Partitions of Minimal Length on Manifolds

We study partitions on three dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a $\Gamma$-convergence result and we show some numerical results. We compare our results to those already present in the literature in the case of the sphere. For general surfaces we provide an optimization algorithm on meshes which can give a good approximation of the optimal cost, starting from the results obtained using the relaxed formulation

A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results

We consider the multiphase shape optimization problem $$\min\Big\{\sum_{i=1}^h\lambda_1(\Omega_i)+\alpha|\Omega_i|:\ \Omega_i\ \hbox{open},\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset\Big\},$$ where $\alpha>0$ is a given constant and $D\subset\Bbb{R}^2$ is a bounded open set with Lipschitz boundary. We give some new results concerning the qualitative properties of the optimal sets and the regularity of the corresponding eigenfunctions. We also provide numerical results for the optimal partitions

On the Blaschke-Lebesgue theorem for the Cheeger constant via areas and perimeters of inner parallel sets

The first main result presented in the paper shows that the perimeters of inner parallel sets of planar shapes having a given constant width are minimal for the Reuleaux triangles. This implies that the areas of inner parallel sets and, consequently, the inverse of the Cheeger constant are also minimal for the Reuleaux triangles. Proofs use elementary geometry arguments and are based on direct comparisons between general constant width shapes and the Reuleaux triangle

Volume computation for Meissner polyhedra and applications

The volume of a Meissner polyhedron is computed in terms of the lengths of its dual edges. This allows to reformulate the Meissner conjecture regarding constant width bodies with minimal volume as a series of explicit finite dimensional problems. A direct consequence is the minimality of the volume of Meissner tetrahedras among Meissner pyramids

Optimization of Neumann Eigenvalues under convexity and geometric constraints

In this paper we study optimization problems for Neumann eigenvalues $\mu_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study the existence of an optimal domain in all considered cases. We also consider the case of the unit disk, giving values of the index $k$ for which it can be or cannot be extremal. We give some numerical examples for small values of $k$ that lead us to state some conjectures.Comment: 23 pages, 19 figure

Polygonal Faber-Krahn inequality: Local minimality via validated computing

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the vertices in $\Bbb R^{2n}$. Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for $n \in \{5,6\}$. All computations, including the finite element computations, are realized using interval arithmetic

Semaine d'Etude Mathématiques et Entreprises 6 : Représentation des fonctions de réponse radiométrique

Ce rapport rassemble les différentes pistes de réponses apportées au problème posé par l'entreprise Kolor lors de la 6ème Semaine d'Étude Maths-Entreprises de juin 2013. Le problème porte sur la représentation de fonctions de réponse radiométrique utilisées dans de nombreux logiciels de manipulation de photographies. Une grande partie du travail effectué a consisté à comprendre le problème et ses enjeux afin de le formaliser de façon mathématique. Après une description formelle des outils envisagés, nous les évaluons par rapport aux contraintes du problème afin de déterminer leurs avantages et inconvénients. D'un point de vue pratique, les outils sont présentés dans l'objectif d'être éventuellement développés et intégrés à un logiciel existant. Nous avons donc tenté, dans la mesure du possible, de prendre en compte la simplicité de ces outils que ce soit du côté développement ou du côté utilisation. Ce rapport s'articule en six parties. Après une description pratique du problème, nous en détaillons les principales caractéristiques. Dans une troisième partie, nous décrivons les trois outils envisagés. Les deux parties suivantes constituent l'étude des outils par rapport aux contraintes du problème. Finalement nous donnons une conclusion de cette étude

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