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

Abstract

We derive global W2,p estimates (with n≤p<∞) for viscosity solutions to fully nonlinear elliptic equations under relaxed structural assumptions on the governing operator that are weaker than convexity and oblique boundary conditions as follows:{F(D2u,Du,u,x)=f(x)inΩβ(x)⋅Du(x)+γ(x)u(x)=g(x)on∂Ω, for f∈Lp(Ω) and under appropriate assumptions on the data β,γ, g and Ω⊂Rn. Our approach makes use of geometric tangential methods, which consist 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 pay special attention to the borderline scenario, i.e., f∈BMOp⊋L∞. In such a setting, we prove that solutions enjoy BMOp type estimates for their second derivatives. Finally, 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 for a suitable class of viscosity solutions will also be addressed.