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Spectral partitions for Sturm–Liouville problems

Part of: Ordinary differential operators Variational methods for eigenvalues of 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)
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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