×

Computational methods for extremal Steklov problems. (English) Zbl 1432.65164


MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P15 Estimates of eigenvalues in context of PDEs
49Q10 Optimization of shapes other than minimal surfaces
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] E. Akhmetgaliyev, Fast Numerical Methods for Mixed, Singular Helmholtz Boundary Value Problems and Laplace Eigenvalue Problems—with Applications to Antenna Design, Sloshing, Electromagnetic Scattering and Spectral Geometry, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 2016.
[2] E. Akhmetgaliyev, O. Bruno, N. Nigam, and N. Tazhimbetov, A high-accuracy boundary integral strategy for the Steklov eigenvalue problem, 2016.
[3] P. R. S. Antunes, Optimization of sums and quotients of Dirichlet–Laplacian eigenvalues, Appl. Math. Comput., 219 (2013), pp. 4239–4254, . · Zbl 06447235
[4] P. R. S. Antunes and P. Freitas, Optimisation of eigenvalues of the Dirichlet Laplacian with a surface area restriction, Appl. Math. Optim., 73 (2016), pp. 313–328, . · Zbl 1339.35338
[5] P. R. S. Antunes and P. Freitas, Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians, J. Optim. Theory Appl., 154 (2012), pp. 235–257, . · Zbl 1252.90076
[6] R. Banuelos, T. Kulczycki, I. Polterovich, and B. Siudeja, Eigenvalue inequalities for mixed Steklov problems, in Operator Theory and Its Applications, Amer. Math. Soc. Transl. Ser. 2 231, AMS, Providence, RI, 2010, pp. 19–34, . · Zbl 1217.35127
[7] B. Bogosel, The method of fundamental solutions applied to boundary eigenvalue problems, J. Comput. Appl.Math., 306 (2016), pp. 265–285, . · Zbl 1338.49068
[8] B. Bogosel, D. Bucur, and A. Giacomini, Optimal Shapes Maximizing the Steklov Eigenvalues, preprint, 2016, . · Zbl 1367.49037
[9] B. Bogosel and É. Oudet, Qualitative and numerical analysis of a spectral problem with perimeter constraint, SIAM J. Control Optim., 54 (2016), pp. 317–340, . · Zbl 1335.49070
[10] J. F. Bonder, P. Groisman, and J. D. Rossi, Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach, Ann. Mat. Pura Appl. (4), 186 (2007), pp. 341–358, . · Zbl 1223.35245
[11] R. Brayton, S. Director, G. Hachtel, and L. Vidigal, A new algorithm for statistical circuit design based on quasi-Newton methods and function splitting, IEEE Trans. Circuits Syst., 26 (1979), pp. 784–794, .
[12] F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, ZAMM Z. Angew. Math. Mech., 81 (2001), pp. 69–71. · Zbl 0971.35055
[13] P. Cheng, J. Huang, and Z. Wang, Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity, Numer. Methods Partial Differential Equations, 28 (2012), pp. 2021–2040, . · Zbl 1306.74063
[14] B. Colbois, A. El Soufi, and A. Girouard, Isoperimetric control of the Steklov spectrum, J. Funct. Anal., 261 (2011), pp. 1384–1399, . · Zbl 1235.58020
[15] D. L. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 1998, . · Zbl 0893.35138
[16] M. Dambrine, D. Kateb, and J. Lamboley, An extremal eigenvalue problem for the Wentzell–Laplace operator, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), pp. 409–450. . · Zbl 1347.35186
[17] B. Dittmar, Sums of reciprocal Stekloff eigenvalues, Math. Nachr., 268 (2004), pp. 44–49, . · Zbl 1054.35041
[18] A. Fraser and R. Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math., 226 (2011), pp. 4011–4030, . · Zbl 1215.53052
[19] A. Girouard, R. S. Laugesen, and B. A. Siudeja, Steklov eigenvalues and quasiconformal maps of simply connected planar domains, Arch. Ration. Mech. Anal., 219 (2016), pp. 903–936, . · Zbl 1333.35274
[20] A. Girouard and I. Polterovich, On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues, Funct. Anal. Appl., 44 (2010), pp. 106–117, . · Zbl 1217.35125
[21] A. Girouard and I. Polterovich, Shape optimization for low Neumann and Steklov eigenvalues, Math. Methods Appl. Sci., 33 (2010), pp. 501–516, . · Zbl 1186.35121
[22] A. Girouard and I. Polterovich, Spectral Geometry of the Steklov Problem, preprint, , 2014. · Zbl 1375.49056
[23] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013. · Zbl 1268.65037
[24] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Springer, New York, 2006, . · Zbl 1109.35081
[25] J. Hersch, L. E. Payne, and M. M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Ration. Mech. Anal., 57 (1974), pp. 99–114, . · Zbl 0315.35069
[26] J. Huang and T. Lü, The mechanical quadrature methods and their extrapolation for solving BIE of Steklov eigenvalue problems, J. Comput. Math., 22 (2004), pp. 719–726. · Zbl 1069.65123
[27] R. A. Ibrahim, Liquid Sloshing Dynamics: Theory and Applications, Cambridge University Press, Cambridge, UK, 2005, . · Zbl 1103.76002
[28] C.-Y. Kao, R. Lai, and B. Osting, Maximization Laplace-Beltrami eigenvalues on closed Riemannian surfaces, ESAIM Control Optim. Calc. Var., 23 (2017), pp. 685–720, . · Zbl 1362.35199
[29] R. Kress, Linear Integral Equations, Appl. Math. Sci. 82, Springer, New York, 1999, . · Zbl 0920.45001
[30] T. Kulczycki, M. Kwaśnicki, and B. Siudeja, The shape of the fundamental sloshing mode in axisymmetric containers, J. Engrg. Math., (2014), pp. 1–27, .
[31] N. Kuznetsov, T. Kulczycki, M. Kwaśnicki, A. Nazarov, S. Poborchi, I. Polterovich, and B. Siudeja, The legacy of Vladimir Andreevich Steklov, Notices Amer. Math. Soc., 61 (2014), pp. 9–22, . · Zbl 1322.01050
[32] H. Mayer and R. Krechetnikov, Walking with coffee: Why does it spill?, Phys. Rev. E, 85 (2012), 046117, .
[33] B. Osting, Optimization of spectral functions of Dirichlet-Laplacian eigenvalues, J. Comput. Phys., 229 (2010), pp. 8578–8590, . · Zbl 1201.65203
[34] B. Osting and C.-Y. Kao, Minimal convex combinations of sequential Laplace–Dirichlet eigenvalues, SIAM J. Sci. Comput., 35 (2013), pp. B731–B750, . · Zbl 1273.35196
[35] B. Osting and C.-Y. Kao, Minimal convex combinations of three sequential Laplace-Dirichlet eigenvalues, Appl. Math. Optim., 69 (2014), pp. 123–139, . · Zbl 1305.49066
[36] E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain, ESAIM Control Optim. Calc. Var., 10 (2004), pp. 315–330, . · Zbl 1076.74045
[37] B. A. Troesch, An isoperimetric sloshing problem, Comm. Pure Appl. Math., 18 (1965), pp. 319–338, . · Zbl 0145.46401
[38] R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), pp. 745–753, . · Zbl 0056.09801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.