Classification of singular sets of solutions to elliptic equations

Haiyun Deng; Hairong Liu; Long Tian

doi:10.3934/cpaa.2020129

Article Contents

2020, Volume 19, Issue 6: 2949-2964. Doi: 10.3934/cpaa.2020129

This issue Previous Article Next Article Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

Classification of singular sets of solutions to elliptic equations

1.
School of Statistics and Mathematics, Nanjing Audit University, Nanjing, Jiangsu, 211815, China
2.
School of Science, Nanjing Forestry University, Nanjing, Jiangsu, 210037, China
3.
School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China

^* Corresponding author
^* Corresponding author

Received: December 2018

Revised: December 2019

Published: March 2020

The work is supported by National Natural Science Foundation of China (No.11401307, No.11401310, No.11771214) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0321). The first author is fully supported by China Scholarship Council(CSC) for visiting Rutgers University(201806840122).

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Abstract

In this paper, we mainly investigate the classification of singular sets of solutions to elliptic equations. Firstly, we define the $ j $-symmetric singular set $ S^j(u) $ of solution $ u $, and show that the Hausdorff dimension of the $ j $-symmetric singular set $ S^j(u) $ is not more than $ j $. Then we prove the generalized $ \varepsilon $-regularity lemma for $ j $-symmetric homogeneous harmonic polynomial $ P $ with origin $ 0 $ as the isolated critical point in $ \mathbb{R}^{n-j} $, and by the generalized $ \varepsilon $-regularity lemma, we show the Hausdorff measure estimate of the $ j $-symmetric singular set $ S^j(u) $. Moreover, we study the geometric structure of interior singular points of solutions $ u $ in a planar bounded domain.

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Mathematics Subject Classification: Primary: 35J15, 28A75; Secondary: 35B38.

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