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Minimising convex combinations of low eigenvalues∗

Published online by Cambridge University Press:  07 March 2014

Mette Iversen  and
Dario Mazzoleni
Mette Iversen
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom. Mette.Iversen@bris.ac.uk
Dario Mazzoleni
Affiliation:
Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy; dario.mazzoleni@unipv.it Department Mathematik, Friedrich−Alexander Universität Erlangen-Nürnberg, Cauerstrasse, 11, 91058 Erlangen, Germany; mazzoleni@math.fau.de
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Abstract

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

EigenvaluesDirichlet–LaplacianShape Optimization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 442 - 459
Copyright
© EDP Sciences, SMAI, 2014

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References

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972).
Ashbaugh, M.S. and Benguria, R., Proof of the Payne−Pölya−Weinberger conjecture. Bull. Amer. Math. Soc. 25 (1991) 19–29. Google Scholar
Ashbaugh, M.S. and Benguria, R., Isoperimetric bound for λ 3/λ 2 for the membrane problem. Duke Math. J. 63 (1991) 333–341. Google Scholar
van den Berg, M., On Rayleigh’s formula for the first Dirichlet eigenvalue of a radial perturbation of a ball. J. Geometric Anal. 23 (2013) 1427–1440. Google Scholar
van den Berg, M. and Iversen, M., On the minimization of Dirichlet eigenvalues of the Laplace operator. J. Geometric Anal. 23 (2013) 660–676. Google Scholar
Brasco, L., Nitsch, C. and Pratelli, A., On the boundary of the attainable set of the Dirichlet spectrum. Z. Angew. Math. Phys. 64 (2013) 591–597. Google Scholar
D. Bucur and G. Buttazzo, Variational methods in shape optimization problems. Prog. Nonlinear Differ. Eq. Appl. Birkhäuser Verlag, Boston (2005).
Bucur, D., Buttazzo, G. and Figueiredo, I., On the attainable eigenvalues of the Laplace operator. SIAM J. Math. Anal. 30 (1999) 527–536. Google Scholar
Bucur, D. and Henrot, A., Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. Roy. Soc. London 456 (2000) 985–996. Google Scholar
Buttazzo, G. and Dal Maso, G., An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993) 183–195. Google Scholar
R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 2. Wiley-VCH, New York (1962).
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers Math. Birkhäuser Verlag, Basel (2006).
Mazzoleni, D. and Pratelli, A., 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. Google Scholar
Osting, B. and Kao, C.-Y., Minimal convex combinations of three sequential Laplace−Dirichlet eigenvalues, Appl. Math. Optim. 69 (2014) 123–139. Google Scholar
S.A. Wolf, Asymptotic and Numerical Analysis of Linear and Nonlinear Eigenvalue Problems, Ph.D. Thesis. Stanford University (1993).
Wolf, S.A. and Keller, J.B., Range of the First Two Eigenvalues of the Laplacian. Proc. R. Soc. London A 447 (1994) 397–412. Google Scholar