Sharp Hessian estimates for fully nonlinear elliptic equations under relaxed convexity assumptions, oblique boundary conditions and applications

TL;DR

This paper develops global Hessian estimates for fully nonlinear elliptic equations with relaxed assumptions, using geometric tangential methods and addressing borderline cases with BMO estimates, applicable to obstacle problems with oblique boundary conditions.

Contribution

It introduces a novel approach using geometric tangential methods to derive Hessian estimates under weaker structural assumptions than convexity and oblique boundary conditions.

Findings

Established global Hessian estimates under relaxed assumptions.

Proved BMO type second derivative estimates in borderline cases.

Applied results to obstacle problems with oblique boundary conditions.

Abstract

In this work we derive global estimates for viscosity solutions to fully nonlinear elliptic equations under relaxed structural assumptions on the governing operator which are weaker than convexity and oblique boundary conditions and under suitable assumptions on the dat. Our approach makes use of geometric tangential methods, which consists of importing "fine regularity estimates" from a limiting profile, i.e., the Recession operator, associated with the original second order one via compactness and stability procedures. As a result, we devote a special attention to the borderline scenario. In such a setting, we prove that solutions enjoy BMO type estimates for their second derivatives. In the end, as another application of our findings, we obtain Hessian estimates to obstacle type problems under oblique boundary conditions and no convexity assumptions, which may have their own mathematical interest. A density result in a suitable class of viscosity solutions will be also addressed.