Poisson formula for harmonic functions

Consider a harmonic function $f : D \rightarrow \mathbf{R}$ defined in a domain $D$ in a Euclidean space ${\bf R} ^ { n }$, $n \geq 2$. Let $B ( x _ { 0 } , r )$ denote the open ball

\begin{equation*} B ( x _ { 0 } , r ) = \{ x \in \mathbf{R} ^ { n } : | x - x _ { 0 } | < r \} \end{equation*}

with centre $x _ { 0 }$ and radius $r > 0$. Assume that the closure of this ball is contained in $D$.

The classical Poisson formula expresses that $f$ can be recovered inside the ball by the values of $f$ on the boundary of the ball integrated against the Poisson kernel $P$ for the ball,

\begin{equation} \tag{a1} P ( x , \xi ) = \frac { r ^ { 2 } - | x - x _ { 0 } | ^ { 2 } } { \omega _ { n } r | x - \xi | ^ { n } }, \end{equation}

\begin{equation*} x \in B ( x _ { 0 } , r ) ,\, \xi \in \partial B ( x _ { 0 } , r ), \end{equation*}

where

\begin{equation*} \omega _ { n } = \frac { 2 \pi ^ { n / 2 } } { \Gamma ( \frac { n } { 2 } ) } \end{equation*}

is the $( n - 1 )$-dimensional surface area of the unit ball in ${\bf R} ^ { n }$.

The Poisson formula is

\begin{equation*} f ( x ) = \int _ { \partial B ( x _ { 0 } , r ) } P ( x , \xi ) f ( \xi ) d \sigma ( \xi ), \end{equation*}

where $\sigma$ is the surface measure of the ball, the total mass of which is $\omega _ { n } r ^ { n - 1 }$.

For $x = x_0$ 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 $\Omega$ with a sufficiently smooth boundary and such that the closure of $\Omega$ is contained in $D$. The Poisson kernel (a1) is replaced by the Poisson kernel $P _ { \Omega } ( x , \xi )$ for $\Omega$ and $\sigma$ is replaced by the surface measure on the boundary of $\Omega$. The Poisson kernel, defined on $\Omega \times \partial \Omega$, is given as

\begin{equation*} P _ { \Omega } ( x , \xi ) = \frac { \partial } { \partial n } G _ { \Omega } ( x , \xi ), \end{equation*}

where the inward normal derivative of the Green function $G _ { \Omega } ( x , y )$ for $\Omega$ with respect to the second variable $y$ is used.

For each $\xi \in \partial \Omega$ the function $P _ { \Omega } ( . , \xi )$ is positive and harmonic in $\Omega$, and for each $x \in \Omega$ the measure

\begin{equation*} \mu _ { x } ^ { \Omega } = P _ { \Omega } ( x , \xi ) d \sigma ( \xi ) \end{equation*}

is a probability, called the harmonic measure for $\Omega$ at $x$.

The Poisson kernel has the properties ($\eta \in \partial \Omega$)

\begin{equation*} \operatorname { lim } _ { x \rightarrow \eta } P _ { \Omega } ( x , \xi ) = 0 , \eta \neq \xi, \end{equation*}

and

\begin{equation*} \operatorname { lim } _ { x \rightarrow \eta } \mu _ { x } ^ { \Omega } = \delta _ { \eta } \end{equation*}

where the last limit is in the weak topology for probability measures on $\partial \Omega$ (cf. also Weak convergence of probability measures) and $\delta _ { \eta }$ is the Dirac distribution at $ \eta $.

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 $\Omega$ is related to the solution of the Dirichlet problem: For a function $f : \partial \Omega \rightarrow \mathbf{R}$, the harmonic continuation in $\Omega$ is (under suitable assumptions) given as

\begin{equation*} x \mapsto \int _ { \partial \Omega } f d \mu _ { x } ^ { \Omega }. \end{equation*}

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

\begin{equation*} \mathbf{R} _ { + } ^ { n } = \left\{ ( x , t ) : x \in \mathbf{R} ^ { n - 1 } , t > 0 \right\}. \end{equation*}

The formula is

\begin{equation*} f ( x , t ) = \frac { 2 } { \omega _ { n } } \int _ { \mathbf{R} ^ { n - 1 } } \frac { t f ( y , 0 ) } { ( | x - y | ^ { 2 } + t ^ { 2 } ) ^ { n / 2 } } d y, \end{equation*}

and it is valid for a harmonic function $f ( x , t )$ 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)