Phase-field approximation for a class of cohesive fracture energies with an activation threshold

We study the $\Gamma$-limit of Ambrosio-Tortorelli-type functionals $D_\varepsilon(u,v)$, whose dependence on the symmetrised gradient $e(u)$ is different in $\mathbb{A} u$ and in $e(u)-\mathbb{A} u$, for a $\mathbb{C}$-elliptic symmetric operator $\mathbb{A}$, in terms of the prefactor depending on the phase-field variable $v$. This is intermediate between an approximation for the Griffith brittle fracture energy and the one for a cohesive energy by Focardi and Iurlano. In particular we prove that $G(S)BD$ functions with bounded $\mathbb{A}$-variation are $(S)BD$

The Stress-Intensity Factor for nonsmooth fractures in antiplane elasticity

Motivated by some questions arising in the study of quasistatic growth in brittle fracture, we investigate the asymptotic behavior of the energy of the solution $u$ of a Neumann problem near a crack in dimension 2. We consider non smooth cracks $K$ that are merely closed and connected. At any point of density 1/2 in $K$, we show that the blow-up limit of $u$ is the usual "cracktip" function $\sqrt{r}\sin(\theta/2)$, with a well-defined coefficient (the "stress intensity factor" or SIF). The method relies on Bonnet's monotonicity formula \cite{b} together with $\Gamma$-convergence techniques.Comment: (version 2 : r\'ef\'erences corrig\'ees

Existence and uniqueness for planar anisotropic and crystalline curvature flow

We prove short-time existence of \phi-regular solutions to the planar anisotropic curvature flow, including the crystalline case, with an additional forcing term possibly unbounded and discontinuous in time, such as for instance a white noise. We also prove uniqueness of such solutions when the anisotropy is smooth and elliptic. The main tools are the use of an implicit variational scheme in order to define the evolution, and the approximation with flows corresponding to regular anisotropies

Guaranteed Minimum Price Contracts for Some, an Insurance for Others?

This paper analyzes the impact of guaranteed minimum price contracts between sub-groups of farmers and a fair trade manufacturer on the spot market price. We focus on the fair trade concept in the coffee supply chain as an example. We analyze a three level vertical chain gathering perfectly competitive farmers upstream who offer their raw product to manufacturers who then sell finished products to a downstream retailer. Without fair trade, all the raw product is sold on the spot market. When a sub-group of farmers benefit from a guaranteed minimum price contract offered by a fair trade certifier, we show that farmers outside of this fair trade agreement may also benefit from a higher spot market price in cases of a limited overproduction.Guaranteed Minimum Price Contracts, Fair Trade, Vertical Chain., Demand and Price Analysis, International Relations/Trade,

Fair Trade: In or Out the Market?

Cet article s'intéresse à l'évolution du concept de commerce équitable. Nous proposons un modèle simple en vue de donner des arguments théoriques dans le débat sur la vente de produits équitables dans la grande distribution. L'hypothèse principale est que certains consommateurs sont prêts à payer un prix plus élevé pour acheter un produit équitable. Nous mettons en évidence que les produits équitables ont plus de chance d'être dans les rayons des supermarchés si le certificateur du label équitable a pour objectif de maximiser les quantités certifiées plutôt que le prix payé aux producteurs. Nous soulignons également que la variable clé dans le choix du revendeur de vendre ou non des produits équitables n'est pas le pourcentage de consommateurs prêts à payer pour un bien équitable, mais combien ces consommateurs sont prêts à payer pour ce type de produit.Différentiation des produits;Discrimination en prix;Revente

Fair Trade Contracts for Some, an Insurance for Others

This article analyzes the impact of Fair Trade contracts between sub-groups of farmers and a Fair Trade organization on the spot market price. We analyze a three level vertical chain gathering perfectly competitive farmers upstream who offer their raw product on a spot market to manufacturers who then sell finished products to a downstream retailer. Absent Fair Trade, the entire raw product is sold on the spot market. When a Fair Trade organization offers a Fair Trade contract to a sub-group of farmers, it gathers a Guaranteed Minimum Price clause and a straight relationship between the sub-group of farmers and the retailer. This article highlights several conditions such that a snowball effect exists, i.e farmers outside of the Fair Trade contract also benefit from a higher spot market price.Guaranteed Minimum Price Contracts, Disintermediation, Fair Trade, Vertical Chain, Two-part Tariff Contracts

A Remark on the Anisotropic Outer Minkowski content

We study an anisotropic version of the outer Minkowski content of a closed set in Rn. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.Comment: We corrected an error in the orignal manuscript, on p. 14 (the boundaries of the regularized sets are not necessarily C^{1,1}