{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T11:24:44Z","timestamp":1774610684396,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2012,1,10]],"date-time":"2012-01-10T00:00:00Z","timestamp":1326153600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2012,7]]},"DOI":"10.1007\/s10957-011-9983-3","type":"journal-article","created":{"date-parts":[[2012,1,9]],"date-time":"2012-01-09T12:19:36Z","timestamp":1326111576000},"page":"235-257","source":"Crossref","is-referenced-by-count":65,"title":["Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians"],"prefix":"10.1007","volume":"154","author":[{"given":"Pedro R. S.","family":"Antunes","sequence":"first","affiliation":[]},{"given":"Pedro","family":"Freitas","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,1,10]]},"reference":[{"key":"9983_CR1","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1080\/10586458.2006.10128966","volume":"15","author":"P. Antunes","year":"2006","unstructured":"Antunes, P., Freitas, P.: New bounds for the principal Dirichlet eigenvalue of planar regions. Exp. Math. 15, 333\u2013342 (2006)","journal-title":"Exp. Math."},{"key":"9983_CR2","unstructured":"Antunes, P., Freitas, P.: A\u00a0numerical study of the spectral gap. J.\u00a0Phys. A 5(055201) (2008) 19\u00a0pp."},{"key":"9983_CR3","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s00205-003-0259-4","volume":"169","author":"A. Henrot","year":"2003","unstructured":"Henrot, A., Oudet, E.: Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 169, 73\u201387 (2003)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9983_CR4","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1051\/cocv:2004011","volume":"10","author":"E. Oudet","year":"2004","unstructured":"Oudet, E.: Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10, 315\u2013330 (2004)","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"9983_CR5","volume-title":"Methods of Mathematical Physics","author":"R. Courant","year":"1953","unstructured":"Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol.\u00a0I. Interscience, New York (1953)"},{"key":"9983_CR6","series-title":"Frontiers in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/3-7643-7706-2","volume-title":"Extremum Problems for Eigenvalues of Elliptic Operators","author":"A. Henrot","year":"2006","unstructured":"Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkh\u00e4user, Basel (2006)"},{"key":"9983_CR7","unstructured":"Faber, G.: Beweis, dass unter allen homogenen membranen von gleicher fl\u00fcche und gleicher spannung die kreisf\u00fcrmige den tiefsten grundton gibt. Sitz. ber. bayer. Akad. Wiss., 169\u2013172 (1923)"},{"key":"9983_CR8","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF01208645","volume":"94","author":"E. Krahn","year":"1924","unstructured":"Krahn, E.: \u00dcber eine von Rayleigh formulierte minimaleigenschaft des kreises. Math. Ann. 94, 97\u2013100 (1924)","journal-title":"Math. Ann."},{"key":"9983_CR9","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/BF01187944","volume":"63","author":"G. P\u00f3lya","year":"1955","unstructured":"P\u00f3lya, G.: On the characteristic frequencies of a symmetric membrane. Math. Z. 63, 331\u2013337 (1955)","journal-title":"Math. Z."},{"key":"9983_CR10","first-page":"1","volume":"A9","author":"E. Krahn","year":"1926","unstructured":"Krahn, E.: \u00dcber Minimaleigenshaften der Kugel in drei und mehr Dimensionen. Acta Comm. Univ. Dorpat. A9, 1\u201344 (1926)","journal-title":"Acta Comm. Univ. Dorpat."},{"key":"9983_CR11","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1002\/sapm195231145","volume":"31","author":"E.T. Kornhauser","year":"1952","unstructured":"Kornhauser, E.T., Stakgold, I.: A\u00a0variational theorem for \u22072 u+\u03bbu=0 and its applications. J. Math. Phys. 31, 45\u201354 (1952)","journal-title":"J. Math. Phys."},{"key":"9983_CR12","first-page":"343","volume":"3","author":"G. Szeg\u00f6","year":"1954","unstructured":"Szeg\u00f6, G.: Inequalities for certain eigenvalues of a membrane of given area. Arch. Ration. Mech. Anal. 3, 343\u2013356 (1954)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9983_CR13","first-page":"633","volume":"5","author":"H.F. Weinberger","year":"1956","unstructured":"Weinberger, H.F.: An isoperimetric inequality for the N-dimensional free membrane problem. Arch. Ration. Mech. Anal. 5, 633\u2013636 (1956)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"3","key":"9983_CR14","doi-asserted-by":"crossref","first-page":"637","DOI":"10.4310\/jdg\/1264601037","volume":"83","author":"A. Girouard","year":"2009","unstructured":"Girouard, A., Nadirashvili, N., Polterovich, I.: Maximization of the second positive Neumann eigenvalue for planar domains. J. Differ. Geom. 83(3), 637\u2013662 (2009)","journal-title":"J. Differ. Geom."},{"key":"9983_CR15","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1098\/rspa.2000.0546","volume":"456","author":"D. Bucur","year":"2000","unstructured":"Bucur, D., Henrot, A.: Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. R. Soc. Lond. 456, 985\u2013996 (2000)","journal-title":"Proc. R. Soc. Lond."