Global Calder\'on-Zygmund estimates for asymptotically convex fully nonlinear Grad-Mercier type equations

TL;DR

This paper establishes global Calderón-Zygmund estimates for fully nonlinear Grad-Mercier type equations under asymptotic convexity, providing existence, regularity, and boundary estimates for solutions.

Contribution

It introduces new methods to handle non-convex operators and nonlocal terms, proving existence and regularity of solutions with sharp boundary estimates.

Findings

Existence of $W^{2,p}$-viscosity solutions under asymptotic convexity.

Global $W^{2,p}$ estimates for solutions.

Refined Calderón-Zygmund estimates with boundary regularity.

Abstract

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 \Omega:u(y)\ge u(x)\}|)+f(x) & \text{in } \Omega, u=\psi &\text{on } \partial \Omega. \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 Calder\'{o}n-Zygmund estimates.