Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

Download PDF
17 pages

Loading next page...
 
/lp/springer-journals/minimal-convex-combinations-of-three-sequential-laplace-dirichlet-tfJLjiRr3l

References (16)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Science+Business Media New York
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
DOI
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 Optimization – Springer Journals

Published: Feb 1, 2014

There are no references for this article.