Poisson integral

An integral representation of the solution of the Dirichlet problem for the Laplace equation in the simplest domains. Thus, the Poisson integral for the ball in the Euclidean space , , of radius and with centre at the coordinate origin has the form

(1)

where is a given continuous function on the sphere of radius ,

is the Poisson kernel for the ball, is the area of the sphere , and is the surface element on .

S.D. Poisson

arrived at formula (1) in the case as an integral formula for the sum of the trigonometric series

where , are the Fourier coefficients of the function , and and are the polar coordinates of the points and , respectively; in this case the Poisson kernel has the form

(2)

(for applications of the Poisson integral in the theory of trigonometric series see [3] and also Abel–Poisson summation method).

The Poisson integral for the half-space

has the form

(3)

where

is the volume element in , is a bounded continuous function on and

is the Poisson kernel for the half-space. The formulas (1) and (3) are particular cases of the Green formula

(4)

giving the solution of the Dirichlet problem for domains with smooth boundary using the derivative of the Green function in the direction of the interior normal to at the point . Sometimes formula (4) is also called the Poisson integral.

The basic properties of the Poisson integral are: 1) is a harmonic function of the coordinates of the point ; and 2) the Poisson integral gives the solution of the Dirichlet problem with boundary data in the class of (bounded) harmonic functions, that is, the function extended to the boundary of the domain by the values is continuous in the closed domain. Applications of the Poisson integral in classical mathematical physics are based on these properties (see [4]).

The Poisson integral understood in the sense of Lebesgue when is a summable function, for example on , is called a Poisson–Lebesgue integral; an integral of the form

(5)

with respect to an arbitrary finite Borel measure concentrated on is called a Poisson–Stieltjes integral. The class of harmonic functions representable by an integral (5) is characterized by the fact that any function is the difference of two non-negative harmonic functions in . The class of functions representable by a Poisson–Lebesgue integral is a proper subclass of the class and contains all bounded harmonic functions in . For almost-all points with respect to Lebesgue measure on , a Poisson–Stieltjes integral (5) has angular boundary values coinciding with the values of the derivative of the measure with respect to Lebesgue measure. The theory of Poisson–Stieltjes and Poisson–Lebesgue integrals has also been constructed for the half-space (see [5]).

Various modifications of the Poisson integral play a large role in the theory of analytic functions of several complex variables and in its applications to quantum field theory. For example, the Poisson kernel for the polydisc

in the complex space is obtained by multiplying the kernels (2):

The corresponding Poisson integral

with respect to the distinguished boundary of the polydisc gives a multi-harmonic function , , taking the continuous values on . Generalizations in the form of Poisson–Lebesgue and Poisson–Stieltjes integrals are also considered (see [6]).

In quantum field theory Poisson integrals are also applied to tube domains in the complex space over a convex open acute cone in the space (with vertex at the origin). These have the form

A Poisson integral of the form (3) for the half-plane when is a particular case of such Poisson integrals for tube domains; the Poisson integral for bounded symmetric domains in the space is the same as the Poisson integral for a tube domain in the space of matrices. Taking the density of the Poisson integral to be a generalized function and the Poisson integral itself to be the convolution of with the Poisson kernel one arrives at the important concept of the Poisson integral for certain classes of generalized functions (see [7]–[9]).

References

[1a]	S.D. Poisson, J. Ecole R. Polytechn. , 11 (1820) pp. 295–341
[1b]	S.D. Poisson, "Suite du mémoire sur les intégrales définies et sur la sommation des séries" J. Ecole R. Polytechn. , 12 (1823) pp. 404–509
[2]	H.A. Schwarz, "Ueber die Integration der partiellen Differentialgleichung für die Fläche eines Kreises" Vierteljahrsschr. Naturforsch. Ges. Zurich , 15 (1870) pp. 113–128
[3]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[4]	A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (Translated from Russian)
[5]	E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" Itogi Nauk. Mat. Anal. Teor. Veroyatnost. Regulirovanie. 1962 (1964) pp. 83–100 (In Russian)
[6]	W. Rudin, "Function theory in polydiscs" , Benjamin (1969)
[7]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)
[8]	L.K. Hua, "Harmonic analysis of functions of several complex variables in the classical domains" , Amer. Math. Soc. (1963) (Translated from Russian)
[9]	E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)

Comments

The Poisson kernel for the exterior of the ball is given by

In case of the unit ball in , there are several Poisson-type kernels, e.g. the classical one, solving the classical Dirichlet problem, and the invariant Poisson kernel, which is invariant under holomorphic automorphisms of the unit ball. It reads

where and . This is the analogue for the ball of the Poisson kernel with respect to the distinguished boundary of the polydisc, described above. The invariant Poisson kernel solves the Dirichlet problem for the so-called invariant Laplacian, see [a1].

References

[a1]	W. Rudin, "Function theory in the unit ball in " , Springer (1980)