Extend Harnack and Hölder estimates to singular drifts

Determine whether the Harnack inequalities and Hölder regularity estimates for fully nonlinear uniformly elliptic equations imposed only when the solution gradient lies outside the union of a closed ball and a closed purely one-unrectifiable exceptional set can be extended to equations with singular drifts.

Background

The paper establishes Harnack inequalities and Hölder regularity estimates for viscosity solutions of fully nonlinear uniformly elliptic equations whose equations are imposed only for gradients outside a prescribed ball and a closed purely one-unrectifiable set, possibly unbounded. The estimates are obtained under a bounded drift coefficient in the structural condition for the fully nonlinear operator.

The authors explicitly identify extending these results to singular drifts as an unresolved problem, referring to related work on equations with singular drifts. Such an extension would test whether the geometric contact-set arguments used to exclude exceptional gradients remain effective when the first-order term is singular or unbounded.