The Robin Laplacian - Spectral conjectures, rectangular theorems

Shape optimization conjectures for the first two eigenvalues of the Robin Laplacian are developed and supported with new results for rectangular boxes. The square minimizes the first eigenvalue among rectangles under area normalization when the Robin parameter α R is scaled by perimeter; the square maximizes the second eigenvalue for a sharp range of α-values; the line segment minimizes the Robin spectral gap under diameter normalization for each α R; and the square maximizes the spectral ratio among rectangles when α > 0. Furthermore, the spectral gap of each fixed rectangle is an increasing function of α; the second eigenvalue is concave, and, except in the Neumann case, the shape of the rectangle can be heard from just its first two frequencies.