{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T01:36:34Z","timestamp":1773538594654,"version":"3.50.1"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PTDC\/MAT-CAL\/4334\/2014"],"award-info":[{"award-number":["PTDC\/MAT-CAL\/4334\/2014"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["IF\/00177\/2013"],"award-info":[{"award-number":["IF\/00177\/2013"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present some new bounds for the first Robin eigenvalue\nwith a negative boundary parameter. These\ninclude the constant volume problem, where the bounds are based on the shrinking coordinate method,\nand a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values\nof the parameter.\nThis is in contrast with what happens in the constant area problem, where\nthe disk is the maximiser only for small values of the boundary parameter. We also present sharp\nupper and lower bounds for the first eigenvalue of the ball\nand spherical shells.\nThese results are complemented by the numerical optimisation of\nthe first four and two eigenvalues\nin two and three dimensions, respectively,\nand an evaluation of the quality of the upper bounds obtained.\nWe also study the bifurcations from the ball as the boundary parameter\nbecomes large (negative).<\/jats:p>","DOI":"10.1515\/acv-2015-0045","type":"journal-article","created":{"date-parts":[[2016,8,3]],"date-time":"2016-08-03T07:54:48Z","timestamp":1470210888000},"page":"357-379","source":"Crossref","is-referenced-by-count":36,"title":["Bounds and extremal domains for Robin eigenvalues with negative boundary parameter"],"prefix":"10.1515","volume":"10","author":[{"given":"Pedro R. S.","family":"Antunes","sequence":"first","affiliation":[{"name":"Group of Mathematical Physics , Faculdade de Ci\u00eancias da Universidade de Lisboa , Campo Grande , Edif\u00edcio C6 1749-016 Lisboa , Portugal"}]},{"given":"Pedro","family":"Freitas","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Faculty of Human Kinetics & Group of Mathematical Physics , Faculdade de Ci\u00eancias da Universidade de Lisboa , Campo Grande, Edif\u00edcio C6 1749-016 Lisboa , Portugal"}]},{"given":"David","family":"Krej\u010di\u0159\u00edk","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics , Nuclear Physics Institute , Academy of Sciences , 25068 \u0158e\u017e , Czech Republic"}]}],"member":"374","published-online":{"date-parts":[[2016,8,2]]},"reference":[{"key":"2023033109443814301_j_acv-2015-0045_ref_001_w2aab3b7e1521b1b6b1ab2ab1Aa","unstructured":"M. S.  Abramowitz and I. A.  Stegun,\nHandbook of Mathematical Functions,\nDover, New York, 1965."},{"key":"2023033109443814301_j_acv-2015-0045_ref_002_w2aab3b7e1521b1b6b1ab2ab2Aa","unstructured":"C. J. S.  Alves and P. R. S.  Antunes,\nThe method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes,\nComput. Mater Con. 2 (2005), 251\u2013266."},{"key":"2023033109443814301_j_acv-2015-0045_ref_003_w2aab3b7e1521b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"C. J. S.  Alves and P. R. S.  Antunes,\nThe method of fundamental solutions applied to some inverse eigenproblems,\nSIAM J. Sci. Comput. 35 (2013), A1689\u2013A1708.","DOI":"10.1137\/110860380"},{"key":"2023033109443814301_j_acv-2015-0045_ref_004_w2aab3b7e1521b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"D. E.  Amos,\nComputation of modified Bessel functions and their ratios,\nMath. Comp. 28 (1974), 239\u2013251.","DOI":"10.1090\/S0025-5718-1974-0333287-7"},{"key":"2023033109443814301_j_acv-2015-0045_ref_005_w2aab3b7e1521b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"P. R. S.  Antunes and P.  Freitas,\nNumerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians,\nJ. Optim. Theory Appl. 154 (2012), 235\u2013257.","DOI":"10.1007\/s10957-011-9983-3"},{"key":"2023033109443814301_j_acv-2015-0045_ref_006_w2aab3b7e1521b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"P. R. S.  Antunes and P.  