We prove the existence of smooth solutions of two-dimensional nonuniformly elliptic curvature equations subject to a nonlinear oblique boundary condition. These are equations whose principal part is given by a suitable symmetric function of the principal curvatures of the graph of the solution $u$. The types of boundary conditions we are able to treat are the same as those we considered in earlier work on Hessian equations.
"Nonlinear oblique boundary value problems for two-dimensional curvature equations." Adv. Differential Equations 1 (3) 301 - 336, 1996. https://doi.org/10.57262/ade/1366896042