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We prove some new upper and lower bounds for the first Dirichlet
eigenvalue of triangles and quadrilaterals. In particular, we improve
Pólya and Szegö's [ (1951)] lower bound for quadrilaterals and extend
Hersch's [ (1966) 457–460] upper bound for parallelograms to general quadrilaterals.
We consider the problem of minimising the th-eigenvalue of the Robin Laplacian in R. Although for = 1,2 and a positive boundary parameter it is known that the minimisers do not depend on , we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on . We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with
, which is in sharp contrast with the Weyl asymptotics for a fixed domain....
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