Regularity for fully nonlinear elliptic equations in generalized Orlicz spaces

TL;DR

The paper develops a global Calderón-Zygmund theory for fully nonlinear elliptic equations in generalized Orlicz (Musielak-Orlicz) spaces, accommodating nonconvex nonlinearities in the Hessian and nonstandard growth. By leveraging a stopping-time framework, an approximation lemma against a convex limiting operator $F^{\star}$, and Vitali-type coverings, the authors obtain global $W^{2,\psi(\cdot)}$-regularity: the Hessian $D^{2}u$ inherits the integrability of the data $f$ in $L^{\psi(\cdot)}(\Omega)$ under a smallness condition on the oscillation of $F^{\star}$. They establish both interior and boundary Hessian estimates, and extend to gradient and global estimates for the full Dirichlet problem, removing dependence on the $L^{\infty}$-norm of the solution via a compactness argument. This work generalizes Calderón-Zygmund theory to nonstandard growth settings and nonconvex nonlinearities, broadening applicability to problems with Orlicz-type and variable-exponent growth.

Abstract

In this paper, we establish an optimal global Calderón-Zygmund type estimate for the viscosity solution to the Dirichlet boundary problem of fully nonlinear elliptic equations with possibly nonconvex nonlinearities. We prove that the Hessian of the solution is as integrable as the nonhomogeneous term in the setting of a given generalized Orlicz space even when the nonlinearity is asymptotically convex with respect to the Hessian of the solution.

Theorems & Definitions (29)

• Lemma 2.1
• Proposition 2.2
• Definition 2.3: Pucci extremal operator
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• Theorem 3.1