Oblique boundary value problems for augmented Hessian equations III

Abstract

In bounded domains, without any geometric conditions, we study the existence and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and classical C^2), solutions of general semilinear oblique boundary value problems for degenerate, (and non-degenerate), augmented Hessian equations, with strictly regular associated matrix functions. By establishing local second derivative estimates at the boundary and proving viscosity comparison principles, we show that the solution is correspondingly smooth near boundary points where the appropriate uniform convexity is satisfied.

This paper continues our study in Part I, when the boundary geometric conditions are not assumed to hold everywhere, leading to interpretation of the oblique boundary condition in a weak viscosity sense