Consider the two-dimensional domain

$$G=\left\{(x, y) \mid R_{1}^{2}<x^{2}+y^{2}<R_{2}^{2}\right\}$$

where $0<R_{1}<R_{2}<\infty$. Solve the Dirichlet boundary value problem for the Laplace equation

$$\begin{gathered} \Delta u=0 \text { in } G, \\ u=u_{1}(\varphi), r=R_{1}, \\ u=u_{2}(\varphi), r=R_{2}, \end{gathered}$$

where $(r, \varphi)$ are polar coordinates. Assume that $u_{1}, u_{2}$ are $2 \pi$-periodic functions on the real line and $u_{1}, u_{2} \in L_{l o c}^{2}(\mathbb{R})$.

[Hint: Use separation of variables in polar coordinates, $u=R(r) \Phi(\varphi)$, with periodic boundary conditions for the function $\Phi$ of the angle variable. Use an ansatz of the form $R(r)=r^{\alpha}$ for the radial function.]