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Spectral partitions for Sturm–Liouville problems

Part of: Variational methods for eigenvalues of operators Ordinary differential operators Boundary value problems

Published online by Cambridge University Press:  22 May 2019

Paolo Tilli  and
Davide Zucco
Paolo Tilli
Affiliation:
Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy (paolo.tilli@polito.it; davide.zucco@polito.it)
Davide Zucco
Affiliation:
Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy (paolo.tilli@polito.it; davide.zucco@polito.it)
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Abstract

We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm–Liouville problems. Via Γ-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm–Liouville operator.

Keywords

Sturm–Liouville eigenvalueΓ-convergenceoptimizationoptimal partitions

MSC classification

Primary: 34B24: Sturm-Liouville theory
Secondary: 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions 49R05: Variational methods for eigenvalues of operators (should also be assigned at least one other classification number in Section 49)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 2155 - 2173
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Atkinson, F. V. and Mingarelli, A. B.. Multiparameter eigenvalue problems. Sturm–Liouville theory (Boca Raton: CRC Press, 2011).Google Scholar
2Bouchitté, G., Jimenez, C. and Mahadevan, R.. Asymptotic analysis of a class of optimal location problems. J. Math. Pures Appl. 95 (2011), 382–419.CrossRefGoogle Scholar
3Brascamp, H. J. and Lieb, E. H.. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 (1976), 366–389.CrossRefGoogle Scholar
4Buttazzo, G., Santambrogio, F. and Varchon, N.. Asymptotics of an optimal compliance-location problem. ESAIM Control Optim. Calc. Var. 12 (2006), 752–769.CrossRefGoogle Scholar
5Caffarelli, L. A. and Lin, F. H.. An optimal partition problem for eigenvalues. J. Sci. Comput. 31 (2007), 5–18.CrossRefGoogle Scholar
6Conti, M., Terracini, S. and Verzini, G.. An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198 (2003), 160–196.CrossRefGoogle Scholar
7Courant, R. and Hilbert, D.. Methods of mathematical physics, vol. I (New York: Interscience Publishers, 1953).Google Scholar
8Dal Maso, G.. An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications (Boston: Birkhäuser, 1993).CrossRefGoogle Scholar
9El Soufi, A. and Harrell, E. M.. On the placement of an obstacle so as to optimize the Dirichlet heat trace. SIAM J. Math. Anal. 48 (2016), 884–894.CrossRefGoogle Scholar
10Evans, L. C. and Gariepy, R. F.. Measure theory and fine properties of functions. Stud. Adv. Math. (Boca Raton: CRC Press, 1992).Google Scholar
11Helffer, B., Hoffmann-Ostenhof, T. and Terracini, S.. Nodal domains and spectral minimal partitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 101–138.CrossRefGoogle Scholar
12Henrot, A. and Zucco, D.. Optimization of the first Dirichlet eigenvalue of the Laplacian with an obstacle. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (doi:10.2422/2036-2145.201702_003).Google Scholar
13Lucardesi, I., Morandotti, M., Scala, R. and Zucco, D.. Upscaling of screw dislocations with increasing tangential strain. Submitted, arXiv:1808.08898.Google Scholar
14Rudin, W.. Real and complex analysis (New York: McGraw-Hill Book Co, 1987).Google Scholar
15Suzuki, A. and Drezner, Z.. The p-center location. Location Sci. 4 (1996), 69–82.CrossRefGoogle Scholar
16Suzuki, A. and Okabe, A.. Using Voronoi diagrams. In Facility location: a survey of applications and methods (Boston, MA: Springer, 1995).CrossRefGoogle Scholar
17Tilli, P. and Zucco, D.. Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length. SIAM J. Math. Anal. 45 (2013), 3266–3282.CrossRefGoogle Scholar
18Tilli, P. and Zucco, D.. Where best to place a Dirichlet condition in an anisotropic membrane? SIAM J. Math. Anal. 47 (2015), 2699–2721.CrossRefGoogle Scholar
19Zettl, A.. Sturm–Liouville Theory. Mathematical Surveys and Monographs (Providence: American Mathematical Society, 2005).Google Scholar