Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions

In this paper we continue the study of critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. In \cite{Lin-Shen-3d}, by controling the "turning" of approximate tangent planes, we show that the $(d-2)$-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period $\e$, provided that doubling indices for solutions are bounded. In this paper we use a different approach, based on the reduction of the doubling indices of $u_\e$, to study the two-dimensional case. The proof relies on the fact that the critical set of a homogeneous harmonic polynomial of degree two or higher in dimension two contains only one point.