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31: Potential theory


Introduction

Potential theory may be viewed as the mathematical treatment of the potential-energy functions used in physics to study gravitation and electromagnetism.

If some electrically charged particles are distributed in space, then a function U is defined on all of space (except right where the particles are) which measures the potential energy at each point. This function is harmonic, that is, it satisfies the Laplace equation

    d^2 U / dx^2  +  d^2 U / dy^2  +  d^2 U / dz^2  = 0,
a condition which, for example, forces the value of U at a point to be the average of its values on a ball centered at that point.

Classical problems include the determination of harmonic functions taking prescribed values at a point, on a sphere, and so on (the Dirichlet problem) -- that is, determining the force field which results from a particular arrangement of force sources.

Harmonic functions in the plane include the real and complex parts of analytic functions, so Potential Theory overlaps Complex Analysis. (Actually potential theory in the plane is rather different from in higher dimensions, since the fundamental solution of the Laplace equation, corresponding to a single point charge, is 1/r^(n-2) in n-dimensional space, but log(r) in the plane. Nonetheless, the results in all dimensions often have cognates in complex analysis.)

The process of determining the potential at points on the interior of a region when the potential on the boundary is known involves integrating on the boundary using a weighting function derived from Green's functions. Clearly this approach suggests close affinities with the areas of measure theory and integration. (Indeed, potential theory also considers capacity, a sort of measure of sets related to capacitance in electronics.)

The related notions of subharmonic and superharmonic functions also arise (e.g. from considering potential functions over regions including the force sources). Determination of the existence of solutions to the Laplacian may involve extremal arguments within these families of functions; thus tools from functional analysis are necessary as well. There is a delicate interplay between the strong harmonic condition in an open set and very weak conditions (e.g. mere continuity) at the boundary.

Interesting terminological tidbits in this area include the term "balayage" (a "sweeping-out" to obtain a certain minimal superharmonic function), and the book title "Complex Manifolds without Potential Theory" (S. S. Chern), which theoretically has the potential to conjure up manifold complex interpretations. :-)

History

Applications and related fields

For probabilistic potential theory, See 60J45 [Schematic of subareas and related areas]

Subfields

This is among the smaller areas in the Math Reviews database.

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

Lukes, Jaroslav; Netuka, Ivan: "What is the right solution of the Dirichlet problem?", Romanian-Finnish Seminar on Complex Analysis (Proc., Bucharest, 1976), pp. 564--572; Lecture Notes in Math., 743; Springer, Berlin, 1979. MR80m:31015

"Reviews in Complex Analysis 1980-1986" (four volumes), American Mathematical Society, Providence, RI, 1989. 3064 pp., ISBN 0-8218-0127-9: Reviews reprinted from Mathematical Reviews published during 1980--1986.

Helms, "Potential Theory" is a reasonable mathematical introduction.

For connections to complex analysis consider Ransford, "Potential Theory in the Complex Plane" (LMS student texts #28)

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