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We study the oblique derivative problem for uniformly elliptic equations oncone domains. Under the assumption of axi-symmetry of the solution, we findsufficient conditions on the angle of the oblique vector for H\"olderregularity of the gradient to hold up to the vertex of the cone. The proof ofregularity is based on the application of carefully constructed barrier methodsor via perturbative arguments. In the case that such regularity does not hold,we give explicit counterexamples. We also give a counterexample to regularityin the absence of axi-symmetry. Unlike in the equivalent two dimensionalproblem, the gradient H\"older regularity does not hold for all axi-symmetricsolutions, but rather the qualitative regularity properties depend on both theopening angle of the cone and the angle of the oblique vector in the boundarycondition.

Oblique Derivative Problems for Elliptic Equations on Conical Domains