On the smoothness of solutions of fully nonlinear second order equations in the plane
Alessandro Goffi
TL;DR
The paper addresses interior regularity for fully nonlinear second-order elliptic equations in the plane without geometric restrictions on the operator F. It combines divergence-form and nondivergence-form plane regularity theories to obtain quantitative $C^{2,\alpha}$ estimates, first showing $u\in C^{2,\bar{\alpha}}$ with an explicit $\bar{\alpha}$ depending on $\lambda/\Lambda$, then improving this to $u\in C^{2,\tilde{\alpha}}$ with $\tilde{\alpha}>\lambda/\Lambda$ using nondivergence techniques. The results provide explicit, dimension-specific Hölder exponents for fully nonlinear equations in $\mathbb{R}^2$, extending Evans-Krylov-type regularity with precise dependence on ellipticity ratios. This advances understanding of how ellipticity controls smoothness in low dimensions without structural assumptions on F, and has implications for Bellman-Isaacs equations in the plane.
Abstract
We study interior $C^{2,α}$ regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form $F(D^2u)=0$ in two independent variables and without any geometric condition on $F$. By means of the theory of divergence form equations we prove that $C^2$ solutions of the previous equation are $C^{2,\barα(λ/Λ)}$ in the interior of the domain, where $0<λ\leqΛ$ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain $C^{2,\tildeα}$ regularity for an explicit exponent $\tildeα=\tildeα(λ/Λ)>λ/Λ$.