Global Calderón-Zygmund estimates for asymptotically convex fully nonlinear Grad-Mercier type equations

In this paper, we consider the following Dirichlet problem for the fully nonlinear elliptic equation of Grad-Mercier type under asymptotic convexity conditions \begin{equation*} \left{ \begin{array}{ll} F(D^2u(x),Du(x),u(x),x)=g(|{y\in Ω:u(y)\ge u(x)}|)+f(x) & $\text{in }$ Ω, u=ψ&$\text{on }$ \partial Ω. \end{array} \right. \end{equation*} In order to overcome the non-convexity of the operator $F$ and the nonlocality of the nonhomogeneous term $g$, we apply the compactness methods and frozen technique to prove the existence of the $W^{2,p}$-viscosity solutions and the global $W^{2,p}$ estimate. As an application, we derive a Cordes-Nirenberg type continuous estimate up to boundary. Furthermore, we establish a global BMO estimate for the second derivatives of solutions by using an asymptotic approach, thereby refining the borderline case of Calderón-Zygmund estimates.