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Nodal Sets and Doubling Conditions in Elliptic Homogenization

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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 {Lε} 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 Lε(uε) = 0 in a ball in ℝd are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of uε by solutions of the homogenized equation.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Fanghua Lin

  2. Department of Mathematics, University of Kentucky, Lexington, Kentucky, 40506, USA

    Zhongwei Shen

Authors
  1. Fanghua Lin
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  2. Zhongwei Shen
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Correspondence to Fanghua Lin.

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Dedicated to Our Teacher Professor Carlos E. Kenig on the Occasion of His 65th Birthday

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)

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Lin, F., Shen, Z. Nodal Sets and Doubling Conditions in Elliptic Homogenization. Acta. Math. Sin.-English Ser. 35, 815–831 (2019). https://doi.org/10.1007/s10114-019-8228-5

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  • DOI: https://doi.org/10.1007/s10114-019-8228-5

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