Critical sets of solutions of elliptic equations in periodic homogenization

Abstract

In this paper we study critical sets of solutions $u_{\epsilon }$ 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 $(d-2)$-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period $\epsilon$, 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 $u_{\epsilon }$ onto the subspace of spherical harmonics of order $\ell$, when the doubling index for $u_{\epsilon }$ on a sphere $\partial B(0,r)$ is trapped between $\ell -\delta$ and $\ell +\delta$, for $r$ between 1 and a minimal radius $r^{\ast }\ge C_0\epsilon$. This estimate is proved by using harmonic approximation successively. With a suitable $L^2$ 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.