# 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

, Volume 69 (2) – Apr 1, 2014
33 pages

/lp/springer_journal/existence-and-regularity-of-minimizers-for-some-spectral-functionals-YWuyQ28h8M
Publisher
Springer Journals
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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