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

Fanghua Lin; Zhongwei Shen

doi:10.1007/s10013-023-00632-4

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

Fanghua Lin
, Zhongwei Shen

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

Abstract

In this paper, we continue the study of critical sets of solutions u_ε 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 ε , 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_ε , 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.

Original language	English
Pages (from-to)	951-961
Number of pages	11
Journal	Vietnam Journal of Mathematics
Volume	51
Issue number	4
DOIs	https://doi.org/10.1007/s10013-023-00632-4
State	Published - Oct 2023

Bibliographical note

Publisher Copyright:
© 2023, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

Funding

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.

Funders	Funder number
National Science Foundation Arctic Social Science Program	1856235, DMS-1955249, DMS-1856235

Keywords

Critical set
Doubling index
Hausdorff measure
Homogenization

ASJC Scopus subject areas

General Mathematics

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10.1007/s10013-023-00632-4

Cite this

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abstract = "In this paper, we continue the study of critical sets of solutions uε 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 ε , 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 uε , 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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T1 - Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions

AU - Lin, Fanghua

AU - Shen, Zhongwei

PY - 2023/10

Y1 - 2023/10

N2 - In this paper, we continue the study of critical sets of solutions uε 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 ε , 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 uε , 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.

AB - In this paper, we continue the study of critical sets of solutions uε 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 ε , 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 uε , 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.

KW - Critical set

KW - Doubling index

KW - Hausdorff measure

KW - Homogenization

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JO - Vietnam Journal of Mathematics

JF - Vietnam Journal of Mathematics

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

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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