Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains

1994 ◽  
Vol 69 (1) ◽  
pp. 142-154 ◽  
Author(s):  
Giovanni Alessandrini
Keyword(s):  
Convex Domains ◽  
Nodal Lines ◽  
Fixed Membrane ◽  
Membrane Problem ◽  
General Convex

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).

On the mean value of the fundamental mode in the fixed membrane problem

1973 ◽  
Vol 3 (3) ◽  
pp. 295-306 ◽  
Author(s):  
L.E. Payne ◽  
I. Stakgold
Keyword(s):  
Fundamental Mode ◽  
Mean Value ◽  
Fixed Membrane ◽  
Membrane Problem ◽  
The Mean
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