Published Paper
Inserted: 5 mar 2013
Last Updated: 30 oct 2017
Journal: Appl. Math. Optim.
Volume: 69
Number: 2
Pages: 199--231
Year: 2014
Abstract:
In this paper we prove that the shape optimisation problem'
\[
\min\Big\{\lambda_k(\Omega): \Omega\subset\mathbb R^d \Omega\ \hbox{open},\
P(\Omega)=1,\
\Omega
<+\infty\Big\},
\]
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 $C^{1,\alpha}$ outside a closed set of Hausdorff dimension
$d-8$. Our results are more general and apply to spectral functionals of the
form $f(\lambda_{k_1}(\Omega),\dots,\lambda_{k_p}(\Omega))$, for increasing
functions $f$ satisfying some suitable bi-Lipschitz type condition.
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