Let $D = \lbrace\left(x_1,x_2,x_3\right) \in \mathbb{R}^3 : x_3 \gt 0\rbrace$. Solve Dirichlet problem for Poisson equlity $-\Delta=f$ in $D$, $u = g$ on $\Gamma$ in space of limited functions, if:
(a) $f=0,\ g\left(x_1,x_2\right)=\cos x_1.\cos x_2$
(b) $f=e^{-x_3}\sin x_1.\cos x_2,\ g\left(x_1,x_2\right)=0$
(c) $f=0,\ g\left(x_1,x_2\right)=\dfrac{1}{\sqrt{1+x_1^2+x_2^2}}$
Do I need to use Poisson formula for problems (a) and (c)? I tried with it, but solving integral is not easy. For problem (b) I have no idea.