Critical sets of solutions of elliptic equations in periodic homogenization

In this paper we study critical sets of solutions (Formula presented.) of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first-order correctors, we show that the (Formula presented.) -dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period (Formula presented.), provided that doubling indices for solutions are bounded. The key step is an estimate of “turning” of an approximate tangent map, the projection of a non-constant solution (Formula presented.) onto the subspace of spherical harmonics of order (Formula presented.), when the doubling index for (Formula presented.) on a sphere (Formula presented.) is trapped between (Formula presented.) and (Formula presented.), for (Formula presented.) between 1 and a minimal radius (Formula presented.). This estimate is proved by using harmonic approximation successively. With a suitable (Formula presented.) renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets.