Some remarks on the Lp regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition

TL;DR

This paper establishes an endpoint regularity result for the Lp integrability of second derivatives of solutions to non-divergence elliptic equations with Dini continuous coefficients, highlighting the optimality of the Dini condition.

Contribution

It proves a new regularity theorem under Dini continuity assumptions and provides counterexamples demonstrating the limits of these regularity results.

Findings

Proved Lp regularity of second derivatives under Dini continuity.

Counterexample showing Dini condition's optimality.

Counterexample related to BMO regularity of second derivatives.

Abstract

In this note we prove an end-point regularity result on the Lp integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations.