RESEARCH ARTICLE

Lower bounds of principal eigenvalue in dimension one

  • Mu-Fa CHEN
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  • School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China

Received date: 30 Nov 2011

Accepted date: 31 May 2012

Published date: 01 Aug 2012

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper.

Cite this article

Mu-Fa CHEN . Lower bounds of principal eigenvalue in dimension one[J]. Frontiers of Mathematics in China, 2012 , 7(4) : 645 -668 . DOI: 10.1007/s11464-012-0223-4

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