# Theory of the potential

## Potential function

The theory of the potential is a branch of the Mathématiques which developed starting from the concept Physique of Newtonian Potentiel introduced by Poisson for the needs for the Newtonian Mécanique.

It relates to the study of the Opérateur Laplacian and in particular of the harmonic functions and under harmonics. In the Plane complex for example, this theory starts with the study of the function in the following way defined potential and its energy:

That is to say $\ mu$ a Measurement of Borel finished with compact support in $\ mathbb \left\{C\right\}$. The associated potential is defined on $\ mathbb \left\{C\right\}$ by

$p_ \left\{\ driven\right\} \left(Z\right) = \ int \ log|z-w|\; \ mathrm D \ driven \left(W\right) = \ driven * log$

Energy $I \left(\ driven\right)$ of $\ mu$ is defined as being the sum of the potentials:

$I \left(\ driven\right) = \ iint \ log|z-w|\; \ mathrm D \ driven \left(W\right) \; \ mathrm D \ driven \left(Z\right)$

The potential is a simple example of function under harmonic. A theorem of representation of Riesz says to us that under certain very simple conditions, the functions under harmonics are the potential functions, modulo the whole of the harmonic functions. This remark thus gives all its interest to the study of the potential functions.

## Capacity

The capacity is a function acting on the units. It is with the theory of the potential, which measurement is with the theory of measurement. It to some extent makes it possible to measure the size of a unit, within the meaning of the theory of the potential. It appears naturally in several fields of mathematics, in particular in theory of the approximation or complex analysis.

If $E$ is a subset of $\ mathbb \left\{C\right\}$, its capacity is defined as being $\ sup_ \left\{\ driven\right\} e^ \left\{I \left(\ driven\right)\right\}$, the sup being taken to all the measures of probability of Borel.

One can modify the definition of the potential slightly, by replacing the Euclidean distance $d \left(Z, W\right) =|z-w|$ by the hyperbolic pseudo-distance, if $E$ is a subset of the disc unit, or by the shperic distance, if $E$ is a subset of the sphere of Riemann. That will then provide new capacities, respectively the hyperbolic capacity and the spherical or elliptic capacity of E.

A $E$ unit is known as polar if it is of null capacity. One can show that a subset of the disc unit is polar if and only if it is polar relative with the capacities hyperbolic and elliptic.

A polar unit is necessarily of null measurement of Lebesgue. The polar units and $F_ \left\{\ sigma\right\}$ are completely disconnexes. One can see that the reciprocal one with these two assertions is false. The triadic whole of Cantor is completely disconnexe and of null measurement, but is not of null capacity.

The capacity is relatively difficult to handle and study, owing to the fact that it is neither additive, nor under or on additive.

## See too

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