Doubling inequalities and nodal sets in periodic elliptic homogenization. Issue 3 (4th March 2022)

• Record Type: Journal Article
• Authors: Kenig, Carlos E.; Zhu, Jiuyi; Zhuge, Jinping
• Is Part Of: Communications in partial differential equations. Volume 47:Issue 3(2022)
• Page Start: 549
• Page End: 584
• Publication Date: 2022-03-04
• Subjects: Doubling inequalities -- nodal sets -- periodic homogenization; 35A02 -- 35B27 -- 35J15; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2021.1989699
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: We prove explicit doubling inequalities and obtain uniform upper bounds (under ( d − 1 ) -dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from certain "analyticity, " and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.