تقرير
Critical Sets of Solutions of Elliptic Equations in Periodic Homogenization
| العنوان: | Critical Sets of Solutions of Elliptic Equations in Periodic Homogenization |
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| المؤلفون: | Lin, Fanghua, Shen, Zhongwei |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, 35J15, 35B27 |
| الوصف: | 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 $\ell -\delta$ and $\ell +\delta$, 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. Comment: Minor revision. Theorem 3.1 is strengthened. The definition of the minimal radius in the proof of Lemma 7.1 is modified |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2203.13393 |
| رقم الأكسشن: | edsarx.2203.13393 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |