Estimates of Dirichlet Eigenvalues for Degenerate Elliptic Operators (Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics)

• Chen, Hua

  School of Mathematics and Statistics and Computational Science Hubei Key Laboratory, Wuhan University

• Luo, Peng

  School of Mathematics and Statistics and Computational Science Hubei Key Laboratory, Wuhan University

抄録

Let Ω be a bounded open domain in Rn with smooth boundary and X = (X1, X2, …, Xm) be a system of real smooth vector fields defined on Ω with the boundary ∂Ω which is non-characteristic for X. If X satisfies the H ormander's condition, then the vector fields is finite degenerate and the sum of square operator △X = Σm j=1 X2 j is a finitely degenerate elliptic operator, otherwise the operator -△X is called infinitely degenerate. If λj is the jth Dirichlet eigenvalue for -△X on Ω, then this paper shall study the lower bound estimates for λj. Firstly, by using the sub-elliptic estimate directly, we shall give a simple lower bound estimates of λj for general finitely degenerate △X which is polynomial increasing in j. Secondly, if △X is so-called Grushin type degenerate elliptic operator, then we can give a precise lower bound estimates for λj. Finally, by using logarithmic regularity estimate, for infinitely degenerate elliptic operator △X we prove that the lower bound estimates of λj will be logarithmic increasing in j.

"Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics". May 27～29, 2016. edited by Hisashi Okamoto, Yoshio Tsutsumi, Naomasa Ueki, Tadayoshi Adachi and Senjo Shimizu. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

収録刊行物

• 数理解析研究所講究録別冊

  数理解析研究所講究録別冊 B67 1-24, 2017-10

  Research Institute for Mathematical Sciences, Kyoto University

キーワード

• 35J70
• 35P15
• Dirichlet eigenvalues
• finitely degenerate elliptic operators
• infinitely degenerate elliptic operators
• Hörmander's condition
• sub-elliptic estimate
• logarithmic regularity estimate

詳細情報

• CRID: 1050282813437267328
• NII論文ID: 120006715426
• NII書誌ID: AA12196120
• ISSN: 18816193
• HANDLE: 2433/243702
• 本文言語コード: en
• 資料種別: departmental bulletin paper
• データソース種別:
  • IRDB
  • CiNii Articles