},{"key":"9983_CR16","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1098\/rspa.1994.0147","volume":"447","author":"S.A. Wolf","year":"1994","unstructured":"Wolf, S.A., Keller, J.B.: Range of the first two eigenvalues of the Laplacian. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 447, 397\u2013412 (1994)","journal-title":"Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci."},{"key":"9983_CR17","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/BF00378167","volume":"122","author":"G. Buttazzo","year":"1993","unstructured":"Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122, 183\u2013195 (1993)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1","key":"9983_CR18","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1006\/jdeq.1995.1109","volume":"120","author":"S.J. Cox","year":"1995","unstructured":"Cox, S.J.: Extremal eigenvalue problems for starlike planar domains. J. Differ. Equ. 120(1), 174\u2013197 (1995)","journal-title":"J. Differ. Equ."},{"key":"9983_CR19","unstructured":"Poliquin, G., Roy-Fortin, G.: Wolf-Keller theorem for Neumann eigenvalues. Ann. Sci. Math. Qu\u00e9bec (to appear)"},{"key":"9983_CR20","unstructured":"Colbois, B., El Soufi, A.: Extremal eigenvalues of the Laplacian on Euclidean domains and Riemannian manifolds (preprint)"},{"issue":"4","key":"9983_CR21","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1137\/0722040","volume":"22","author":"A. Bogomolny","year":"1985","unstructured":"Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22(4), 644\u2013669 (1985)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"9983_CR22","first-page":"251","volume":"2","author":"C.J.S. Alves","year":"2005","unstructured":"Alves, C.J.S., Antunes, P.R.S.: The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. Comput. Mater. Continua 2(4), 251\u2013266 (2005)","journal-title":"Comput. Mater. Continua"},{"key":"9983_CR23","volume-title":"Genetic Algorithms","author":"D. Goldberg","year":"1988","unstructured":"Goldberg, D.: Genetic Algorithms. Addison Wesley, Reading (1988)"},{"key":"9983_CR24","doi-asserted-by":"crossref","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J. Nocedal","year":"1999","unstructured":"Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999)"},{"key":"9983_CR25","volume-title":"Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms","author":"J.A. Snyman","year":"2005","unstructured":"Snyman, J.A.: Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms. Springer, Berlin (2005)"},{"key":"9983_CR26","series-title":"Springer Series in Computational Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-58106-9","volume-title":"Introduction to Shape Optimization: Shape Sensitivity Analysis","author":"J. Sokolowski","year":"1992","unstructured":"Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol.\u00a010. Springer, Berlin (1992)"},{"key":"9983_CR27","doi-asserted-by":"crossref","first-page":"7003","DOI":"10.1016\/j.jcp.2008.04.008","volume":"227","author":"A.H. Barnett","year":"2008","unstructured":"Barnett, A.H., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J.\u00a0Comput. Phys. 227, 7003\u20137026 (2008)","journal-title":"J.\u00a0Comput. Phys."},{"key":"9983_CR28","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1137\/0705004","volume":"5","author":"C.B. Moler","year":"1968","unstructured":"Moler, C.B., Payne, L.E.: Bounds for eigenvalues and eigenfunctions of symmetric operators. SIAM J. Numer. Anal. 5, 64\u201370 (1968)","journal-title":"SIAM J. Numer. Anal."},{"key":"9983_CR29","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1112\/plms\/s3-11.1.419","volume":"11","author":"G. P\u00f3lya","year":"1961","unstructured":"P\u00f3lya, G.: On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. 11, 419\u2013433 (1961)","journal-title":"Proc. Lond. Math. Soc."},{"key":"9983_CR30","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.enganabound.2004.10.005","volume":"29","author":"J.T. Chen","year":"2005","unstructured":"Chen, J.T., Chen, I.L., Lee, Y.T.: Eigensolutions of multiply connected membranes using the method of fundamental solutions. Eng. Anal. Bound. Elem. 29, 166\u2013174 (2005)","journal-title":"Eng. Anal. Bound. Elem."},{"issue":"6","key":"9983_CR31","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1002\/num.20594","volume":"27","author":"P.R.S. Antunes","year":"2011","unstructured":"Antunes, P.R.S.: Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions. Numer. Methods Partial Differ. Equ. 27(6), 1525\u20131550 (2011)","journal-title":"Numer. Methods Partial Differ. Equ."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-011-9983-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-011-9983-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-011-9983-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,22]],"date-time":"2019-06-22T03:41:51Z","timestamp":1561174911000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-011-9983-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,10]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,7]]}},"alternative-id":["9983"],"URL":"https:\/\/doi.org\/10.1007\/s10957-011-9983-3","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1,10]]}}}