A variational method for second order shape derivatives

We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functional for p grater than or equal to 2.Comment: Submitted paper. 29 page

Body of constant width with minimal area in a given annulus

In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in \cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons.Comment: to appear in Journal Ecole Polytechniqu

About the Blaschke-Santalo diagram of area, perimeter and moment of inertia

We study the Blaschke-Santal\'o diagram associated to the area, the perimeter, and the moment of inertia. We work in dimension 2, under two assumptions on the shapes: convexity and the presence of two orthogonal axis of symmetry. We discuss topological and geometrical properties of the diagram. As a by-product we address a conjecture by P\'olya, in the simplified setting of double symmetry

Crack growth by vanishing viscosity in planar elasticity

We show the existence of quasistatic evolutions in a fracture model for brittle materials by a vanishing viscosity approach, in the setting of planar linearized elasticity. The crack is not prescribed a priori and is selected in a class of (unions of) regular curves. To prove the result, it is crucial to analyze the properties of the energy release rate

Optimisation de la compliance de structures élastiques minces

The main topic of the Thesis is the optimization of the compliance of thin elastic structures.The problem consists in finding the most robust configurations, when an infinitesimal amount of elastic material is subjected to a fixed force, and contained within a region having infinitesimal volume.The resistance to a load can be measured by computing a shape functional, the compliance, in which the shape represents the volume occupied by the elastic material. Thus we are led to study a minimization problem of a shape functional, under suitable constraints.In particular, we treat the case in which the design region is a thin rod, represented by a cylinder with infinitesimal cross section. The study finds its motivation in engineering problems: thin structures are very convenient to be used in practical applications.The approach we adopt draws inspiration from some recent works by I. Fragalà, G. Bouchitté and P. Seppecher, in which the authors deal with the case of thin elastic plates [G. Bouchitté, I. Fragalà, P. Seppecher: Structural optimization of thin plates: the three dimensional approach., Arch. Rat. Mech. Anal. (2011)]. We point out that these two problems are not merely technical variants one of the other, due to the substantial difference between the limit passages 3d-1d and 3d-2d, namely from 3 to 1 and from 3 to 2 dimensions.The study of optimal configurations led us to face another interesting variational problem: actually to establish whether homogenization phenomena occur in bars in pure torsion regime turns out to be equivalent to solve a nonstandard free boundary problem in the plane. This new problem is very challenging and, besides the link with torsion rods, it has mathematical interest in itself. One of the tools which can be employed to attack the problem is shape derivative for minima of integral functionals. The theory of shape derivatives is a widely studied topic (see e.g. the monograph by A. Henrot and M.Pierre Variations et Optimisation de Formes. Une Analyse Géométrique, Springer Berlin (2005), and the references therein), but the approach we propose in new and relies on assumptions which are weaker that the classical ones.Le sujet principal de la Thèse est l’optimisation de la compliance des structures élastiques minces. Le problème consiste en déterminer la configuration la plus résistante, lorsqu'une quantité infinitésimale de matériau élastique est soumis à une force fixée et est confinée dans une région de volume infinitésimal.La résistance au chargement peut être mesurée en calculant une fonctionnelle de forme, la compliance, dans laquelle la forme représente le volume occupé par le matériau élastique. Donc nous sommes conduits à étudier un problème de minimisation d'une fonctionnelle de forme, sous une contrainte appropriée.Nous nous intéresserons plus particulièrement au cas où la région de design est un fil fin, représenté par un cylindre de section transverse infinitésimale. L'étude est motivée par des problèmes d'ingénierie: les structures minces sont très intéressantes d'un point de vue pratique.La stratégie utilisée tire son inspiration des travaux récents par I. Fragalà, G. Bouchitté et P. Seppecher, dans lesquels les auteurs considèrent des plaques élastiques [G. Bouchitté, I. Fragalà, P. Seppecher: Structural optimization of thin plates: the three dimensional approach., Arch. Rat. Mech. Anal. (2011)]. Cependant il faut souligner que le cas du cylindre est loin de se résumer à une variante technique du cas des plaques. Comme nous le verrons en effet, le modèle limite obtenu dans l'analyse asymptotique 3d-1d est plus riche et subtile que celui correspondant à une analyse asymptotique 3d-2d.L'étude des configurations optimales pour le modèle limite obtenu nous a conduit à une problématique nouvelle: l'existence de vraies forme optimales (donc sans apparition de structures composites) pour une barre en régime de pure torsion est liée à l'existence de solutions pour un problème non standard de frontière libre dans le plan. Ce problème représente un challenge et nous nous contenterons de donner quelques premiers résultats et perspectives.Par ailleurs, en liaison avec ce problème, nous développerons une stratégie nouvelle pour caractériser la dérivée de forme pour le minimum d'une fonction intégrale. La théorie de dérivées de forme est un sujet très largement étudié (voir e.g. la monographie de A. Henrot et M.Pierre Variations et Optimisation de Formes. Une Analyse Géométrique, Springer Berlin (2005), et les références qui y sont contenues), mais les techniques classiques qui y sont utilisées s'appuient sur des hypothèses de régularité non vérifiées dans notre cas

Sensitivity of the Compliance and of the Wasserstein Distance with Respect to a Varying Source

We show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover, we provide an integral representation formula for the derivative as a linear functional of the deformation vector field. The result holds true as well for the p-compliance in the scalar case of conductivity. Then we study the limit problem as p→ + ∞, which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans. When the latter contains only one element, we prove that the derivative of the p-compliance converges to the derivative of the Wasserstein distance in the limit as p→ + ∞