integral of Poisson Kernel P(x,y) (on the boundary) for a bounded domain

Question

show that integral of Poisson Kernel P(x,y) (on the boundary) for any bounded domain in $R^n$ is equal to 1 , in general

i.e
How can I show

$\int_{\delta b} \mathrm{P(x,y)}\, ds(y) = 1$

Answer 1

The Poisson kernel provides the solution of the harmonic Dirichlet problem: $$ \left\{\begin{array}{ccc} \Delta u=0 \quad\text{in}\,\,\Omega, \tag{1}\\ u=f \quad\text{on}\,\,\,\partial\Omega. \end{array} \right. $$ That is $$ u(x)=\int_{\partial\Omega} P(x,y)\,f(y)\,dy, $$ satisfies $(1)$. In particular $u(x)=\int_{\partial\Omega} P(x,y)\,dy,$ corresponds to the case when $f\equiv 1$, in which case the solution of $(1)$ is equal to $1$ as well, i.e., $u\equiv 1$.