Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the laplacian on thin planar domains
Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals
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.
Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian
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....
New bounds for the principal Dirichlet eigenvalue of planar regions.
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