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In this paper we give new results on complete abstract second order differential equations of elliptic type in the framework of Holder spaces, extending those given in [4] and [5]. More precisely we study $u^{\prime \prime }+2Bu^{\prime }+Au=f$ in the case when $f$ is Holder continuous and under some natural assumptions on the operators $A$ and $B$. We give necessary and sufficient conditions of compatibility to obtain a strict solution $u$ and also to ensure that the strict solution has the maximal regularity property.

Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces