It's well known that: Given a continuous function defined on the boundary of the disk, then there exists a unique harmonic function in the interior of the disk. What if we replace the disk by a cone? 

Take $\Omega$ as 
$$z^2=x^2+y^2,x^2+y^2\leq1$$ for example.

What's the regularity of boundary function $f$ that can guarantee the existence of a harmonic function of $u$? If $u$ exists, what's the regularity of $u$? 
Does $u$ satisfy the Green's formula? i.e.,
$$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$