Find stationary temperature distribution in the hemisphere of radius 1 in $\mathbb{R}^3$ with conditions $$\left. \frac{\partial T}{\partial n}\right|_{z=0}=Q=const,\;\;\; \left. T\right|_{side surface}=0.$$
MY ATTEMPT: Let $u$ be the solution. $v=u+Qz$. And continue it to the lower half-space to get even function.
Then $v$ is the solution to the problem $$\left. \frac{\partial T}{\partial n}\right|_{z=0}=0,\;\;\; \left. T\right|_{side surface, z\geq 0}=Qz, \left.T\right|_{side surface, z\leq 0}=-Qz.$$ Then I rewrite it in polar coordinates $$Q|z|=Qr|cos(\theta)|.$$ Then I write standart decomposition into the linear combination of Legendre polynomials and find corresponding coeffitients.
Say please am I on the right path? There are terrible calculations that come to a dead end. Please help me.