Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

Buoso, D.; Freitas, P.

doi:10.1090/proc/14792

Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles
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by D. Buoso and P. Freitas PDF

Proc. Amer. Math. Soc. 148 (2020), 1109-1120 Request permission

Abstract:

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed $1.066459$. We then consider the sequence formed by the minimal $k$th eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as $k$ goes to infinity.

References

Carlos J. S. Alves and Pedro R. S. Antunes, The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates, Internat. J. Numer. Methods Engrg. 77 (2009), no. 2, 177–194. MR 2485079, DOI 10.1002/nme.2404
Pedro R. S. Antunes and Pedro Freitas, Optimal spectral rectangles and lattice ellipses, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), no. 2150, 20120492, 15. MR 3001382, DOI 10.1098/rspa.2012.0492
Pedro Ricardo Simão Antunes, Pedro Freitas, and James Bernard Kennedy, Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian, ESAIM Control Optim. Calc. Var. 19 (2013), no. 2, 438–459. MR 3049718, DOI 10.1051/cocv/2012016
Sinan Ariturk and Richard S. Laugesen, Optimal stretching for lattice points under convex curves, Port. Math. 74 (2017), no. 2, 91–114. MR 3734407, DOI 10.4171/PM/1994
Mark S. Ashbaugh and Rafael D. Benguria, On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J. 78 (1995), no. 1, 1–17. MR 1328749, DOI 10.1215/S0012-7094-95-07801-6
Mark S. Ashbaugh and Richard S. Laugesen, Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 2, 383–402. MR 1433428
M. van den Berg, D. Bucur, and K. Gittins, Maximising Neumann eigenvalues on rectangles, Bull. Lond. Math. Soc. 48 (2016), no. 5, 877–894. MR 3556370, DOI 10.1112/blms/bdw049
M. van den Berg and K. Gittins, Minimizing Dirichlet eigenvalues on cuboids of unit measure, Mathematika 63 (2017), no. 2, 469–482. MR 3706591, DOI 10.1112/S0025579316000413
B. M. Brown, E. B. Davies, P. K. Jimack, and M. D. Mihajlović, A numerical investigation of the solution of a class of fourth-order eigenvalue problems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), no. 1998, 1505–1521. MR 1808762, DOI 10.1098/rspa.2000.0573
Dorin Bucur and Pedro Freitas, Asymptotic behaviour of optimal spectral planar domains with fixed perimeter, J. Math. Phys. 54 (2013), no. 5, 053504, 6. MR 3098942, DOI 10.1063/1.4803140
Davide Buoso, Analyticity and criticality results for the eigenvalues of the biharmonic operator, Geometric properties for parabolic and elliptic PDE’s, Springer Proc. Math. Stat., vol. 176, Springer, [Cham], 2016, pp. 65–85. MR 3571822, DOI 10.1007/978-3-319-41538-3_{5}
Davide Buoso and Pier Domenico Lamberti, Eigenvalues of polyharmonic operators on variable domains, ESAIM Control Optim. Calc. Var. 19 (2013), no. 4, 1225–1235. MR 3182687, DOI 10.1051/cocv/2013054
Victor I. Burenkov, Sobolev spaces on domains, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 137, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. MR 1622690, DOI 10.1007/978-3-663-11374-4
Charles V. Coffman, On the structure of solutions $\Delta ^{2}u=\lambda u$ which satisfy the clamped plate conditions on a right angle, SIAM J. Math. Anal. 13 (1982), no. 5, 746–757. MR 668318, DOI 10.1137/0513051
Bruno Colbois and Ahmad El Soufi, Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces, Math. Z. 278 (2014), no. 1-2, 529–546. MR 3267589, DOI 10.1007/s00209-014-1325-3
Pedro Freitas, Asymptotic behaviour of extremal averages of Laplacian eigenvalues, J. Stat. Phys. 167 (2017), no. 6, 1511–1518. MR 3652523, DOI 10.1007/s10955-017-1789-8
P. Freitas and J. B. Kennedy, Extremal domains and Pólya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles, arXiv:1805.10075, to appear in Int. Math. Res. Not. IMRN.
Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. MR 2667016, DOI 10.1007/978-3-642-12245-3
Katie Gittins and Simon Larson, Asymptotic behaviour of cuboids optimising Laplacian eigenvalues, Integral Equations Operator Theory 89 (2017), no. 4, 607–629. MR 3735512, DOI 10.1007/s00020-017-2407-5
V. A. Kozlov, V. A. Kondrat′ev, and V. G. Maz′ya, On sign variability and the absence of “strong” zeros of solutions of elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 2, 328–344 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 2, 337–353. MR 998299, DOI 10.1070/IM1990v034n02ABEH000649
Hsü Tung Ku, Mei Chin Ku, and Ding-Yi Tang, Inequalities for eigenvalues of elliptic equations and the generalized Pólya conjecture, J. Differential Equations 97 (1992), no. 1, 127–139. MR 1161315, DOI 10.1016/0022-0396(92)90087-4
S. Larson, Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains, arXiv:1611.05680.
Richard S. Laugesen and Shiya Liu, Optimal stretching for lattice points and eigenvalues, Ark. Mat. 56 (2018), no. 1, 111–145. MR 3800462, DOI 10.4310/ARKIV.2018.v56.n1.a8
N. Marshall, Stretching convex domains to capture many lattice points, Int. Math. Res. Notices, to appear.
Nicholas F. Marshall and Stefan Steinerberger, Triangles capturing many lattice points, Mathematika 64 (2018), no. 2, 551–582. MR 3798612, DOI 10.1112/S0025579318000219
Nikolai S. Nadirashvili, Rayleigh’s conjecture on the principal frequency of the clamped plate, Arch. Rational Mech. Anal. 129 (1995), no. 1, 1–10. MR 1328469, DOI 10.1007/BF00375124
M. P. Owen, Asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle, J. Differential Equations 136 (1997), no. 1, 166–190. MR 1443328, DOI 10.1006/jdeq.1996.3235
Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, Translations of Mathematical Monographs, vol. 155, American Mathematical Society, Providence, RI, 1997. Translated from the Russian manuscript by the authors. MR 1414899, DOI 10.1090/mmono/155
D. G. Vasil′ev, Two-term asymptotic behavior of the spectrum of a boundary value problem in the case of a piecewise smooth boundary, Dokl. Akad. Nauk SSSR 286 (1986), no. 5, 1043–1046 (Russian). MR 829600
C. Wieners, Bounds for the $N$ lowest eigenvalues of fourth-order boundary value problems, Computing 59 (1997), no. 1, 29–41. MR 1465309, DOI 10.1007/BF02684402