Hessian Continuity at Degenerate Points in Nonvariational Elliptic Problems

Abstract

Established in the 1930s, Schauder a priori estimates are among the most classical and powerful tools in the analysis of problems ruled by second-order elliptic partial differential equations. Since then, a central problem in regularity theory has been to understand Schauder-type estimates fashioning particular borderline scenarios. In such context, it has been a common accepted aphorism that the continuity of the Hessian of a solution could never be superior than the continuity of the medium. Notwithstanding, in this article, we show that solutions to uniformly elliptic, linear equations with |$C^{0,\bar {\epsilon }}$| coefficients are of class |$C^{2,\alpha }$|⁠, for any |$0 <\bar {\epsilon } \ll \alpha <1$|⁠, at Hessian degenerate points, |$\mathscr {H}(u):={\{ }X \,|\, D^2u(X) = 0{\} }$|⁠. In fact, we develop a more general regularity result at such Hessian degenerate points, featuring into the theory of fully nonlinear equations. The key, innovative idea in the proof is to interpret Hessian degenerate sets as abstract (nonphysical) free boundary points. This novel insight has a number of further applications.