potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...

, a mathematical discipline, balayage (from French: ''

balayage In potential theory, a mathematical discipline, balayage (from French: '' balayage'' "scanning, sweeping") is a method devised by Henri Poincaré for reconstructing a harmonic function In mathematics, mathematical physics and the theory of s ...

'' "scanning, sweeping") is a method devised by

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

for reconstructing a

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...

in a domain from its values on the boundary of the domain. In modern terms, the balayage operator maps a

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

''μ'' on a closed domain ''D'' to a measure ''ν'' on the boundary ''∂ D'', so that the

Newtonian potential In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object ...

s of ''μ'' and ''ν'' coincide outside

\bar D

. The procedure is called balayage since the mass is "swept out" from ''D'' onto the boundary. For ''x'' in ''D'', the balayage of ''δ''_''x'' yields the

harmonic measure In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the ...

''ν''_''x'' corresponding to ''x''. Then the value of a harmonic function ''f'' at ''x'' is equal to :

f(x) = \int_ f(y) \, d\nu_x(y).

References

Potential theory {{Mathanalysis-stub