Oblique Derivative Problems for Elliptic Equations on Conical Domains

TL;DR

This paper investigates the regularity of solutions to oblique derivative problems for elliptic equations on conical domains, identifying conditions under which gradient regularity holds and providing counterexamples where it fails.

Contribution

It establishes sufficient conditions for gradient Hölder regularity in axi-symmetric solutions and presents explicit counterexamples, highlighting the dependence on cone and oblique vector angles.

Findings

Gradient regularity depends on cone and oblique vector angles.

Counterexamples show regularity failure without axi-symmetry.

Regularity results differ from two-dimensional cases.

Abstract

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 H\"older 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 H\"older 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.