View source for Poisson formula for harmonic functions

Consider a [[Harmonic function|harmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300901.png" /> defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300902.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300904.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300905.png" /> denote the open ball

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300906.png" /></td> </tr></table>

with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300907.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300908.png" />. Assume that the closure of this ball is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p1300909.png" />.

The classical Poisson formula expresses that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009010.png" /> can be recovered inside the ball by the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009011.png" /> on the boundary of the ball integrated against the Poisson kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009012.png" /> for the ball,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009014.png" /></td> </tr></table>

where

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is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009016.png" />-dimensional surface area of the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009018.png" />.

The Poisson formula is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009020.png" /> is the surface measure of the ball, the total mass of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009021.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009023.png" /> with a sufficiently smooth boundary and such that the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009024.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009025.png" />. The Poisson kernel (a1) is replaced by the Poisson kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009028.png" /> is replaced by the surface measure on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009029.png" />. The Poisson kernel, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009030.png" />, is given as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009031.png" /></td> </tr></table>

where the inward normal derivative of the [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009033.png" /> with respect to the second variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009034.png" /> is used.

For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009035.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009036.png" /> is positive and harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009037.png" />, and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009038.png" /> the measure

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009039.png" /></td> </tr></table>

is a probability, called the harmonic measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009040.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009041.png" />.

The Poisson kernel has the properties (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009042.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009043.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009044.png" /></td> </tr></table>

where the last limit is in the weak topology for probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009045.png" /> (cf. also [[Weak convergence of probability measures|Weak convergence of probability measures]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009046.png" /> is the [[Dirac distribution|Dirac distribution]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009047.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009048.png" /> is related to the solution of the [[Dirichlet problem|Dirichlet problem]]: For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009049.png" />, the harmonic continuation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009050.png" /> is (under suitable assumptions) given as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009051.png" /></td> </tr></table>

There is also a Poisson formula for unbounded domains, the simplest of which is for the upper half-space

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009052.png" /></td> </tr></table>

The formula is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009053.png" /></td> </tr></table>

and it is valid for a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130090/p13009054.png" /> in the upper half-space provided it has a continuous extension to the closure and satisfies some growth condition. See [[Hardy spaces|Hardy spaces]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Doob,   "Classical potential theory and its probabilistic counterpart" , Springer  (1984)</TD></TR></table>