@article{b75685f70b6541259bae914c515fcdc3,

title = "The Robin Laplacian - Spectral conjectures, rectangular theorems",

abstract = "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.",

author = "Laugesen, {Richard S.}",

note = "Funding Information: This research was supported by grants from the Simons Foundation (Grant No. 429422 to Richard Laugesen) and the University of Illinois Research Board (No. RB19045) and by travel support from the University of Illinois Scholars{\textquoteright} Travel Fund. Conversations with Dorin Bucur were particularly helpful, at the conference “Results in Contemporary Mathematical Physics” in honor of Rafael Benguria (Santiago, Chile, December 2018) and the conference on “Shape Optimization and Isoperimetric and Functional Inequalities” (Levico Terme, Italy, September 2019). I am grateful to Derek Kielty for carrying out numerical investigations in support of this research and pointing out relevant literature, and to Pedro Freitas for many informative conversations about Robin eigenvalues. Publisher Copyright: {\textcopyright} 2019 Author(s).",

year = "2019",

month = dec,

day = "1",

doi = "10.1063/1.5116253",

language = "English (US)",

volume = "60",

journal = "Journal of Mathematical Physics",

issn = "0022-2488",

publisher = "American Institute of Physics Publising LLC",

number = "12",

}