Regularity theory for $L^n$-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

Abstract

We develop interior $W^{2,p,\mu }$ and $W^{2,\text{BMO}}$ regularity theories for $L^n$-viscosity solutions to fully nonlinear elliptic equations $T(D^{2}u,x)=f(x)$, where $T$ is approximately convex at infinity. Particularly, $W^{2,\text{BMO}}$ regularity theory holds if operator $T$ is locally semiconvex near infinity and all eigenvalues of $D^{2}T(M)$ are at least $-C\Vert M{\Vert }^{-(1+{\sigma }_0)}$ as $M\to \infty$. $W^{2,\text{BMO}}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W^{2,\text{BMO}}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.