Consider $$\left\{\begin{array}{rll} - \Delta u & = & 0, \ \ \mbox{in} \ \mathbb{R}^2_+\\ u(x,0) & = & e^{-\pi x^2}, \ \ x \in \mathbb{R}. \end{array} \right.$$ Show that $$\max \{u(x, y); \ (x, y) \in [0,1] \times [0,1]\} = 1.$$
Outline of the proof: To solve this problem, I tried to use the fact that if $u$ is the solution to the problem $$\left\{\begin{array}{rll} - \Delta u & = & 0, \ \ \mbox{in} \ \mathbb{R}^2_+\\ u & = & g, \ \ \mbox{on} \ \partial \mathbb{R}^2_+, \end{array} \right.$$ we have $$u (x_1, x_2) = \dfrac{2x_2}{2\omega_2} \int_{\partial \mathbb{R}^2_+} \dfrac{g(y)}{|x-y|^2} dy,$$ with $x = (x_1, x_2) \in \mathbb{R}^2 _+$ and $y \in \partial \mathbb{R}^2_+$. However, I was confused on how to solve this integral under the given conditions... Help me, I have difficulty in this type of solution!