Let us consider the Laplace problem $\Delta u=0$ with the unitary circle as domain.
The boundary conditions are: \begin{cases} u=1 & \quad \text{if } \rho=1 \text{ and } \theta\in[\pi/2-\alpha; \pi/2+\alpha]\\ u=-1 & \quad \text{if } \rho=1 \text{ and } \theta\in[3\pi/2-\alpha; 3\pi/2+\alpha]\\ \nabla u\cdot \hat{\rho}=0 & \quad \text{if } \rho=1 \text{ and } \theta\in\{[0; \pi/2-\alpha]\cup[\pi/2+\alpha; 3\pi/2-\alpha]\cup[3\pi/2+\alpha; 2\pi]\} \end{cases} where $\hat{\rho}$ is the unit radial versor.
I am interested in finding an analytical solution over the whole domain.