Chapter
MSC:
31-03
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Summary:
Since the time of Poisson and Green it has been known that a charge placed inside a ball can be replaced by a charge distributed on the sphere $S$ in such a way that the potentials of the original charge and of the swept out charge on $S$ coincide outside the ball. More generally, given a domain in $R^3$ bounded by a sufficiently smooth surface $S$ and a mass or charge $μ$ inside $S$, Gauss pointed out in 1839 that $μ$ can be swept out onto $S$, that is, a distribution $μ$ on $S$ can be found such that the Newtonian potentials of $μ$ and $μ$ coincide outside the domain. Such a transformation of mass is called balayage. This contribution, The central notion of potential theory: balayage, is a survey of the role played by balayage in the historical development of potential theory with particular emphasis on the first half of the 20th century. In particular, we discuss connections with the Dirichlet problem and notions such as capacity, harmonic measure, energy, exceptional sets, thinness etc. We also mention the place of balayage in probabilistic potential theory related to Brownian motion and in abstract potential theory
Search
Partner of
