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

Producción científica: Article › revisión exhaustiva

11 Citas (Scopus)

Resumen

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.

Idioma original	English
Páginas (desde-hasta)	815-831
Número de páginas	17
Publicación	Acta Mathematica Sinica, English Series
Volumen	35
N.º	6
DOI	https://doi.org/10.1007/s10114-019-8228-5
Estado	Published - jun 1 2019

Nota bibliográfica

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

Financiación

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)

Financiadores	Número del financiador
National Science Foundation Arctic Social Science Program	DMS-1501000, 1501000

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1007/s10114-019-8228-5

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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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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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Nodal Sets and Doubling Conditions in Elliptic Homogenization

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