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Regularity of optimal transport maps on multiple products of spheres

Alessio Figalli, Young-Heon Kim, Robert McCann (2013)

Journal of the European Mathematical Society

This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps....

Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type ☆ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type ☆ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step...

Relating phase field and sharp interface approaches to structural topology optimization

Luise Blank, Harald Garcke, M. Hassan Farshbaf-Shaker, Vanessa Styles (2014)

ESAIM: Control, Optimisation and Calculus of Variations

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for...

Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

Heikki Hakkarainen, Juha Kinnunen, Panu Lahti, Pekka Lehtelä (2016)

Analysis and Geometry in Metric Spaces

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean...

Relaxation of an optimal design problem in fracture mechanic: the anti-plane case

Arnaud Münch, Pablo Pedregal (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In the framework of the linear fracture theory, a commonly-used tool to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called energy release rate defined as the variation of the mechanical energy with respect to the crack dimension. Precisely, the well-known Griffith's criterion postulates the evolution of the crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we consider the shape design problem which consists in optimizing...

Removing holes in topological shape optimization

Maatoug Hassine, Philippe Guillaume (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The gradient based topological optimization tools introduced during the last ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remain unchanged, at least during the topological phase of the optimization algorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not for improving the cost function. Then, two numerical examples...

Removing holes in topological shape optimization

Philippe Guillaume, Maatoug Hassine (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The gradient based topological optimization tools introduced during the last ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remain unchanged, at least during the topological phase of the optimization algorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not for improving the cost function. Then, two numerical...

Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion

Bernard Bonnard, Olivier Cots, Jean-Baptiste Pomet, Nataliya Shcherbakova (2014)

ESAIM: Control, Optimisation and Calculus of Variations

The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S2 associated to the...

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