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Documents  Buttazzo, Giuseppe | enregistrements trouvés : 9

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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.

49Q10 ; 52B60 ; 35P15 ; 52A40 ; 52A10 ; 35A15

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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...

49Q10 ; 35R35 ; 47A75 ; 49R05

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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.

49Q10 ; 49K20 ; 35P15 ; 35J05 ; 47J30

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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.

35Q74 ; 49Q10 ; 74K20

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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.

49Q20 ; 53C17 ; 49N60

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Research talks;Control Theory and Optimization;Partial Differential Equations

We consider spectral optimization problems of the form

$\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$

where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation

$\lambda_1(\Omega;D)=\min\lbrace\int_\Omega|\nabla u|^2dx+k\int_{\partial D}u^2d\mathcal{H}^{d-1}:$

$u\in H^1(D),u=0$ on $\partial\Omega\cap D,||u||_{L^2(\Omega)}=1\rbrace$

reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains. The case of Dirichlet condition on a $\textit{fixed}$ part and of Neumann condition on the $\textit{free}$ part of the boundary is also considered
We consider spectral optimization problems of the form

$\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$

where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation

$\lam...

49Q10 ; 49J20 ; 49N45

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- xii; 793 p.
ISBN 978-1-611973-47-1

MOS-SIAM series on optimization

Localisation : Ouvrage RdC (ATTO)

analyse variationnelle # espace de Sobolev # espace BV # équation différentielle # optimisation # relaxation # mesure de Young # optimisation de forme

49-01 ; 49K20 ; 49J45 ; 49J52 ; 49Q10 ; 49Q20 ; 90C48

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- x; 150 p.
ISBN 978-3-540-85798-3

Lecture notes in mathematics , 1961

Localisation : Collection 1er étage

optimisation # équation de transport # réseau # flux en réseau # semi-continuité # convergence # mesure géométrique # traffic autoroutier

49J45 ; 49Q10 ; 49Q15 ; 49Q20 ; 90B06 ; 90B10 ; 90B20

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- 262 p.
ISBN 978-0-19-850465-8

Oxford lecture series in mathematics and its applications , 0015

Localisation : Ouvrage RdC (BUTT)

calcul des variations # contrôl optimal # optimisation # problème aux frontières # problème aux valeurs limites non linéaire # théorie d'existence # équation différentielle ordinaire

34B15 ; 49Jxx

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