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Minimal convex combinations of three sequential Laplace-Dirichlet eigenvalues. (English) Zbl 1305.49066

Summary: We study the shape optimization problem where the objective function is a convex combination of three sequential Laplace-Dirichlet eigenvalues. That is, for \(\alpha\geq 0, \beta\geq 0\), and \(\alpha+\beta\leq1\), we consider \(\inf\{\alpha\lambda_k(\Omega)+\beta\lambda_{k+1}(\Omega)+(1-\alpha-\beta)\lambda_{k+2}(\Omega):\Omega \;\text{open\;set\;in}\;\mathbb{R}^2 \;\text{and}\;|\Omega|\leq 1\}\). Here \(\lambda_k(\Omega)\) denotes the \(k\)-th Laplace-Dirichlet eigenvalue and \(|\cdot|\) denotes the Lebesgue measure. For \(k=1,2\), the minimal values and minimizers are computed explicitly when the set of admissible domains is restricted to the disjoint union of balls. For star-shaped domains, we show that for \(k=1\) and \(\alpha+2\beta\leq1\), the ball is a local minimum. For \(k=1,2\), several properties of minimizers are studied computationally, including uniqueness, connectivity, symmetry, and eigenvalue multiplicity.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
35J25 Boundary value problems for second-order elliptic equations

Software:

MPSpack; HANSO
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Full Text: DOI

References:

[1] Antunes, P.R.S., Freitas, P.: Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154(1), 235-257 (2012) · Zbl 1252.90076 · doi:10.1007/s10957-011-9983-3
[2] Antunes, P.R.: Optimization of sums and quotients of Dirichlet-Laplacian eigenvalues. Appl. Math. Comput. 219(9), 4239-4254 (2013) · Zbl 1515.35163 · doi:10.1016/j.amc.2012.10.095
[3] Betcke, T., Barnett, A.: mpspack, a MATLAB toolbox to solve Helmholtz PDE, wave scattering, and eigenvalue problems using particular solutions and integral equations. http://code.google.com/p/mpspack/ (2012)
[4] Bucur, D.: Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073-1083 (2012) · Zbl 1254.35165 · doi:10.1007/s00205-012-0561-0
[5] Courant, R., Hilbert, D.: Methods of Mathematical Physics. Vol. I. Interscience, New York (1953) · Zbl 0051.28802
[6] Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser, Basel (2006) · Zbl 1109.35081
[7] Iversen, M., Mazzoleni, D.: Minimizing convex combinations of low eigenvalues ESAIM Control Optim. Calc. Var., to appear · Zbl 1290.49096
[8] Kuttler, J.R., Sigillito, V.G.: Eigenvalues of the Laplacian in two dimensions. SIAM Rev. 26(2), 163-193 (1984) · Zbl 0574.65116 · doi:10.1137/1026033
[9] Lewis, A.S., Overton, M.L.: Nonsmooth optimization via quasi-Newton methods. Math. Program., 1-29 (2012) · Zbl 0816.35097
[10] Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. 100(3), 433-453 (2013) · Zbl 1296.35100 · doi:10.1016/j.matpur.2013.01.008
[11] Osting, B.: Optimization of spectral functions of Dirichlet-Laplacian eigenvalues. J. Comput. Phys. 229(22), 8578-8590 (2010) · Zbl 1201.65203 · doi:10.1016/j.jcp.2010.07.040
[12] Osting, B., Kao, C.-Y.: Minimal convex combinations of sequential Laplace-Dirichlet eigenvalues. SIAM J. Sci. Comput. 35(3), B731-B750 (2013) · Zbl 1273.35196 · doi:10.1137/120881865
[13] Oudet, E.: Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10, 315-335 (2004) · Zbl 1076.74045 · doi:10.1051/cocv:2004011
[14] Overton, M.L.: HANSO: Hybrid algorithm for non-smooth optimization. http://www.cs.nyu.edu/overton/software/hanso/ (2012) · Zbl 1201.65203
[15] Wolf, S.A., Keller, J.B.: Range of the first two eigenvalues of the Laplacian. Proc. R. Soc. Lond. A 447, 397-412 (1994) · Zbl 0816.35097 · doi:10.1098/rspa.1994.0147
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