Dirichlet Problem

Question

I have to solve the following Dirichlet Problem

$$\Delta u=0\quad\text{in}\,\,\, D,$$ $$u(\mathrm{e}^{it})=\frac{1}{2}(\mathrm{e}^{it}+\mathrm{e}^{-it}),$$ for $$u \in C^2(D)\cap C(\overline{D}).$$

Now should I calculate Laplacian of u than to equal it with zero or what should be first step?

Answer 1

Hint. If you translate the boundary condition in cartesian coordinates, then it looks like $$ u(x,y)=x, \quad \text{whenever}\,\,x^2+y^2=1. $$ Also note, that $u(x,y)=x$ is a harmonic function in $\mathbb R^2$.