Potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century
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
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 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
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 symmetries of the -dimensional
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
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
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
One important use of these inequalities is to prove
Spaces of harmonic functions
Since the Laplace equation is linear, the set of harmonic functions defined on a given domain is, in fact, a
See also
References
- JSTOR 1969517.
- A.I. Prilenko, E.D. Solomentsev (2001) [1994], "Potential theory", Encyclopedia of Mathematics, EMS Press
- E.D. Solomentsev (2001) [1994], "Abstract potential theory", Encyclopedia of Mathematics, EMS Press
- ISBN 0-387-95218-7.
- ISBN 0-486-60144-7.
- L. L. Helms (1975). Introduction to potential theory. R. E. Krieger ISBN 0-88275-224-3.
- ISBN 3-540-41206-9.
- arXiv:math/0001057.
- This article incorporates material from potentialtheory on Creative Commons Attribution/Share-Alike License.