{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T07:00:41Z","timestamp":1763535641539},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"18","license":[{"start":{"date-parts":[[2019,10,28]],"date-time":"2019-10-28T00:00:00Z","timestamp":1572220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia, Portugal","award":["IF\/01461\/2015","PTDC\/MAT-CAL\/4334\/2014"],"award-info":[{"award-number":["IF\/01461\/2015","PTDC\/MAT-CAL\/4334\/2014"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,9,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate the question of whether the eigenvalues of the Laplacian with Robin boundary conditions can satisfy inequalities of the same type as those in P\u00f3lya\u2019s conjecture for the Dirichlet and Neumann Laplacians and, if so, what form these inequalities should take. Motivated in part by P\u00f3lya\u2019s original approach and in part by recent analogous works treating the Dirichlet and Neumann Laplacians, we consider rectangles and unions of rectangles and show that for these two families of domains, for any fixed positive value $\\alpha$ of the boundary parameter, P\u00f3lya-type inequalities do indeed hold, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in both cases, showing that they are different in the two situations. Our approach to proving these results includes a characterization of the corresponding extremal domains for the $k^{\\textrm{}}$th eigenvalue in regions of the $(k,\\alpha )$-plane, which in turn supports recent conjectures on the nature of the extrema among all bounded domains.<\/jats:p>","DOI":"10.1093\/imrn\/rnz204","type":"journal-article","created":{"date-parts":[[2019,7,24]],"date-time":"2019-07-24T11:24:12Z","timestamp":1563967452000},"page":"13730-13782","source":"Crossref","is-referenced-by-count":1,"title":["Extremal Domains and P\u00f3lya-type Inequalities for the Robin Laplacian on Rectangles and Unions of Rectangles"],"prefix":"10.1093","volume":"2021","author":[{"given":"Pedro","family":"Freitas","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais 1, P-1049-001 Lisboa, Portugal"},{"name":"Grupo de F\u00edsica Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade de Lisboa, Campo Grande, Edif\u00edcio C6, P-1749-016 Lisboa, Portugal"}]},{"given":"James B","family":"Kennedy","sequence":"additional","affiliation":[{"name":"Grupo de F\u00edsica Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade de Lisboa, Campo Grande, Edif\u00edcio C6, P-1749-016 Lisboa, Portugal"}]}],"member":"286","published-online":{"date-parts":[[2019,10,28]]},"reference":[{"key":"2021092211315869700_ref1","article-title":"Non-concavity of Robin eigenfunctions","author":"Andrews","year":"2017"},{"key":"2021092211315869700_ref2","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s00245-015-9304-6","article-title":"Optimisation of eigenvalues of the Dirichlet Laplacian with a surface area restriction","volume":"73","author":"Antunes","year":"2016","journal-title":"Appl. Math. Optim."},{"key":"2021092211315869700_ref3","article-title":"Optimal spectral rectangles and lattice ellipses","volume":"469","author":"Antunes","year":"2013","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"2021092211315869700_ref4","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1051\/cocv\/2012016","article-title":"Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian","volume":"19","author":"Antunes","year":"2013","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"2021092211315869700_ref5","article-title":"Maximal spectral surfaces of revolution converge to a catenoid","volume":"472","author":"Ariturk","year":"2016","journal-title":"Proc. A"},{"key":"2021092211315869700_ref6","doi-asserted-by":"crossref","first-page":"91","DOI":"10.4171\/PM\/1994","article-title":"Optimal stretching for lattice points under convex curves","volume":"74","author":"Ariturk","year":"2017","journal-title":"Port. 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