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Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk

Published online by Cambridge University Press:  20 November 2018

Eveline Legendre*
Affiliation:
Université de Montréal, Montréal, QC H3C 3J7, e-mail: egendre@dms.umontreal.ca
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Abstract

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We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

References

[A] Alessandrini, G., Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Comment. Math. Helv. 69(1994), no. 1, 142–154. doi:10.1007/BF02564478Google Scholar
[BD] Burchard, A. and Denzler, J., On the geometry of optimal windows, with special focus on the square. SIA M J. Math. Anal. 37(2006), no. 6, 1800–1827. doi:10.1137/S0036141004444184Google Scholar
[C1] Chavel, I., Eigenvalue in Riemannian Geometry. Pure and Applied Mathematics, 115, Academic Press Inc., Orlando, FL, 1984.Google Scholar
[CU] Cox, S. J. and Uhlig, P. X., Where best to hold a drum fast. SIA M J. Optimi. 9(1999), no. 4, 948–964. doi:10.1137/S1052623497326083Google Scholar
[D1] Denzler, J., Bounds for the heat diffusion through windows of given area. J. Math. Anal. Appl. 217(1998), no. 2, 405–422. doi:10.1006/jmaa.1997.5716Google Scholar
[D2] Denzler, J., Windows of given area with minimal heat diffusion. Trans. Amer. Math. Soc. 351(1999), no. 2, 569–580. doi:10.1090/S0002-9947-99-02207-2Google Scholar
[Gr] Graham, M. K., Optimisation of some eigenvalue problems. Ph.D. Thesis, Heriot–Watt University, Edinburgh, 2007.Google Scholar
[HO2] Henrot, A. and Pierre, M., Variation et optimisation de formes. Une analyse géométrique. Mathématiques et Applications, 48, Springer, Berlin, 2005.CrossRefGoogle Scholar
[L] Lin, C. S., On the second eigenfunctions of the Laplacian in R2. Comm. Math. Phys. 111(1987), no. 2, 161–166. doi:10.1007/BF01217758Google Scholar
[Me] Melas, A. D., On the nodal line of the second eigenfunction of the Laplacian in R2. J. Differential Geom. 35(1992), no. 1, 255–263.Google Scholar
[M] Miranda, C., Partial differential equations of elliptic type. Second ed., Springer-Verlag, New-York–Berlin, 1970.Google Scholar
[P] Payne, L. E., On two conjectures in the fixed membrane eigenvalue problem. Z. Angew. Math. Phys. 24(1973), 721–729. doi:10.1007/BF01597076Google Scholar
[PS] Polyà, G. and Szegö, G., Isoperimetrical inequalities in mathematical physics. Annals of Mathematics Studies, 27, Princeton University Press, Princeton, NJ, 1951.CrossRefGoogle Scholar
[SY] Schoen, R. and Yau, S.-T., Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology, International Press, Cambridge, MA, 1994.Google Scholar
[Sp] Sperner, E., Spherical symmetrization and eigenvalue estimates. Math. Z. 176(1981), no. 1, 75–86. doi:10.1007/BF01258906Google Scholar
[Sv] Šverak, V., On optimal shape design. J. Math. Pure Appl. 72(1993), no. 6, 537–551.Google Scholar
[Za] Zaremba, S., Sur un problème toujours possible comprenant à titre de cas particuliers, le problème de Dirichlet et celui de Neumann, J. Math. Pure Appl. 6(1927), 127–163.Google Scholar
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