Geometric sets of elliptic equations with Dini coefficients

Abstract: We study divergence-form elliptic equations with Dini coefficients, extending the frameworks of \cite{NV} for Lipschitz coefficients and \cite{HJ1} for Hölder coefficients. We first establish an almost monotonicity property for the doubling index under merely continuous coefficients. This leads to weak volume estimates for critical sets and uniform volume estimates for nodal sets. The Dini assumption is essential here since we construct an example with continuous coefficients and bounded doubling index whose nodal set has infinite measure. We also prove polynomial growth estimates for sub-level sets. Using a different approach, we establish quantitative uniqueness of tangent maps and a cone-splitting principle, which yield explicit Minkowski-type estimates for critical sets. Finally, combined with the results of Kenig--Zhao \cite{KZ4}, our estimates give measure bounds for boundary critical sets.