Freitas,\nOptimal spectral rectangles and lattice ellipses,\nProc. R. Soc. Lond. Ser. A 469 (2013), Article ID 20120492.","DOI":"10.1098\/rspa.2012.0492"},{"key":"2023033109443814301_j_acv-2015-0045_ref_007_w2aab3b7e1521b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"P. R. S.  Antunes and P.  Freitas,\nOptimisation of eigenvalues of the Dirichlet Laplacian with a surface area restriction,\nAppl. Math. Optim. 73 (2016), no. 2, 313\u2013328.","DOI":"10.1007\/s00245-015-9304-6"},{"key":"2023033109443814301_j_acv-2015-0045_ref_008_w2aab3b7e1521b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"P. R. S.  Antunes, P.  Freitas and J. B.  Kennedy,\nAsymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian,\nESAIM Control Optim. Calc. Var. 19 (2013), 438\u2013459.","DOI":"10.1051\/cocv\/2012016"},{"key":"2023033109443814301_j_acv-2015-0045_ref_009_w2aab3b7e1521b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"M.  Bareket,\nOn an isoperimetric inequality for the first eigenvalue of a boundary value problem,\nSIAM J. Math. Anal. 8 (1977), 280\u2013287.","DOI":"10.1137\/0508020"},{"key":"2023033109443814301_j_acv-2015-0045_ref_010_w2aab3b7e1521b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"A.  Berger,\nThe eigenvalues of the Laplacian with Dirichlet boundary condition in \u211d2{\\mathbb{R}^{2}} are almost never minimized by disks,\nAnn. Global Anal. Geom. 47 (2015), 285\u2013304.","DOI":"10.1007\/s10455-014-9446-9"},{"key":"2023033109443814301_j_acv-2015-0045_ref_011_w2aab3b7e1521b1b6b1ab2ac11Aa","unstructured":"M.-H.  Bossel,\nMembranes \u00e9lastiquement li\u00e9es: Extension du th\u00e9or\u00e9me de Rayleigh\u2013Faber\u2013Krahn et de l\u2019in\u00e9galit\u00e9 de Cheeger,\nC. R. Acad. Sci. Paris S\u00e9r. I 302 (1986), 47\u201350."},{"key":"2023033109443814301_j_acv-2015-0045_ref_012_w2aab3b7e1521b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"D.  Bucur and P.  Freitas,\nAsymptotic behaviour of optimal spectral planar domains with fixed perimeter,\nJ. Math. Phys. 54 (2013), Article ID 053504.","DOI":"10.1063\/1.4803140"},{"key":"2023033109443814301_j_acv-2015-0045_ref_013_w2aab3b7e1521b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"B.  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Math. 280 (2015), 322\u2013339.","DOI":"10.1016\/j.aim.2015.04.023"},{"key":"2023033109443814301_j_acv-2015-0045_ref_019_w2aab3b7e1521b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"T.  Giorgi and R. G.  Smits,\nMonotonicity results for the principal eigenvalue of the generalized Robin problem,\nIllinois J. Math. 49 (2005), 1133\u20131143.","DOI":"10.1215\/ijm\/1258138130"},{"key":"2023033109443814301_j_acv-2015-0045_ref_020_w2aab3b7e1521b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"T.  Giorgi and R. G.  Smits,\nEigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity,\nZ. Angew. Math. Phys. 58 (2007), 1224\u2013245.","DOI":"10.1007\/s00033-005-0049-y"},{"key":"2023033109443814301_j_acv-2015-0045_ref_021_w2aab3b7e1521b1b6b1ab2ac21Aa","doi-asserted-by":"crossref","unstructured":"T.  Giorgi and R. G.  Smits,\nBounds and monotonicity for the generalized Robin problem,\nZ. Angew. Math. 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Kato,\nPerturbation Theory for Linear Operators,\nSpringer, Berlin, 1966.","DOI":"10.1007\/978-3-662-12678-3"},{"key":"2023033109443814301_j_acv-2015-0045_ref_025_w2aab3b7e1521b1b6b1ab2ac25Aa","doi-asserted-by":"crossref","unstructured":"J. B.  Kennedy,\nAn isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions,\nProc. Amer. Math. Soc. 137 (2009), 627\u2013633.","DOI":"10.1090\/S0002-9939-08-09704-9"},{"key":"2023033109443814301_j_acv-2015-0045_ref_026_w2aab3b7e1521b1b6b1ab2ac26Aa","doi-asserted-by":"crossref","unstructured":"E.  Krahn,\n\u00dcber eine von Rayleigh formulierte Minimaleigenschaft des Kreises,\nMath. Ann. 94 (1924), 97\u2013100.","DOI":"10.1007\/BF01208645"},{"key":"2023033109443814301_j_acv-2015-0045_ref_027_w2aab3b7e1521b1b6b1ab2ac27Aa","unstructured":"E.  Krahn,\n\u00dcber Minimaleigenschaft der Kugel in drei und mehr Dimensionen,\nActa Comm. Univ. 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