### Abstract

We study variational problems of the form inf{lk(W): W open in \mathbbRm, T(W) £ 1 },infk(): open in Rm T()1 where λ k (Ω) is the k-th eigenvalue of the Dirichlet Laplacian acting in L 2(Ω), and where T is a non-negative set function defined on the open sets in ℝ m , which is invariant under isometries, additive on disjoint families of open sets, and is such that the ball with T(B)=1 is a minimizer for k=1. Upper bounds are obtained for the number of components of any bounded minimizer if T satisfies a scaling relation. For example, we show that if T is Lebesgue measure and if k≤m+1 then any bounded minimizer has at most 7 components. We also consider variational problems over open sets Ω in ℝ m involving the (m−1)-dimensional Hausdorff measure of ∂Ω.

Translated title of the contribution | On the minimization of Dirichlet eigenvalues |
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Original language | English |

Pages (from-to) | 660-676 |

Number of pages | 17 |

Journal | Journal of Geometric Analysis |

Volume | 23 |

DOIs | |

Publication status | Published - Apr 2013 |

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## Cite this

van den Berg, M., & Iversen, M. (2013). On the minimization of Dirichlet eigenvalues of the Laplace operator.

*Journal of Geometric Analysis*,*23*, 660-676. https://doi.org/10.1007/s12220-011-9258-0