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The least Steklov eigenvalue
for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the
-norm of harmonic functions. These applications suggest to address the problem of minimizing
in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular,...
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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