Best constant in Sobolev trace inequalities on the half-space

Using a mass transportation method, we study optimal Sobolev trace inequalities on the half-space and prove a conjecture made by Escobar in 1988 about the minimizers.     

A new class of transport distances between measures

We introduce a new class of distances between nonnegative Radon measures in . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given. J. Dolbeault and B. Nazaret have been partially supported by the ANR project IFO. The second author has also been partially supported by the ANR project OTARIE. G. Savaré has been partially supported by grants of M.I.U.R., PRIN ’06. Part of this research was carried out while the third author was visiting professor at Ceremade, Université Paris-Dauphine, whose hospitality and support are also gratefully acknowledged.     

Geodesics for a class of distances in the space of probability measures

In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savaré. We first prove the existence of a potential function and then give necessary and sufficient optimality conditions that take the form of a coupled system of PDEs somehow similar to the Mean-Field-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves.     

Use of 3-D Digital Image Correlation to Characterize the Mechanical Behavior of a Fiber Reinforced Refractory Castable

Refractory castables exhibit very low fracture strain levels when subjected to tension or bending. The main objective of this work is to show that 3-D digital image correlation (3-D DIC) allows such low strain levels to be measured. Compared to mechanical extensometer measurements, 3-D DIC makes it possible to reach similar strain resolution levels and to avoid the problem of position dependance related to the heterogeneous nature of the strain and to strain localization phenomena. First, the 3-D DIC method and the experimental set-up are presented. Secondly, an analysis of the 3-D DIC method is performed in order to evaluate the resolution, the standard uncertainty and the spatial resolution for both displacement and strain measurements. An optimized compromise between strain spatial resolution and standard uncertainty is reached for the configuration of the experimental bending test. Finally, the macroscopic mechanical behavior of a fiber reinforced refractory castable (FRRC) is studied using mechanical extensometry and 3-D DIC in the case of tensile and four-point bending tests. It is shown that similar results are obtained with both methods. Furthermore, in the case of bending tests on damaged castable, 3-D DIC results demonstrate the ability to determine Young’s modulus from heterogeneous strain fields better than by using classical beam deflection measurements.     

Synthesis of pseudopeptidic [(5H)-6-oxodibenzo[b,f][1,5]diazocine-5-yl]arylglycinamides by an Ugi 4CC/Staudinger/aza-Wittig sequence

Sequential Ugi reaction between p-substituted benzaldehydes, 2-acetyl or 2-benzoylanilines, cyclohexyl or benzyl isocyanides, and 2-azidobenzoic acid, followed by a Staudinger/aza-Wittig cyclization in the presence of triphenylphosphine afforded pseudopeptidic [(5H)-6-oxodibenzo[b,f][1,5]diazocine-5-yl]arylglycinamides.     

A system of boundary integral equations for polygonal plates with free edges

We consider the problem of a polygonal plate with free edges. It is a boundary value problem for the biharmonic operator on a polygon with Neumann boundary conditions. Its resolution is studied via boundary integral equations. A variational formulation of the boundary problem obtained by a double-layer potential is given. Finally, we implement the method and give numerical results. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

The Embedding W n,1 into C 0 on Compact Riemannian Manifolds

We study the best constant in the inequality corresponding to the Sobolev embedding W ^n,1(R ⁿ ) into the space of bounded continuous functions C ₀(R ⁿ ). Then, we adapt this inequality on compact Riemannian manifolds and discuss on its optimality.