[1] Anedda, C., Cuccu, F., Porru, G.:Minimization of the first eigenvalue in problems involving the bi-Laplacian. Revista de Mat. Teor. Apl. 16(1), 127-136 (2009)
[2] Arnold, D.N., David, G., Filoche, M., Jerison, D., Mayboroda, S.:Localization of eigenfunctions via an effective potential. Comm. Partial Differential Equations 44(11), 1186-1216 (2019)
[3] Bakharev, F., Matveenko, S.:Localization of eigenfunctions in a narrow Kirchhoff plate. Russ. J. Math. Phys. 28(2), 156-178 (2021)
[4] Banks, D.O.:Bounds for the eigenvalues of some vibrating systems. Pac. J. Math. 10(2), 439-474 (1960)
[5] Banks, D.O.:Bounds for the eigenvalues of nonhomogeneous hinged vibrating rods. J. Math. Mech. 16, 949-966 (1967)
[6] Beesack, P.R.:Isoperimetric inequalities for the nonhomogeneous clamped rod and plate. J. Math. Mech. 8, 471-482 (1959)
[7] Bjørstad, P.E., Tjøstheim, B.P.:High precision solutions of two fourth order eigenvalue problems. Computing 63(2), 97-107 (1999)
[8] Brenner, S.C., Monk, P., Sun, J.:C0 interior penalty Galerkin method for biharmonic eigenvalue problems. In:Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, pp. 3-15. Springer, Switzerland (2015)
[9] Chen, W., Chou, C.-S., Kao, C.-Y.:Minimizing eigenvalues for inhomogeneous rods and plates. J. Sci. Comput. 69(3), 983-1013 (2016)
[10] Chen, W., Diest, K., Kao, C.-Y., Marthaler, D.E., Sweatlock, L.A., Osher, S.:Gradient based optimization methods for metamaterial design. In:Numerical Methods for Metamaterial Design, pp. 175-204. Springer, Dordrecht (2013)
[11] Chladni, E.F.F.:Die Akustik. Breitkopf & Härtel, Leipzig (1802)
[12] Ciarlet, P.G., Raviart, P.-A.:A mixed finite element method for the biharmonic equation. In:Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 125-145. Academic Press, New York (1974)
[13] Colasuonno, F., Vecchi, E.:Symmetry in the composite plate problem. Commun. Contemp. Math. 21(02), 1850019 (2019)
[14] Cuccu, F., Emamizadeh, B., Porru, G.:Optimization of the first eigenvalue in problems involving the bi-Laplacian. Proceed. Am. Math. Soc. 137(5), 1677-1687 (2009)
[15] Cuccu, F., Porru, G.:Maximization of the first eigenvalue in problems involving the bi-Laplacian. Nonlinear Anal. Theory Methods Appl. 71(12), 800-809 (2009)
[16] Dobson, D.C., Santosa, F.:Optimal localization of eigenfunctions in an inhomogeneous medium. SIAM J. Appl. Math. 64(3), 762-774 (2004)
[17] Elishakoff, I.:Eigenvalues of Inhomogeneous Structure. Unusual Closed-Form Solutions. CRC Press, Boca Raton (2004)
[18] Gander, M.J., Kwok, F.:Chladni figures and the Tacoma bridge:motivating PDE eigenvalue problems via vibrating plates. SIAM Rev. 54(3), 573-596 (2012)
[19] Gautschi, W.:Leonhard Euler:his life, the man, and his works. SIAM Rev. 50(1), 3-33 (2008)
[20] Henrot, A.:Extremum Problems for Eigenvalues of Elliptic Operators. Springer, Basel-Boston-Berlin (2006)
[21] Kao, C.-Y., Su, S.:Efficient rearrangement algorithms for shape optimization on elliptic eigenvalue problems. J. Sci. Comput. 54(2/3), 492-512 (2013)
[22] Kirchhoff, G.:Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Die Reine Angew. Math. (Crelles Journal) 1850(40), 51-88 (1850)
[23] Krein, M.G.:On Certain Problems on the Maximum and Minimum of Characteristic Values and on the Lyapunov Zones of Stability. American Mathematical Society translations. American Mathematical Society, Boston (1955)
[24] Kropinski, M., Lindsay, A., Ward, M.:Asymptotic analysis of localized solutions to some linear and nonlinear biharmonic eigenvalue problems. Stud. Appl. Math. 126(4), 347-408 (2011)
[25] Lagrange, J.-L.:Sur la figure des colonnes. Miscellanea Taurinensia 5, 123-166 (1770)
[26] Lamberti, P.D., Provenzano, L.:A maximum principle in spectral optimization problems for elliptic operators subject to mass density perturbations. arXiv:1205.5624 (2012)
[27] Leissa, A.W.:Vibration of Plates. Scientific and Technical Information Division, National Aeronautics and Space Adminstration, Washington, D.C. (1969)
[28] Lindsay, A.E., Quaife, B., Wendelberger, L.:A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points. Adv. Comput. Math. 44(4), 1249-1273 (2018)
[29] Meng, J., Mei, L.:The optimal order convergence for the lowest order mixed finite element method of the biharmonic eigenvalue problem. J. Comput. Appl. Math. 402, 113783 (2022)
[30] Mohammadi, S., Bahrami, F.:Extremal principal eigenvalue of the bi-Laplacian operator. Appl. Math. Model. 40(3), 2291-2300 (2016)
[31] Monk, P.:A mixed finite element method for the biharmonic equation. SIAM J. Numer. Anal. 24(4), 737-749 (1987)
[32] Nazarov, S.A., Sweers, G.:A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners. J. Differential Equations 233(1), 151-180 (2007)
[33] Schwarz, B.:Some results on the frequencies of nonhomogeneous rods. J. Math. Anal. Appl. 5(2), 169-175 (1962)
[34] Stöckmann, H.-J.:Chladni meets Napoleon. Eur. Phys. J. Spec. Top. 145(1), 15-23 (2007)
[35] Wang, C.Y., Wang, C.:Structural Vibration:Exact Solutions for Strings, Membranes, Beams, and Plates. CRC Press, Boca Raton (2013)
