JournalsrlmVol. 28, No. 1pp. 49–63

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

  • Luis Escauriaza

    Universidad del País Vasco, Bilbao, Spain
  • Santiago Montaner

    Basque Center for Applied Mathematics, Bilbao, Spain
Some remarks on the $L^p$ regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition cover
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Abstract

In this note we prove an end-point regularity result on the LpL^p 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].

Cite this article

Luis Escauriaza, Santiago Montaner, Some remarks on the LpL^p 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. 49–63

DOI 10.4171/RLM/751