Harmonic approximation and improvement of flatness in a singular perturbation problem

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• 作者：Kelei Wang (1)
  
  1. Wuhan Institute of Physics and Mathematics ; Chinese Academy of Sciences ; Wuhan ; 430071 ; China
  
• 关键词：35B06 ; 35B08 ; 35B25 ; 35J91
• 刊名：manuscripta mathematica
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：146
• 期：1-2
• 页码：281-298
• 全文大小：207 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebraic Geometry
  Topological Groups and Lie Groups
  Geometry
  Number Theory
  Calculus of Variations and Optimal Control
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1785

文摘

We study the De Giorgi type conjecture, that is, one dimensional symmetry problem for entire solutions of a two components elliptic system in \({\mathbb{R}^n}\) , for all \({n \geq 2}\) . We prove that, if a solution (u, v) has a linear growth at infinity, then it is one dimensional, that is, depending only on one variable. The main ingredient is an improvement of flatness estimate, which is achieved by the harmonic approximation technique adapted in the singularly perturbed situation.