We consider the variational problem inf{αλ1(Ω) + βλ2(Ω) + (1 - α - β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1}, for α, β ∈ [0, 1], α + β ≤ 1, where λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.
Keywords: eigenvalues, Dirichlet-Laplacian, shape optimization
@article{COCV_2014__20_2_442_0,
author = {Iversen, Mette and Mazzoleni, Dario},
title = {Minimising convex combinations of low eigenvalues},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {442--459},
year = {2014},
publisher = {EDP Sciences},
volume = {20},
number = {2},
doi = {10.1051/cocv/2013070},
mrnumber = {3264211},
zbl = {1290.49096},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2013070/}
}
TY - JOUR AU - Iversen, Mette AU - Mazzoleni, Dario TI - Minimising convex combinations of low eigenvalues JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 442 EP - 459 VL - 20 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2013070/ DO - 10.1051/cocv/2013070 LA - en ID - COCV_2014__20_2_442_0 ER -
%0 Journal Article %A Iversen, Mette %A Mazzoleni, Dario %T Minimising convex combinations of low eigenvalues %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 442-459 %V 20 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2013070/ %R 10.1051/cocv/2013070 %G en %F COCV_2014__20_2_442_0
Iversen, Mette; Mazzoleni, Dario. Minimising convex combinations of low eigenvalues. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 2, pp. 442-459. doi: 10.1051/cocv/2013070
[1] and , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972). | Zbl
[2] and , Proof of the Payne−Pölya−Weinberger conjecture. Bull. Amer. Math. Soc. 25 (1991) 19-29. | Zbl | MR
[3] and , Isoperimetric bound for λ3/λ2 for the membrane problem. Duke Math. J. 63 (1991) 333-341. | Zbl | MR
[4] , On Rayleigh's formula for the first Dirichlet eigenvalue of a radial perturbation of a ball. J. Geometric Anal. 23 (2013) 1427-1440. | MR
[5] and , On the minimization of Dirichlet eigenvalues of the Laplace operator. J. Geometric Anal. 23 (2013) 660-676. | Zbl | MR
[6] , and , On the boundary of the attainable set of the Dirichlet spectrum. Z. Angew. Math. Phys. 64 (2013) 591-597. | Zbl | MR
[7] and , Variational methods in shape optimization problems. Prog. Nonlinear Differ. Eq. Appl. Birkhäuser Verlag, Boston (2005). | Zbl | MR
[8] , and , On the attainable eigenvalues of the Laplace operator. SIAM J. Math. Anal. 30 (1999) 527-536. | Zbl | MR
[9] and , Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. Roy. Soc. London 456 (2000) 985-996. | Zbl | MR
[10] and , An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993) 183-195. | Zbl | MR
[11] and , Methods of Mathematical Physics, vol. 2. Wiley-VCH, New York (1962). | Zbl
[12] , Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers Math. Birkhäuser Verlag, Basel (2006). | Zbl | MR
[13] and , Existence of minimizers for spectral problems. J. Math. Pures Appl. 100 (2013) 433-453. DOI: http://dx.doi.org/10.1016/j.matpur.2013.01.008. | Zbl | MR
[14] and , Minimal convex combinations of three sequential Laplace−Dirichlet eigenvalues, Appl. Math. Optim. 69 (2014) 123-139. | MR
[15] , Asymptotic and Numerical Analysis of Linear and Nonlinear Eigenvalue Problems, Ph.D. Thesis. Stanford University (1993). | MR
[16] and , Range of the First Two Eigenvalues of the Laplacian. Proc. R. Soc. London A 447 (1994) 397-412. | Zbl | MR
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