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# Documents  Terracini, Susanna | enregistrements trouvés : 1

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## Regularity of the optimal sets for spectral functionals. Part I: sum of eigenvalues Terracini, Susanna | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations

In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
$\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},$
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure.
We prove that any minimizer $_{opt}$ has a regular part of the topological boundary which is relatively open and
$C^{\infty}$ and that the singular part has Hausdorff dimension smaller than $d-d^*$, where $d^*\geq 5$ is the minimal
dimension allowing the existence of minimal conic solutions to the blow-up problem.

We mainly use techniques from the theory of free boundary problems, which have to be properly extended to the case of
vector-valued functions: nondegeneracy property, Weiss-like monotonicity formulas with area term; finally through the
properties of non tangentially accessible domains we shall be in a position to exploit the ''viscosity'' approach recently proposed by De Silva.

This is a joint work with Dario Mazzoleni and Bozhidar Velichkov.
In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
$\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},$
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue m...

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##### Audience

Ressources Electroniques (Depuis le CIRM)

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