Nodal Sets and Doubling Conditions in Elliptic Homogenization

Fanghua Lin; Zhongwei Shen

doi:10.1007/s10114-019-8228-5

Nodal Sets and Doubling Conditions in Elliptic Homogenization

Fanghua Lin
, Zhongwei Shen

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L _ε } in divergence form with rapidly oscillating and periodic coefficients. We show that the (d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L _ε (u _ε ) = 0 in a ball in ℝ ^d are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u _ε by solutions of the homogenized equation.

Original language	English
Pages (from-to)	815-831
Number of pages	17
Journal	Acta Mathematica Sinica, English Series
Volume	35
Issue number	6
DOIs	https://doi.org/10.1007/s10114-019-8228-5
State	Published - Jun 1 2019

Bibliographical note

Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany & The Editorial Office of AMS.

Funding

Received May 24, 2018, accepted October 26, 2018 The first author is supported in part by NSF (Grant No. DMS-1501000); the second author is supported in part by NSF (Grant No. DMS-1600520)

Funders	Funder number
National Science Foundation Arctic Social Science Program	DMS-1501000, 1501000

Keywords

35B27
35J15
Nodal sets
doubling condition
homogenization

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1007/s10114-019-8228-5

Cite this

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abstract = " This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators \{L ε \} in divergence form with rapidly oscillating and periodic coefficients. We show that the (d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L ε (u ε ) = 0 in a ball in ℝ d are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u ε by solutions of the homogenized equation.",

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N2 - This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L ε } in divergence form with rapidly oscillating and periodic coefficients. We show that the (d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L ε (u ε ) = 0 in a ball in ℝ d are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u ε by solutions of the homogenized equation.

AB - This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L ε } in divergence form with rapidly oscillating and periodic coefficients. We show that the (d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L ε (u ε ) = 0 in a ball in ℝ d are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u ε by solutions of the homogenized equation.

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KW - Nodal sets

KW - doubling condition

KW - homogenization

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JF - Acta Mathematica Sinica, English Series

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Nodal Sets and Doubling Conditions in Elliptic Homogenization

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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