Is the Dirichlet problem
\begin{cases} \Delta u = 0 \text{ in} \mathbb{R}^n_+ \\ u\vert_{\partial \mathbb{R}^n_+} = \varphi \end{cases}
always solvable for arbitrary continuous and bounded $\varphi$? Do we have to impose some decaying behaviour on $\varphi$?
Thanks!
In $\mathbb{R}^2$, a solution is $$ u(x,y)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y}{(x-x')^2+y^2}\varphi(x')dx',\;\;\; y > 0. $$ This the Poisson representation. Such a solution is not unique because there are solutions of Laplace's equation in the upper half-plane that vanish on the real axis--namely $v(x,y)=y$. However, if you require $u$ to be bounded throughout, then the solution is unique. There is a Poisson solution for a half space of $\mathbb{R}^{n}$, too, using the Poisson kernel (Wikipedia link.)
