Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. We seek to solve the Dirichlet problem on the unit disk; in other words, we need to determine, if any exist, the function or functions $f$ defined and continuous on $\overline{D(0,1)}$ (closed disk), of class $\mathcal{C}^2$ on $D(0,1)$, and such that $$\left\{\begin{array}{l} \Delta f = 0 \text{ on } D(0,1) \\ \forall t \in \mathbb{R},\ f(\cos(t), \sin(t)) = h(t) \end{array}\right.$$ Show the existence and uniqueness of the solution to this Dirichlet problem.
Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. We seek to solve the Dirichlet problem on the unit disk; in other words, we need to determine, if any exist, the function or functions $f$ defined and continuous on $\overline{D(0,1)}$ (closed disk), of class $\mathcal{C}^2$ on $D(0,1)$, and such that
$$\left\{\begin{array}{l} \Delta f = 0 \text{ on } D(0,1) \\ \forall t \in \mathbb{R},\ f(\cos(t), \sin(t)) = h(t) \end{array}\right.$$
Show the existence and uniqueness of the solution to this Dirichlet problem.