Balayage method
A method for solving the Dirichlet problem for the Laplace equation, developed by H. Poincaré ([1], [2], see also [4]), which will now be described. Let be a bounded domain of the Euclidean space
,
, let
be the boundary of
. Let
be the Dirac measure concentrated at the point
, let
be the Newton potential of the measure
for
, or the logarithmic potential of the measure
if
. A balayage (or sweeping) of the measure
from the domain
to the boundary
is a measure
on
whose potential
coincides outside
with
and is not larger than
inside
; this measure
is unique and coincides with the harmonic measure on
for the point
. The balayage of an arbitrary positive measure, concentrated on
, is defined in a similar manner. If
is a sphere, the density of the mass distribution
, i.e. the derivative of the measure
, is identical with the Poisson kernel (cf. Poisson integral). In general, if the boundary
is sufficiently smooth, the measure
is absolutely continuous, and the density of the mass distribution
coincides with the normal derivative of the Green function for
. The measure
serves to write down the solution of the Dirichlet problem as the so-called formula of de la Vallée-Poussin:
![]() |
where is a function defined on
.
In his original publication on the balayage method, Poincaré began by demonstrating the geometrical construction of the process for a sphere. Then, basing himself on Harnack's theorems (cf. Harnack theorem) and on the fact that it is possible to exhaust the domain by a sequence of spheres
, he constructed an infinite sequence of potentials
in which each potential
is obtained from the preceding one,
, by the balayage method of moving the masses from the domain
to its boundary, and which reduces to solving the Dirichlet problem for a sufficiently smooth domain
(for a detailed discussion of the conditions of applicability of the balayage method, see [3]).
In modern potential theory [5], [6] the balayage problem is treated as an independent problem, resembling the Dirichlet problem, and it turns out that the balayaged measure can be considered on sets of a general nature. For instance, the balayage problem in its simplest form is to find, for a given mass distribution inside a closed domain
, a mass distribution
on
such that the potentials of both distributions coincide outside
. If the boundary
is smooth, the solution of the balayage problem for
will be an absolutely continuous measure
. Its density, or the derivative
,
, may be written down in terms of the Green function
of the domain
in the form
![]() | (*) |
where is the derivative of
in the direction of the interior normal to
at the point
. Inside the domain
the potentials satisfy the inequality
, i.e. balayage inside the domain results in a decrease of the potential. If
is the Dirac measure at the point
, formula (*) yields
, i.e. the normal derivative of the Green function is the density of the measure obtained by balayage of the unit mass concentrated at the point
. Generalization of formula (*) yields an expression for the balayaged measure
of an arbitrary Borel set
for an arbitrary domain
:
![]() |
where is the harmonic measure of
with respect to the domain
at the point
.
If is an arbitrary compact set in
and
is a bounded positive Borel measure, the balayage (or sweeping) of the measure
onto the compact set
is a measure
on
such that
everywhere, and such that quasi-everywhere on
, i.e. with the possible exception of a set of points of exterior capacity zero,
. Such a formulation of the balayage problem, which is more general than balayage from a domain, may also be extended to potentials of other types, e.g. Bessel potentials or Riesz potentials (cf. Bessel potential; Riesz potential). Balayage of measures onto arbitrary Borel sets
is also considered.
The problem of balayage for superharmonic functions (cf. Superharmonic function) has been similarly formulated. Let be a non-negative superharmonic function on a domain
. The balayage of the function
onto a compact set
is the largest superharmonic function
such that 1) its associated measure is concentrated on
; 2)
everywhere; and 3)
quasi-everywhere on
.
In abstract potential theory (cf. Potential theory, abstract) the balayage problem in both its formulations is solved for sets in an arbitrary harmonic space
, i.e. in a locally compact topological space
which permits the isolation of an axiomatically defined sheaf of harmonic functions. This axiomatic approach makes it possible to consider the balayage problem for potentials connected with partial differential equations of a more general nature [7]. For the balayage method in stochastics cf. [8].
References
[1] | H. Poincaré, "Sur les équations aux dérivees partielles de la physique mathématique" Amer. J. Math. , 12 : 3 (1890) pp. 211–294 |
[2] | H. Poincaré, "Theorie du potentiel Newtonien" , Paris (1899) |
[3] | Ch.J. de la Vallée-Poussin, "Le potentiel logarithmique, balayage et répresentation conforme" , Gauthier-Villars (1949) |
[4] | L.N. Sretenskii, "Theory of the Newton potential" , Moscow-Leningrad (1946) (In Russian) |
[5] | N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) |
[6] | M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1965) |
[7] | C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) |
[8] | P.A. Meyer, "Probability and potentials" , Blaisdell (1966) |
Comments
Balayage is also referred to as sweeping of a measure. A classic reference for problems in potential theory related to Green functions is [a1].
In probabilistic potential theory the swept measure on
of a probability measure
concentrated on
turns out to be the distribution of a standard Brownian motion on
, which has initial distribution
, at the moment of first hitting
.
Another link with probabilistic potential theory is provided by the fact that, for each sufficiently nice harmonic space, there exists a Hunt process whose excessive functions are the positive hyper-harmonic functions. If
denotes the hitting distribution of a compact set
, then
for positive superharmonic functions
, and the balayage of a measure
on
is given by
. Therefore, the notion of balayage of a function or a measure can also be defined in terms of the potential kernel of a semi-group of kernels, see [a3].
References
[a1] | M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975) |
[a2] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 |
[a3] | C. Dellacherie, P.A. Meyer, "Probabilités et potentiel" , 1–2 , Hermann (1975–1983) |
Balayage method. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Balayage_method&oldid=11819