Hessian continuity at degenerate points in nonvariational elliptic problems

TL;DR

This paper demonstrates that solutions to certain elliptic PDEs with limited coefficient regularity can have highly regular Hessians at degenerate points, surpassing traditional regularity limits.

Contribution

It establishes new regularity results for the Hessian at degenerate points in elliptic equations, exceeding classical Schauder estimate limitations.

Findings

Hessian regularity at degenerate points can be higher than expected.

Solutions are of class C^{2,α} at Hessian degenerate points.

Surpasses traditional Schauder regularity bounds.

Abstract

Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. 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, $\mathscr{H}(u):=\{X \suchthat D^2u(X) = 0\}$. In fact we develop a more general regularity result at such Hessian degenerate points, featuring into the theory of fully nonlinear equations. Insofar as the optimal modulus of continuity for the Hessian is concerned, the result of this paper is the first one in the literature to surpass the inborn obstruction from the sharp Schauder {\it a priori} regularity theory.