论文标题
定期均质化中椭圆方程的临界解决方案的关键解决方案集
Critical Sets of Solutions of Elliptic Equations in Periodic Homogenization
论文作者
论文摘要
在本文中，我们研究了以差异形式的二阶椭圆方程的关键解决方案$ u_ \ e $，并具有迅速振荡和周期性系数。我们表明，$（d-2）$ - 尺寸的Hausdorff测量值相对于周期$ \ e $均匀地界定了限制，前提是解决方案的倍增指数是有限的。关键步骤是估计“转动”的估计，以预测非固定解决方案$ u_ \ e $ $ $ \ ell $的球形谐波的子空间，当$ \ e \ e $ $ u_ \ e $上的双倍索引上的$ \ e $ $ \ partial $ \ partial b（0，r）$ \ ell -ul -ul -uim $ \ ell -uim $ \ ell + + + + + + + + + $ r^*\ ge c_0 \ e $。通过依次使用谐波近似来证明此估计值。通过合适的$ l^2 $重新归一化以及重新缩放，我们能够控制同质化和投影引入的累积错误。我们的证明还为关键集的Minkowski内容提供了统一的界限。
In this paper we study critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. We show that the $(d-2)$-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period $\e$, provided that 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 the subspace of spherical harmonics of order $\ell$, when the doubling index for $u_\e$ on a sphere $\partial B(0, r)$ is trapped between $\ell -δ$ and $\ell +δ$, for $r$ between $1$ and a minimal radius $r^*\ge C_0\e$. 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.
