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# Documents  Henrot, Antoine | enregistrements trouvés : 7

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## Some new inequalities for the Cheeger constant Fragalà, Ilaria | CIRM H

Post-edited

Research talks;Control Theory and Optimization;Partial Differential Equations;Geometry

We discuss some new results for the Cheeger constant in dimension two, including:
- a polygonal version of Faber-Krahn inequality;
- a reverse isoperimetric inequality for convex bodies;
- a Mahler-type inequality in the axisymmetric setting;
- asymptotic behaviour of optimal partition problems.
Based on some recent joint works with D.Bucur,
and for the last part also with B.Velichkov and G.Verzini.

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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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## On the stability of the Bossel-Daners inequality Trombetti, Cristina | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations

The Bossel-Daners is a Faber-Krahn type inequality for the first Laplacian eigenvalue with Robin boundary conditions. We prove a stability result for such inequality.

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## Isoperimetry with density Morgan, Frank | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Geometry

In 2015 Chambers proved the Log-convex Density Conjecture, which says that for a radial density f on $R^n$, spheres about the origin are isoperimetric if and only if log f is convex (the stability condition). We discuss recent progress and open questions for other densities, unequal perimeter and volume densities, and other metrics.

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## A minimaxmax problem for improving the torsional stability of rectangular plates Gazzola, Filippo | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations;Mathematical Physics

We introduce a new function which measures the torsional instability of a partially hinged rectangular plate. By exploiting it, we compare the torsional performances of different plates reinforced with stiffening trusses. This naturally leads to a shape optimization problem which can be set up through a minimaxmax procedure.

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## Variation et optimisation de formes :une analyse géométrique Henrot, Antoine ; Pierre, Michel | Springer 2005

Ouvrage

- 333 p.
ISBN 978-3-540-26211-4

Mathématiques & applications , 0048

Localisation : Collection 1er étage

optimisation de forme # variation de domaine # opérateur de Laplace # capacité classique # dérivation par rapport à une forme # topologie # forme optimale # propriétés géométriques # homogénisation

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## Shape variation and optimization:a geometrical analysis Henrot, Antoine ; Pierre, Michel | European Mathematical Society 2018

Ouvrage

- xi; 365 p.
ISBN 978-3-03719-178-1

Tracts in mathematics , 0028

Localisation : Ouvrage RdC (HENR)

optimisation des formes # design optimum # calcul des variations # variation des domaines # convergence de Hausdorff # $\Gamma$-convergence # dérivée de forme # géométrie des formes optimales # problème de Laplace-Dirichlet # problème de Neumann # problème surdéterminé # inégalité isopérimétrique # capacité # théorie du potentiel # théorie spectrale # homogénéisation

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