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Placement of an Obstacle for Optimizing the Fundamental Eigenvalue of Divergence Form Elliptic Operators

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Variational Views in Mechanics

Part of the book series: Advances in Mechanics and Mathematics ((ACM,volume 46))

Abstract

In this chapter, we consider an obstacle placement problem inside a disk, represented by a shape optimization problem to find the maximum or the minimum fundamental eigenvalue of a general divergence form elliptic operator. The obstacle is invariant under the action of a dihedral group, which is usually the case with regular polygons and ellipses. This is a generalization of a previous work by the same authors concerning the fundamental Dirichlet eigenvalue optimization. We show that when the order of symmetry of the obstacle is even, the extremal configurations for the fundamental eigenvalue, with respect to rotations of the obstacle, correspond to the cases where an axis of symmetry of the obstacle coincides with an axis of symmetry of the disk. For the case of odd order of symmetry, we provide conjectures about the extremal configurations. Furthermore, for both even and odd symmetry cases, we characterize the global extremal configurations with respect to rotations and translations of the obstacle. Finally, we provide results of several numerical experiments for obstacles with different orders of symmetries and two different types of elliptic operators, which validates our theoretical findings.

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Authors and Affiliations

  1. Indian Institute of Science Education and Research Pune, Pune, India

    Anisa M. H. Chorwadwala

  2. University of Texas at Arlington, Arlington, TX, USA

    Souvik Roy

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  1. Anisa M. H. Chorwadwala
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  2. Souvik Roy
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Correspondence to Souvik Roy .

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Editors and Affiliations

  1. DICEA, University of Florence, Florence, Italy

    Paolo Maria Mariano

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Chorwadwala, A.M.H., Roy, S. (2021). Placement of an Obstacle for Optimizing the Fundamental Eigenvalue of Divergence Form Elliptic Operators. In: Mariano, P.M. (eds) Variational Views in Mechanics. Advances in Mechanics and Mathematics(), vol 46. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-90051-9_6

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