Let $ \Omega \subseteq \mathbb{R}^2 $ be a bounded and simply connected domain with its boundary $\partial \Omega \in C^2 $, $f \in C(\partial \Omega) $ be some given function, then for $$ \Delta u = 0 \ \textrm{in} \ \Omega, \quad u = f \ \textrm{on} \ \partial \Omega $$ the solution can be represented as $$ u(x) = \int_{\partial \Omega} \psi(y) \ln \vert x-y \vert ds(y), \quad x \in \bar \Omega $$ if and only if $ \psi \in C(\partial \Omega) $ solves $$ \int_{\partial \Omega} \psi(y) \ln \vert x-y \vert ds(y) = f(x), \quad x \in \partial \Omega $$ The above proposition should be a classical result in potential theory. Could anyone recommend some reference?
Moreover, we would like to know if the condition "simply connected" can be relaxed to multiply connected?For example, from a disc to an annulus? Thanks in advance!