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. Applied Mathematics and Optimization Springer Journals

Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues

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Springer US
Copyright © 2014 by Springer Science+Business Media New York
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
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