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Spectral partitions for Sturm–Liouville problems

  • Paolo Tilli (a1) and Davide Zucco (a1)

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.

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1Atkinson, F. V. and Mingarelli, A. B.. Multiparameter eigenvalue problems. Sturm–Liouville theory (Boca Raton: CRC Press, 2011).
2Bouchitté, G., Jimenez, C. and Mahadevan, R.. Asymptotic analysis of a class of optimal location problems. J. Math. Pures Appl. 95 (2011), 382419.
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), 366389.
4Buttazzo, G., Santambrogio, F. and Varchon, N.. Asymptotics of an optimal compliance-location problem. ESAIM Control Optim. Calc. Var. 12 (2006), 752769.
5Caffarelli, L. A. and Lin, F. H.. An optimal partition problem for eigenvalues. J. Sci. Comput. 31 (2007), 518.
6Conti, M., Terracini, S. and Verzini, G.. An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198 (2003), 160196.
7Courant, R. and Hilbert, D.. Methods of mathematical physics, vol. I (New York: Interscience Publishers, 1953).
8Dal Maso, G.. An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications (Boston: Birkhäuser, 1993).
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), 884894.
10Evans, L. C. and Gariepy, R. F.. Measure theory and fine properties of functions. Stud. Adv. Math. (Boca Raton: CRC Press, 1992).
11Helffer, B., Hoffmann-Ostenhof, T. and Terracini, S.. Nodal domains and spectral minimal partitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 101138.
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).
13Lucardesi, I., Morandotti, M., Scala, R. and Zucco, D.. Upscaling of screw dislocations with increasing tangential strain. Submitted, arXiv:1808.08898.
14Rudin, W.. Real and complex analysis (New York: McGraw-Hill Book Co, 1987).
15Suzuki, A. and Drezner, Z.. The p-center location. Location Sci. 4 (1996), 6982.
16Suzuki, A. and Okabe, A.. Using Voronoi diagrams. In Facility location: a survey of applications and methods (Boston, MA: Springer, 1995).
17Tilli, P. and Zucco, D.. Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length. SIAM J. Math. Anal. 45 (2013), 32663282.
18Tilli, P. and Zucco, D.. Where best to place a Dirichlet condition in an anisotropic membrane? SIAM J. Math. Anal. 47 (2015), 26992721.
19Zettl, A.. Sturm–Liouville Theory. Mathematical Surveys and Monographs (Providence: American Mathematical Society, 2005).

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