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We present some new bounds for the first Robin eigenvalue with a negativeboundary parameter. These include the constant volume problem, where the boundsare based on the shrinking coordinate method, and a proof that in the fixedperimeter case the disk maximises the first eigenvalue for all values of theparameter. This is in contrast with what happens in the constant area problem,where the disk is the maximiser only for small values of the boundaryparameter. We also present sharp upper and lower bounds for the firsteigenvalue of the ball and spherical shells.  These results are complemented by the numerical optimisation of the firstfour and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluationof the quality of the upper bounds obtained. We also study the bifurcationsfrom the ball as the boundary parameter becomes large (negative).

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter