An isoperimetric inequality for the first eigenfunction in the fixed membrane problem

1972 ◽  
Vol 23 (1) ◽  
pp. 13-15 ◽  
Author(s):  
Lawrence E. Payne ◽  
Margaret E. Rayner
Keyword(s):  
Isoperimetric Inequality ◽  
Fixed Membrane ◽  
Membrane Problem ◽  
First Eigenfunction

THE RANGE OF VALUES OF λ2/λ1 AND λ3/λ1 FOR THE FIXED MEMBRANE PROBLEM

1994 ◽  
Vol 06 (05a) ◽  
pp. 999-1009 ◽  
Author(s):  
MARK S. ASHBAUGH ◽  
RAFAEL D. BENGURIA
Keyword(s):  
Bounded Domain ◽  
The Other ◽  
Dirichlet Laplacian ◽  
Fixed Membrane ◽  
De Vries ◽  
Membrane Problem ◽  
Range Of Values

We investigate the region of the plane in which the point (λ2/λ1, λ3/λ1) can lie, where λ1, λ2, and λ3 are the first three eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain Ω ⊂ ℝ2. In particular, by making use of a technique introduced by de Vries we obtain the best bounds to date for the quantities λ3/λ1 and (λ2 + λ3)/λ1. These bounds are λ3/λ1 ≤ 3.90514+ and (λ2 + λ3)/λ1 ≤ 5.52485+ and give small improvements over previous bounds of Marcellini. Where Marcellini used a bound due to Brands in his argument we use a better version of this bound which we obtain by incorporating deVries' idea. The other bounds that yield the greatest information about the region where points (λ2/λ1, λ3/λ1) can (possibly) lie are those due to Marcellini, Hile and Protter, and us (of which there are several, with two of them being new with this paper).

Isoperimetric bounds for higher eigenvalue ratios for the n-dimensional fixed membrane problem

1993 ◽  
Vol 123 (6) ◽  
pp. 977-985 ◽  
Author(s):  
Mark S. Ashbaugh ◽  
Rafael D. Benguria
Keyword(s):  
Bessel Function ◽  
Bounded Domain ◽  
Eigenvalue Problems ◽  
Positive Zero ◽  
Dirichlet Laplacian ◽  
Nodal Domains ◽  
Good Bound ◽  
Fixed Membrane ◽  
Membrane Problem ◽  
Elliptic Eigenvalue Problems

SynopsisWe give several results which extend our recent proof of the Payne-Pólya–Weinberger conjecture to ratios of higher eigenvalues. In particular, we show that for a bounded domain Ω⊂ℝn the eigenvalues of its Dirichlet Laplacian obey where λm denotes the mth eigenvalue and jp,k denotes the kth positive zero of the Bessel function Jp(x). Certain extensions of this result are given, the most general being the bound where k≧2 and l(m) denotes the number of nodal domains of an mth eigenfunction. Our results imply certain further conjectures of Payne, Pólya, and Weinberger concerning λ3/λ2 and λ4/λ3. In addition, we find a resonably good bound on λ4/λ1. We also briefly discuss extensions to Schrödinger operators and other elliptic eigenvalue problems.

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