Definition and comments

A term "potential theory" arises from either a fact that, within 19th century physics, the fundamental forces of nature were believed to become from either potentials which satisfied Laplace's equation. Hence, expected theory was a learn of functions which could help when potentials. Today, i personally understand that nature and severity is other complicated -- a equations which describe forces come systems of non-linear partial differential equations such as a Einstein equations and the Yang-Mills equations and that the Laplace equation is merely valid as a limiting out break. All a same, the term "potential theory" has remained as a handy term for describing a learn of functions which satisfy a Laplace equation.

Manifestly, there exists considerable overlap between expected theory & a theory of the Laplace equation. To the extent that these are conceivable to draw the distinction between these 2 fields, the difference is extra one of emphasis than subject matter & rests on the when punishment distinction -- likely theory focuses on the properties of the functions as opposed to the properties of the equation. For instance, a effect all about the singularities of harmonic functions would exist as said to belong to expected theory when a effect in how else the guide depends on the boundary information would be said to belong to the theory of the Laplace equation. Course, this is non the stiff & convenient distinction &, within practice there exists considerable overlap between them fields, by having methods & final result from either of these existence utilized in the more.

Symmetry

The utile starting point & organizing principle in the survey of harmonic functions occurs as consideration of the symmetries of the Laplace equation. Although these are non a symmetry in a common feel of the term, you may run by owning the observation that the Laplace equation is linear. This means that a fundamental frequency object of survey inside likely theory occurs as linear space of functions. This observation might prove especially significant after i assume work space approaches to the subject within the late subdivision.

Whenorth for symmetry in a common feel of a term, you might begin by having the theorem that the symmetries of the $n$-dimensional Laplace equation come exactly a conformal symmetries of the $n$-dimensional Euclidean space. This fact has many implications. First, 1 may assume harmonic functions which transform under irreducible representations of the conformal group or of its subgroups (such as a class action or even even rotations or translations). Proceeding therein fashion, a single consistently obtains a solutions of the Laplace equation which arise from either separation of variables like spherical harmonic solutions and Fourier series. By ingesting linear superpositions 1 solutions, one might make big classes of harmonic functions which may become shown to be heavy in the space of completely harmonic functions under suitable topologies.

2nd, of these might apply conformal symmetry to realize such authoritative tricks & techniques for giving harmonic functions when a Kelvin transform and the method of images.

Third, 1 might utilize conformal transforms to map harmonic functions inside of these domain to harmonic functions in another domain. the usual time of such the constructiin is to relate harmonic functions on a disk to harmonic functions on the half-plane.

4th, of these potty have conformal symmetry to extend harmonic functions to harmonic functions in conformally flat Riemannian manifolds. Mayhap the simplest such extension is to assume a harmonic work defined tout ensemble of R^north (using the conceivable exception of a discrete set of singular points) as a harmonic functionorth on the $n$-dimensional sphere. Further complicated situations can too happen. E.g., 1 potty obtainorth the higher-dimensional analogue of Riemann superficial theory by expressing the multiply valued harmonic work as the only-valued work in a branched handle of Rⁿ or even of these could regard harmonic functions which are then invariant under the distinct subclass actiin of the conformal group when functions on the multiply-attached manifold or orbifold.

Two dimensions

From either a fact that a class action of conformal transforms is infinite miscreate around 2 dimensions & finite miscreate for extra than deuce dimensions, of these potty surmise that likely theory inside both dimensions is different than expected theory around more dimensions. This is right &, as the matter of fact, after 1 understands that any both-planar harmonic work is the really a portion of a complex analytic function, one understands that a subject of both-planar likely theory is substantially a equivalent when that of complex analysis. For even this cause, whilst speaking of expected theory, 1 focuses attentiaround on theorems which hang on to in triplet or supplementary dimensions. inside that connection, the surprising fact is that numbers of outcomes & conception originally found in complex analysis (like 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 shopping for which theorems of complex analysis come favorite subjects of theorems of likely theory inside any dimension, the single may obtain a condole with exactly what is favorite just about complex analysis in 2 dimensions & what is just them-planar time of other general resolutions.

Local behavior

An crucial topic within likely theory is the learn of the local behavior of harmonic functions. Possibly a virtually all fundamental theorem all about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions come analytic. There are resolutions which describe a local structure of level sets of harmonic functions. There exists Bôcher's theorem, which characterizes the behavior of isolated singularities of positive harmonic functions. Every bit alluded to in a previous division, a single may classify the isolated singularities of harmonic functions when obliterable singularities, poles, & essential singularities.

Inequalities

The profitable approach to the survey of harmonic functions is the consideration of inequalities it satisfy. Maybe a virtually all basic such inequality, from either which virtually all more inequalities can be derived, is the maximum principle. A second significant symptom is Liouville's theorem, which states the only bounded harmonic functions defined on the whole of Rⁿ are, in fact, constant functions. Additionally to these basic inequalities, of these hwhen such inequalities as Cauchy's estimate, Harnack's inequality, and the Schwarz lemma.

1 crucial apply one inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions. These convergence theorems may typically exist as utilized to prove being of harmonic functions getting particular properties.

Spaces of harmonic functions

Since the Laplace equatiin is linear, the placed of harmonic functions defined on a given domain is, in point of fact, a vector space. By defining suitable norms and/or inner products, one might exhibit sets of harmonic functions which form Hilbert or Banach spaces. Therein fashion, 1 obtains such spaces when a Hardy space, Bloch space, and Bergmann space.