Oblique derivative problems for elliptic equations on conical domains

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