Critical sets of solutions of elliptic equations in periodic homogenization
Critical sets of solutions of elliptic equations in periodic homogenization
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周期均匀化中椭圆方程的临界解集
DOI：
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发表时间：
2022
影响因子：
3
通讯作者：
Zhongwei Shen
中科院分区：
文献类型：
--
作者：
F. Lin;Zhongwei Shen
本文研究了具有快速振荡和周期系数的二阶散度型椭圆型方程的临界解集uε$u_vareps $。在一阶修正的一定条件下，证明了临界集的（d−2）$（d-2）$维Hausdorff测度关于周期ε$vareps $一致有界，条件是解的加倍指数有界.关键的一步是近似切映射的“转向”估计，即非常数解uε$u_varepsilon $在球B（0，r）$部分B（0，r）$上的加倍指数被困在ε −δ$ell -delta$和ε +δ$ell +delta$之间时，对于r$r$在1和最小半径r ≥ C 0 ε$r^*ge C_0 vareprint $之间。利用调和逼近证明了这一估计。通过适当的L2 $L^2 $重正化以及重新标度，我们能够控制由均匀化和投影引入的累积误差。我们的证明也给出了临界集的Minkowski内容的统一界。
In this paper we study critical sets of solutions uε$u_varepsilon$ 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)$(d-2)$ ‐dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε$varepsilon$ , 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 uε$u_varepsilon$ onto the subspace of spherical harmonics of order ℓ$ell$ , when the doubling index for uε$u_varepsilon$ on a sphere ∂B(0,r)$partial B(0, r)$ is trapped between ℓ−δ$ell -delta$ and ℓ+δ$ell +delta$ , for r$r$ between 1 and a minimal radius r∗≥C0ε$r^*ge C_0varepsilon$ . This estimate is proved by using harmonic approximation successively. With a suitable L2$L^2$ 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.
二维快速振荡系数椭圆方程组临界
影响因子：
0.8
作者：
Lin, Fanghua;Shen, Zhongwei
通讯作者：
Shen, Zhongwei
影响因子：
2
作者：
Kenig, Carlos;Zhu, Jiuyi
通讯作者：
Zhu, Jiuyi