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 language | English |
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Pages (from-to) | 951-961 |
Number of pages | 11 |
Journal | Vietnam Journal of Mathematics |
Volume | 51 |
Issue number | 4 |
DOIs | |
State | Published - Oct 2023 |
Bibliographical note
Publisher Copyright:© 2023, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
Keywords
- Critical set
- Doubling index
- Hausdorff measure
- Homogenization
ASJC Scopus subject areas
- General Mathematics