Upper bounds for the ratio of the first two eigenvalues of the Robin Laplacian

Abstract

We sharpen upper bounds for the ratio of the first two Robin eigenvalues by matching the first eigenvalue on a comparison ball with an independently chosen boundary parameter. The bound depends on the interior superlevel sets of the first eigenfunction and improves the fixed-parameter comparison. For the conditional ball inequality, we also prove that equality holds only for a ball. On balls in every dimension $N\ge 2$, we establish strict increase of the first spectral gap and strict decrease of the ratio of the first two eigenvalues with respect to the positive Robin parameter, verifying the ball case of Laugesen's spectral-ratio monotonicity conjecture. Both statements follow from a comparison of the lowest radial eigenvalues of any two spherical harmonic degrees. Finally, building on the known disconnected examples, we give an explicit construction showing that the ratio is unbounded among smooth connected domains of prescribed volume at every fixed positive parameter; in two dimensions the domains can be chosen simply connected. The equality case yields a conditional circularity criterion for elastically supported membranes.