemma. In mathematics, the maximum principle in harmonic analysis states that if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f. ... Liouvilles theorem has various meanings: In complex analysis, see Liouvilles theorem (complex analysis). ... moo cow ... In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open unit disk. ...

One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions. These convergence theorems can often be used to prove existence of harmonic functions having particular properties. The limit of a sequence is one of the oldest concepts in mathematical analysis. ...

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 spaces. In this fashion, one obtains such spaces as the Hardy space, Bloch space, and Bergman space. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In complex analysis, the Hardy spaces are analogues of the Lp spaces of functional analysis. ... The Bloch space, named after André Bloch, is the space of holomorphic functions defined on the open complex unit disc such that the function is bounded. ... In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a function space in which pointwise evaluation is a continuous linear functional. ...

See also

• Probabilistic potential theory

References

This article incorporates material from Potential Theory on PlanetMath, which is licensed under the GFDL. The Encyclopaedia of Mathematics is a large reference work in mathematics. ... The Encyclopaedia of Mathematics is a large reference work in mathematics. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...