Applied Mathematics & Optimization

, Volume 69, Issue 2, pp 199–231 | Cite as

Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint

  • Guido De Philippis
  • Bozhidar VelichkovEmail author


In this paper we prove that the shape optimization problem
$$\min \bigl\{\lambda_k(\varOmega):\ \varOmega\subset \mathbb{R}^d,\ \varOmega\ \hbox{open},\ P(\varOmega)=1,\ |\varOmega|<+\infty \bigr\}, $$
has a solution for any \(k\in \mathbb{N}\) and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d−8. Our results are more general and apply to spectral functionals of the form \(f(\lambda_{k_{1}}(\varOmega),\dots,\lambda_{k_{p}}(\varOmega))\), for increasing functions f satisfying some suitable bi-Lipschitz type condition.


Shape optimization Eigenvalues Free boundary Concentration-compactness 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Hausdorff Center for MathematicsBonnGermany
  2. 2.Scuola Normale Superiore di PisaPisaItaly

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