# Regularity theory for Ln-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

A subscription is required to access this article.

## 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 $−C\|M\|^{−(1 + \sigma _{0})}$ as $M\rightarrow \infty$. $W^{2,\text{BMO}}$ 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.

## Cite this article

Qingbo Huang, Regularity theory for Ln-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