Optimal shapes maximizing the Steklov eigenvalues. (English) Zbl 1367.49037


49R05 Variational methods for eigenvalues of operators
49J35 Existence of solutions for minimax problems
58C40 Spectral theory; eigenvalue problems on manifolds
26A45 Functions of bounded variation, generalizations
49J45 Methods involving semicontinuity and convergence; relaxation
49M30 Other numerical methods in calculus of variations (MSC2010)
Full Text: DOI


[1] E. Akhmetgaliyev, C.-Y. Kao, and B. Osting, Computational Methods for Extremal Steklov Problems, SIAM J. Control. Optim., to appear. · Zbl 1432.65164
[2] C. J. S. Alves and P. R. S. Antunes, The method of fundamental solutions applied to the calculation of eigensolutions for \(2\)D plates, Internat. J. Numer. Methods Engrg., 77 (2009), pp. 177–194, . · Zbl 1257.74096
[3] L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc. (JEMS), 3 (2001), pp. 39–92, . · Zbl 0981.49024
[4] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon, New York, 2000. · Zbl 0957.49001
[5] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, Oxford Lecture Ser. Math. Appl. 25, Oxford University Press, Oxford, 2004. · Zbl 1080.28001
[6] 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
[7] B. Bogosel, Shape Optimization and Spectral Problems, Ph.D. thesis, Universite Grenoble Alpes, Grenoble, France, 2015.
[8] B. Bogosel, The method of fundamental solutions applied to boundary eigenvalue problems, J. Comput. Appl. Math., 306 (2016), pp. 265–285, . · Zbl 1338.49068
[9] F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, ZAMM Z. Angew. Math. Mech., 81 (2001), pp. 69–71. · Zbl 0971.35055
[10] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progr. Nonlinear Differential Equations Appl. 65, Birkhäuser, Boston, 2005. · Zbl 1117.49001
[11] D. Bucur and A. Giacomini, A variational approach to the isoperimetric inequality for the Robin eigenvalue problem, Arch. Ration. Mech. Anal., 198 (2010), pp. 927–961, . · Zbl 1228.49049
[12] G. Buttazzo and G. Dal Maso, An existence result for a class of shape optimization problems, Arch. Rational Mech. Anal., 122 (1993), pp. 183–195, . · Zbl 0811.49028
[13] A. Chambolle and F. Doveri, Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets, Comm. Partial Differential Equations, 22 (1997), pp. 811–840, . · Zbl 0901.35019
[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] B. Colbois, A. El Soufi, and A. Girouard, Isoperimetric control of the spectrum of a compact hypersurface, J. Reine Angew. Math., 683 (2013), pp. 49–65. · Zbl 1282.58017
[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] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001
[18] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math. 85, Cambridge University Press, Cambridge, 1986.
[19] 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
[20] A. Grigor’yan, Y. Netrusov, and S.-T. Yau, Eigenvalues of elliptic operators and geometric applications, in Eigenvalues of Laplacians and Other Geometric Operators, Surv. Differential Geom. IX, International Press, Somerville, MA, 2004, pp. 147–217, . · Zbl 1061.58027
[21] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Front. Math., Birkhäuser, Basel, 2006. · Zbl 1109.35081
[22] J. Hersch, L. E. Payne, and M. M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Ration. Mech. Anal., 57 (1975), pp. 99–114. · Zbl 0315.35069
[23] B. Osting, Optimization of spectral functions of Dirichlet-Laplacian eigenvalues, J. Comput. Phys., 229 (2010), pp. 8578–8590, . · Zbl 1201.65203
[24] R. Petrides, Bornes sur des valeurs propres et métriques extrémales, Thèse de doctorat, Université Claude Bernard de Lyon, Lyon, France, 2015.
[25] C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss. 299, Springer, Berlin, 1992, . · Zbl 0762.30001
[26] R. Weinstock, Inequalities for a classical eigenvalue problem, J. Ration. 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.