Let $\Gamma$ be a rectifiable Jordan curve in the finite complex plane $\mathbb C$ which is regular in the sense of Ahlfors and David. Denote by $L^2_C (\Gamma)$ the space of all complex-valued functions on $\Gamma$ which are square integrable w.r. to the arc-length on $\Gamma$. Let $L^2(\Gamma)$ stand for the space of all real-valued functions in $L^2_C (\Gamma)$ and put
\[ L^2_0 (\Gamma) = \lbrace h \in L^2 (\Gamma)\; \int _{\Gamma} h(\zeta ) |\mathrm{d}\zeta | =0\rbrace. \] Since the Cauchy singular operator is bounded on $L^2_C (\Gamma)$, the Neumann-Poincaré operator $C_1^{\Gamma}$ sending each $h \in L^2 (\Gamma)$ into \[ C_1^{\Gamma} h(\zeta _0) := \Re (\pi \mathrm{i})^{-1} \mathop {\mathrm P. V.}\int _{\Gamma} \frac{h(\zeta )}{\zeta -\zeta _0} \mathrm{d}\zeta , \quad \zeta _0 \in \Gamma , \] is bounded on $L^2(\Gamma)$. We show that the inclusion
\[ C_1^{\Gamma} (L^2_0 (\Gamma)) \subset L^2_0 (\Gamma)
\] characterizes the circle in the class of all $AD$-regular Jordan curves $\Gamma$.
For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.