{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T05:56:59Z","timestamp":1776664619418,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"3","license":[{"start":{"date-parts":[[2020,10,18]],"date-time":"2020-10-18T00:00:00Z","timestamp":1602979200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00c3\u00a7\u00c3\u00a3o para a Ci\u00c3\u00aancia 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"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1.066459\">\n                        <mml:semantics>\n                          <mml:mn>1.066459<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1.066459<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We then consider the sequence formed by the minimal\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    th eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    goes to infinity.\n                  <\/p>","DOI":"10.1090\/proc\/14792","type":"journal-article","created":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T11:13:54Z","timestamp":1571224434000},"page":"1109-1120","source":"Crossref","is-referenced-by-count":7,"special_numbering":"729","title":["Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles"],"prefix":"10.1090","volume":"148","author":[{"given":"D.","family":"Buoso","sequence":"first","affiliation":[]},{"given":"P.","family":"Freitas","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2019,10,18]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1002\/nme.2404","article-title":"The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates","volume":"77","author":"Alves, Carlos J. 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