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Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$.
Is the Poisson integral of $f$, $$ Pf(x)=\int_{\partial D} f(t) \frac{c_x}{\left|x-t\right|^n}dt, $$ with $c_x$ an appropriate constant, always a solution to the Dirichlet problem?

I know the general solution of the Dirichlet problem is given in term of the Green function.
But I can't decide if, for arbitrary domain, the Green function is always the Poisson kernel or not.

As the Poisson kernel is an approximation to the identity, I guess the answer is yes.

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1 Answer 1

up vote 2 down vote accepted

No, the Poisson integral is only for balls.

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