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 ε, 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 ℓ, 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 L² 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.

中文翻译：

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周期均匀化中椭圆方程的临界解集

在本文中，我们研究关键解决方案集${你}_{\epsilon }$具有快速振荡和周期系数的发散形式的二阶椭圆方程。在一阶校正器的某些条件下，我们证明$（d-2）$临界集的维 Hausdorff 测度关于周期 ε 统一有界，前提是解的倍数指数有界。关键步骤是估计近似切线图的“转动”，即非恒定解的投影${你}_{\epsilon }$到ℓ阶球谐函数的子空间上，当倍数指数为${你}_{\epsilon }$在一个球体上$\partial 乙（0,r）$被困在之间$\ell -\delta$和$\ell +\delta$，对于r介于 1 和最小半径之间$r^{*}\ge C_0\epsilon$。相继使用调和近似证明了这一估计。通过适当的L ²重整化以及重新缩放，我们能够控制由均质化和投影引入的累积误差。我们的证明还给出了关键集的闵可夫斯基内容的统一界限。