Let $H=\{(x,y,z)\in\mathbb R^3\,|\,z\geq 0\}$, let $f:H\to\mathbb R$ be harmonic on the interior of $H$, and let $f$ satisfy the boundary condition $f(x,y,0) = a$ for some $a\in\mathbb R$.
One easily verifies that $f(x,y,z) = a+bz$ satisfies all of these conditions for any $b\in\mathbb R$.
Is this the most general function that does so, or is there a larger class of harmonic functions satisfying the above boundary condition?
Thoughts. If one assumes that $f$ is translation invariant in $x$ and $y$, then I believe that the function I wrote down is the most general, but I can't see why the solution would need to possess this invariance. Perhaps it's possible to show that translation-invariance of the boundary condition leads to translation invariance of the solution on the entire half space $H$?