From qualitative smoothness to uniform estimates for fully nonlinear elliptic equations

Abstract

We establish a general mechanism that turns qualitative smoothness into uniform, quantitative regularity estimates for fully nonlinear elliptic equations. The central conclusion is that, for compact classes preserved by the natural rescalings of the equation, qualitative and quantitative regularity have the same critical threshold. At second order, mere twice differentiability throughout the corresponding centered hull already forces a uniform $C^{2,α}$-theory for some $α>0$. A one-sided tangent version gives a new partial regularity criterion for a single operator: every point at which a solution is twice differentiable is regular, whereas the singular set has universal positive codimension. The proof combines compactness of the full renormalized class with a scale-adaptive improvement of flatness, converting information available separately at each profile into estimates uniform across the entire class. The argument requires no quantitative control of the assumed smoothness and opens a path in problems where smoothness is visible, but estimates remain out of reach.