On the two-dimensional rotational body of maximal Newtonian resistance

We investigate, by means of computer simulations, shapes of nonconvex bodies that maximize resistance to their motion through a rarefied medium, considering that bodies are moving forward and at the same time slowly rotating. A two-dimensional geometric shape that confers to the body a resistance very close to the theoretical supremum value is obtained, improving previous results.Comment: This is a preprint version of the paper published in J. Math. Sci. (N. Y.), Vol. 161, no. 6, 2009, 811--819. DOI:10.1007/s10958-009-9602-

Phase field approach to optimal packing problems and related Cheeger clusters

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions

An example of non-convex minimization and an application to Newton's problem of the body of least resistance

We study the minima of the functional $int_Omega f(nabla u)$. The function $f$ is not convex, the set $Omega$ is a domain in $R^2$ and the minimum is sought over all convex functions on $Omega$ with values in a given bounded interval. We prove that a minimum $u$ is almost everywhere `on the boundary of convexity', in the sense that there exists no open set on which $u$ is strictly convex. In particular, wherever the Gaussian curvature is {finite, it is zero. An important application of this result is the problem of the body of least resistance as formulated by Newton (where f(p) = 1/(1+abs{p^2) and $Omega$ is a ball), implying that the minimizer is not radially symmetric. This generalizes a result in~cite{bro

Minimisation de fonctionnelles dans un ensemble de fonctions convexes [Minimizing functionals on a set of convex functions]

We investigate the minima of functionals of the form ¿gWƒ(u), where O 2 is a bounded domain and ƒ a smooth function. The admissible functions are convex and satisfy on O, where and are fixed functions on O. An important example is the problem of the body of least resistance formulated by Newton (see [2]). If ƒ is convex or concave, we show that the minimum is attained by either or if these functions are equal on ¿O. In the case where ƒ is nonconvex, we prove that any minimizer u has a special structure in the region where it is different from and : in any open set where u is differentiable, u is not strictly convex. Convex functions with this property are ‘rare’ in the sense of Baire (see [8]). A consequence of this result is that the radial minimizer calculated by Newton does not attain the global minimum for this problem

The Minimum of Quadratic Functionals of the Gradient on the Set of Convex Functions

We study the infimum of functionals of the form R\Omega Mru \Delta ru among all convex functions u 2 H 1 0(Ω) such that R \Omega jruj 2 = 1. (\Omega is a convex open subset of R N , and M is a given symmetric N \Theta N matrix.) We prove that this infimum is the smallest eigenvalue of M if Ω is C¹. Otherwise the picture is more complicated. We also study the case of an x-dependent matrix M