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

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

Sharp regularity estimates for quasilinear evolution equations

We establish sharp geometric $C^{1+\alpha}$ regularity estimates for bounded weak solutions of evolution equations of $p$-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.Comment: 18 page