# Regularity theory for $L_{n}$-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

### Qingbo Huang

Department of Mathematics & Statistics, Wright State University, Dayton, OH 45435, United States of America

## Abstract

We develop interior $W_{2,p,μ}$ and $W_{2,BMO}$ 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, $W_{2,BMO}$ regularity theory holds if operator $T$ is locally semiconvex near infinity and all eigenvalues of $D_{2}T(M)$ are at least $−C∥M∥_{−(1+σ_{0})}$ as $M→∞$. $W_{2,BMO}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W_{2,BMO}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.

## Cite this article

Qingbo Huang, Regularity theory for $L_{n}$-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 1869–1902

DOI 10.1016/J.ANIHPC.2019.06.001