{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,4]],"date-time":"2025-10-04T14:34:20Z","timestamp":1759588460722},"reference-count":32,"publisher":"Canadian Mathematical Society","issue":"4","license":[{"start":{"date-parts":[[2019,3,7]],"date-time":"2019-03-07T00:00:00Z","timestamp":1551916800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Can. J. Math.-J. Can. Math."],"published-print":{"date-parts":[[2020,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0008414X19000154_inline1.png\" \/><jats:tex-math>$\\unicode[STIX]{x1D6FC}\/L(\\unicode[STIX]{x1D6FA})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0008414X19000154_inline2.png\" \/><jats:tex-math>$\\unicode[STIX]{x1D6FC}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>lies between<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0008414X19000154_inline3.png\" \/><jats:tex-math>$-2\\unicode[STIX]{x1D70B}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0008414X19000154_inline4.png\" \/><jats:tex-math>$2\\unicode[STIX]{x1D70B}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Corollaries include Szeg\u0151\u2019s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock\u2019s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.<\/jats:p><jats:p>The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.<\/jats:p>","DOI":"10.4153\/s0008414x19000154","type":"journal-article","created":{"date-parts":[[2019,3,7]],"date-time":"2019-03-07T08:03:47Z","timestamp":1551945827000},"page":"1024-1043","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":11,"title":["From Steklov to Neumann and Beyond, via Robin: The Szeg\u0151 Way"],"prefix":"10.4153","volume":"72","author":[{"given":"Pedro","family":"Freitas","sequence":"first","affiliation":[]},{"given":"Richard S.","family":"Laugesen","sequence":"additional","affiliation":[]}],"member":"2643","published-online":{"date-parts":[[2019,3,7]]},"reference":[{"key":"S0008414X19000154_r32","first-page":"745","article-title":"Inequalities for a classical eigenvalue problem","volume":"3","author":"Weinstock","year":"1954","journal-title":"J. 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Preprint, arxiv:1905.07658."},{"key":"S0008414X19000154_r30","first-page":"343","article-title":"Inequalities for certain eigenvalues of a membrane of given area","volume":"3","author":"Szeg\u0151","year":"1954","journal-title":"J. Rational Mech. Anal."},{"key":"S0008414X19000154_r20","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-08-09704-9"},{"key":"S0008414X19000154_r16","volume-title":"Shape optimization and spectral theory","author":"Girouard","year":"2017"},{"key":"S0008414X19000154_r14","unstructured":"[14] Freitas, P. and Laugesen, R. S. , From Neumann to Steklov and beyond, via Robin: the Weinberger way. arxiv:1810.07461"},{"key":"S0008414X19000154_r12","article-title":"Extremal domains and P\u00f3lya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles","author":"Freitas","journal-title":"Int. Math. Res. Not. 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