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

Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

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Publisher
Springer US
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
D.O.I.
10.1007/s00245-013-9222-4
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2014

References

  • Existence and regularity for a minimum problem with free boundary
    Alt, H.W.; Caffarelli, L.A.
  • Lipschitz continuity of state functions in some optimal shaping
    Briançon, T.; Hayouni, M.; Pierre, M.

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