{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:40:27Z","timestamp":1680298827622},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"name":"FNS","award":["20-137696\/1"],"award-info":[{"award-number":["20-137696\/1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014,7,1]]},"abstract":"<jats:title>Abstract.<\/jats:title>\n               <jats:p>Let (<jats:italic>M<\/jats:italic>,<jats:italic>g<\/jats:italic>) be a smooth and complete surface,\n<jats:inline-formula id=\"eq1_w2aab2b8c10b1b7b1aab1c13b1b5Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0009_3009fa76896db5c9b8d7a318a9fa4ab7.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03a9<\/m:mi>\n                              <m:mo>\u2282<\/m:mo>\n                              <m:mi>M<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$\\Omega \\subset M$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> be a domain in <jats:italic>M<\/jats:italic>, and <jats:inline-formula id=\"eq2_w2aab2b8c10b1b7b1aab1c13b1b9Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0009_bccde85003dfdaece9050f9cb99cb909.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>\u0394<\/m:mi>\n                              <m:mi>g<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>$\\Delta _g$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> be the Laplace operator on <jats:italic>M<\/jats:italic>. The spectrum of the Dirichlet\u2013Laplace operator on \u03a9 is a sequence <jats:inline-formula id=\"eq3_w2aab2b8c10b1b7b1aab1c13b1c13Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0009_02411d9b70bff536f4774f453be8633c.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>0<\/m:mn>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:msub>\n                                 <m:mi>\u03bb<\/m:mi>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:msub>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>\u03a9<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>\u2264<\/m:mo>\n                              <m:msub>\n                                 <m:mi>\u03bb<\/m:mi>\n                                 <m:mn>2<\/m:mn>\n                              <\/m:msub>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>\u03a9<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>\u2264<\/m:mo>\n                              <m:mo>\u22ef<\/m:mo>\n                              <m:mo>\u2197<\/m:mo>\n                              <m:mi>\u221e<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$0 &amp;lt; \\lambda _1(\\Omega ) \\le \\lambda _2(\\Omega ) \\le \\cdots \\nearrow \\infty $<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. A classical question is to ask what is the domain <jats:inline-formula id=\"eq4_w2aab2b8c10b1b7b1aab1c13b1c15Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0009_65a7868aa402fbdc4dca76e45a57dcb2.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>\u03a9<\/m:mi>\n                              <m:mo>*<\/m:mo>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>$\\Omega ^*$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> which minimizes <jats:inline-formula id=\"eq5_w2aab2b8c10b1b7b1aab1c13b1c17Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0009_3fefcca1908b75395faeb5af9bde1554.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi>\u03bb<\/m:mi>\n                                 <m:mi>m<\/m:mi>\n                              <\/m:msub>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mi>\u03a9<\/m:mi>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$\\lambda _m(\\Omega )$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> among all domains of a given area, and what is the value of the corresponding <jats:inline-formula id=\"eq6_w2aab2b8c10b1b7b1aab1c13b1c19Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0009_2ab775659973fa24e50518bba4e007cc.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi>\u03bb<\/m:mi>\n                                 <m:mi>m<\/m:mi>\n                              <\/m:msub>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:msubsup>\n                                    <m:mi>\u03a9<\/m:mi>\n                                    <m:mi>m<\/m:mi>\n                                    <m:mo>*<\/m:mo>\n                                 <\/m:msubsup>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$\\lambda _m(\\Omega _m^*)$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for <jats:inline-formula id=\"eq7_w2aab2b8c10b1b7b1aab1c13b1c21Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0009_6b2f56614beb02513a59eae91a0fe886.png\" \/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msubsup>\n                              <m:mi>\u03a9<\/m:mi>\n                              <m:mi>m<\/m:mi>\n                              <m:mo>*<\/m:mo>\n                           <\/m:msubsup>\n                        <\/m:math>\n                        <jats:tex-math>$\\Omega _m^*$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. Some verifications with existing numerical results are carried out for the first eigenvalues of domains in \u211d<jats:sup>2<\/jats:sup>. Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.<\/jats:p>","DOI":"10.1515\/cmam-2014-0009","type":"journal-article","created":{"date-parts":[[2014,5,21]],"date-time":"2014-05-21T12:49:39Z","timestamp":1400676579000},"page":"393-409","source":"Crossref","is-referenced-by-count":0,"title":["Numerical Optimization of Eigenvalues of the Dirichlet\u2013Laplace Operator on Domains in Surfaces"],"prefix":"10.1515","volume":"14","author":[{"given":"R\u00e9gis","family":"Straubhaar","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques, Universit\u00e9 de Neuch\u00e2tel, Rue Emile-Argand 11, Case postale 158, 2009 Neuch\u00e2tel, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2014,4,24]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/14\/3\/article-p393.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0009\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0009\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:08:47Z","timestamp":1680296927000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0009\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,4,24]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,4,24]]},"published-print":{"date-parts":[[2014,7,1]]}},"alternative-id":["10.1515\/cmam-2014-0009"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2014-0009","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,4,24]]}}}