Continuity of the first eigenvalue for a family of degenerate eigenvalue problems

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• 作者：Maria Fărcăşeanu ; Mihai Mihăilescu
• 关键词：Degenerate elliptic operator ; Eigenvalue problem ; \({\Gamma}\) ; convergence
• 刊名：Archiv der Mathematik
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：107
• 期：6
• 页码：659-667
• 全文大小：486 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8938
• 卷排序：107

文摘

For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\).