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)