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

Take $\Omega$ as $$z=x^2+y^2,x^2+y^2\leq1$$ 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? ie $$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$