Asymptotic First Eigenvalue Estimates for the Biharmonic Operator on a Rectangle

1998/03/05 by Mark P. Owen
Mathematics · #math.SP #math.FA #msc:35K25

paper · pdf

published as Journal of Differential Equations. Vol. 136, No. 1, (1997) · 27 pages, 4 diagrams, 2 tables

arxiv created 1998/03/05 · arxiv updated 2009/11/30

Abstract

We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small.

Related

• Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles
• Lower order eigenvalues of the biharmonic operator
• Estimates on eigenvalues for the biharmonic operator on a bounded domain in Hⁿ(-1)
• Neumann eigenvalues of the biharmonic operator on domains: geometric bounds and related results
• Lower bounds of eigenvalues of the biharmonic operators by the rectangular Morley element methods