Poisson formula for harmonic functions
Consider a harmonic function 
defined in a domain 
in a Euclidean space 
, 
. Let 
denote the open ball
 |
with centre 
and radius 
. Assume that the closure of this ball is contained in 
.
The classical Poisson formula expresses that 
can be recovered inside the ball by the values of 
on the boundary of the ball integrated against the Poisson kernel 
for the ball,
 | (a1) |
 |
where
 |
is the 
-dimensional surface area of the unit ball in 
.
The Poisson formula is
 |
where 
is the surface measure of the ball, the total mass of which is 
.
For 
the formula reduces to the mean-value theorem for harmonic functions, stating that the value at the centre of the ball is the average over the boundary of the ball.
The same type of formula holds when the ball is replaced by a bounded domain 
with a sufficiently smooth boundary and such that the closure of 
is contained in 
. The Poisson kernel (a1) is replaced by the Poisson kernel 
for 
and 
is replaced by the surface measure on the boundary of 
. The Poisson kernel, defined on 
, is given as
 |
where the inward normal derivative of the Green function 
for 
with respect to the second variable 
is used.
For each 
the function 
is positive and harmonic in 
, and for each 
the measure
 |
is a probability, called the harmonic measure for 
at 
.
The Poisson kernel has the properties (
)
 |
and
 |
where the last limit is in the weak topology for probability measures on 
(cf. also Weak convergence of probability measures) and 
is the Dirac distribution at 
.
There are only a few cases where the Poisson kernel can be given in closed form as for the ball.
The Poisson formula for a domain 
is related to the solution of the Dirichlet problem: For a function 
, the harmonic continuation in 
is (under suitable assumptions) given as
 |
There is also a Poisson formula for unbounded domains, the simplest of which is for the upper half-space
 |
The formula is
 |
and it is valid for a harmonic function 
in the upper half-space provided it has a continuous extension to the closure and satisfies some growth condition. See Hardy spaces.
References
| [a1] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) |
Poisson formula for harmonic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_formula_for_harmonic_functions&oldid=50409