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$$D^2=\{(x,y)\in\mathbb{R}^2\vert x^2+y^2 < 1\}$$

be the unit disc in $\mathbb{R}^2$ and

$$\mathcal{C}_{2\pi}(\mathbb{R})=\{f:\mathbb{R}\longrightarrow\mathbb{R}\vert f \text{ is continuous and \(2\pi\)-periodic}\}$$

the set of continuous and $2\pi$-periodic real functions. The Dirichlet problem for the Laplace equation in the unit disc is stated as

$$ \begin{cases} \Delta u = 0 & \text{ in \(D^2\)}\\ u=f & \text{ in \(\partial D^2\)}\\ \end{cases}, \qquad u\in\mathcal{C}^2(D^2)\cup\mathcal{C}(\overline{D^2}) $$

for $\Delta=\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}$ the Laplace operator and some $f\in\mathcal{C}_{2\pi}(\mathbb{R})$ inducing $f\in\mathcal{C}(\partial D^2)$ by the parameterization $\mathbb{R}\longrightarrow\partial D^2$, $t\longmapsto(\cos t,\sin t)$