# Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues

Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues We study the shape optimization problem where the objective function is a convex combination of three sequential Laplace-Dirichlet eigenvalues. That is, for α ≥0, β ≥0, and α + β ≤1, we consider $\inf\{ \alpha\lambda_{k}(\varOmega)+\beta\lambda _{k+1}(\varOmega)+(1-\alpha-\beta) \lambda_{k+2}(\varOmega)\colon\varOmega\mbox { open set in } \mathbb{R}^{2} \mbox{ and } |\varOmega|\leq1\}$ . Here λ k ( Ω ) denotes the k -th Laplace-Dirichlet eigenvalue and |⋅| denotes the Lebesgue measure. For k =1,2, the minimal values and minimizers are computed explicitly when the set of admissible domains is restricted to the disjoint union of balls. For star-shaped domains, we show that for k =1 and α +2 β ≤1, the ball is a local minimum. For k =1,2, several properties of minimizers are studied computationally, including uniqueness, connectivity, symmetry, and eigenvalue multiplicity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues

, Volume 69 (1) – Feb 1, 2014
17 pages

/lp/springer_journal/minimal-convex-combinations-of-three-sequential-laplace-dirichlet-tfJLjiRr3l
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9219-z
Publisher site
See Article on Publisher Site

### Abstract

We study the shape optimization problem where the objective function is a convex combination of three sequential Laplace-Dirichlet eigenvalues. That is, for α ≥0, β ≥0, and α + β ≤1, we consider $\inf\{ \alpha\lambda_{k}(\varOmega)+\beta\lambda _{k+1}(\varOmega)+(1-\alpha-\beta) \lambda_{k+2}(\varOmega)\colon\varOmega\mbox { open set in } \mathbb{R}^{2} \mbox{ and } |\varOmega|\leq1\}$ . Here λ k ( Ω ) denotes the k -th Laplace-Dirichlet eigenvalue and |⋅| denotes the Lebesgue measure. For k =1,2, the minimal values and minimizers are computed explicitly when the set of admissible domains is restricted to the disjoint union of balls. For star-shaped domains, we show that for k =1 and α +2 β ≤1, the ball is a local minimum. For k =1,2, several properties of minimizers are studied computationally, including uniqueness, connectivity, symmetry, and eigenvalue multiplicity.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2014

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