Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions

Fanghua Lin, Zhongwei Shen

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we continue the study of critical sets of solutions uε of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. In [18], by controlling the “turning” of approximate tangent planes, 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. In this paper we use a different approach, based on the reduction of the doubling indices of uε , to study the two-dimensional case. The proof relies on the fact that the critical set of a homogeneous harmonic polynomial of degree two or higher in dimension two contains only one point.

Original languageEnglish (US)
Pages (from-to)951-961
Number of pages11
JournalVietnam Journal of Mathematics
Volume51
Issue number4
DOIs
StatePublished - Oct 2023

Keywords

  • Critical set
  • Doubling index
  • Hausdorff measure
  • Homogenization

ASJC Scopus subject areas

  • General Mathematics

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