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Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions

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Abstract

In this paper, we continue the study of critical sets of solutions \(u_{\varepsilon }\) 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 \(\varepsilon \), 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 \(u_{\varepsilon }\), 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.

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Acknowledgements

Fanghua Lin is supported in part by NSF grant DMS-1955249. Zhongwei Shen is supported in part by NSF grant DMS-1856235 and by Simons Fellowship.

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Authors and Affiliations

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

    Fanghua Lin

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

    Zhongwei Shen

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  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 Kenig on the occasion of his 70th birthday.

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Lin, F., Shen, Z. Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions. Vietnam J. Math. 51, 951–961 (2023). https://doi.org/10.1007/s10013-023-00632-4

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