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_ε of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. In [18], by controlling 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 ε , 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_ε , 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.