Oblique derivative problems for elliptic equations on conical domains

We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under assumption of axi-symmetry solution, we find sufficient conditions angle vector Hölder regularity gradient to hold up vertex cone. The proof is based application carefully constructed barrier methods or via perturbative arguments. In case that such does not hold, give explicit counterexamples. also a counterexample in absence axi-symmetry. Unlike equivalent two-dimensional problem, all axi-symmetric solutions, but rather qualitative properties depend both opening and boundary condition.