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In this paper we study critical sets of solutions $u_\e$ of second-orderelliptic equations in divergence form with rapidly oscillating and periodiccoefficients. We show that the $(d-2)$-dimensional Hausdorff measures of thecritical sets are bounded uniformly with respect to the period $\e$, providedthat doubling indices for solutions are bounded. The key step is an estimate of"turning " for the projection of a non-constant solution $u_\e$ onto thesubspace of spherical harmonics of order $\ell$, when the doubling index for$u_\e$ on a sphere $\partial B(0, r)$ is trapped between $\ell -\delta$ and$\ell +\delta$, for $r$ between $1$ and a minimal radius $r^*\ge C_0\e$. Thisestimate is proved by using harmonic approximation successively. With asuitable $L^2$ renormalization as well as rescaling we are able to control theaccumulated errors introduced by homogenization and projection. Our proof alsogives uniform bounds for Minkowski contents of the critical sets.

Critical Sets of Solutions of Elliptic Equations in Periodic Homogenization