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Publication
Bounds and extremal domains for Robin eigenvalues with negative boundary parameter
Advisor(s)
Abstract(s)
We present some new bounds for the first Robin eigenvalue with a negative boundary parameter.
These include the constant volume problem, where the bounds are based on the shrinking coordinate
method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values
of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the
maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds
for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical
optimisation of the first four and two eigenvalues in two and three dimensions, respectively, and an evaluation
of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary
parameter becomes large (negative).
Description
Keywords
Eigenvalue optimisation Robin Laplacian Negative boundary parameter Bareket’s conjecture