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In this note we prove an end-point regularity result on the 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. These results are analogous to corresponding results for divergence form elliptic equations in [3, 15].
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Luis Escauriaza, Santiago Montaner, Some remarks on the regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 1, pp. 49–63DOI 10.4171/RLM/751