Weighted Orlicz regularity for fully nonlinear elliptic equations with oblique derivative at the boundary via asymptotic operators

Abstract

We prove weighted Orlicz-Sobolev regularity for fully nonlinear elliptic equations with oblique boundary condition under asymptotic conditions of the following problem: $F(D^{2}u,Du,u,x)=f(x)$ in the bounded domain $\Omega \subset {\mathbb{R}}^n$($n\ge 2$) and $\beta \cdot Du+\gamma u=g$ on $\partial \Omega$, under suitable assumptions on the source term $f$, data $\beta ,\gamma$ and $g$. Our approach guarantees such estimates under conditions where the governing operator $F$ does not require a convex (or concave) structure. We also deal with weighted Orlicz-type estimates for the obstacle problem with oblique derivative condition on the boundary. As a final application, the developed methods provide weighted Orlicz-BMO regularity for the Hessian, provided that the source lies in that space and in weighted Orlicz space associated.

27 pages. arXiv admin note: substantial text overlap with arXiv:2302.09177