Pointwise Regularity for Fully Nonlinear Elliptic Equations in General Forms
Yuanyuan Lian, Lihe Wang, Kai Zhang
TL;DR
This work develops a systematic theory of pointwise regularity for fully nonlinear elliptic equations in general forms, allowing quadratic growth in the gradient, by proving interior and boundary $C^{k,\alpha}$ regularity and endpoint $C^k$ and $C^{k,\mathrm{lnL}}$ regularity for $L^n$-viscosity solutions. The authors deploy a perturbation-compactness framework that relies only minimal structure conditions (notably SC2 or SC1) and a carefully designed oscillation control of the operator near a point, together with a hierarchy of model-problem regularity results and ABP-type maximum principles. Central to the approach are the key lemmas In-l-C1a-mu, In-l-C2a-mu, and their higher-order analogues, which enable iterative approximation of solutions by polynomials of increasing degree through scale-by-scale rescaling. The results yield new, sharp pointwise regularity statements, including higher regularity for linear equations and explicit estimates in several regimes, and provide a versatile toolkit for boundary and interior problems under weak, near-critical data assumptions.
Abstract
In this paper, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are covered. We obtain a series of interior and boundary pointwise $C^{k,α}$ regularity ($k\geq 1$ and $0<α<1$). In addition, we also derive the pointwise $C^k$ regularity ($k\geq 1$) and $C^{k,\mathrm{lnL}}$ regularity ($k\geq 0$), which correspond to the end points $α=0$ and $α=1$ respectively. Some regularity results are new even for the linear equations. Moreover, the minimum requirements are imposed to obtain above regularity and our proofs are simple.