Turbulent boundary layer equationsÉquations de la couche limite turbulente

We study a boundary layer problem for the Navier–Stokes-alpha model obtaining a generalization of the Prandtl equations which we conjecture to represent the averaged flow in a turbulent boundary layer. We study the equations for the semi-infinite plate, both theoretically and numerically. Solutions agree with some experimental data in a part of the turbulent boundary layer. To cite this article: A. Cheskidov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 423–427.     

Groupes de Neretin et propriété (T) de KazhdanNeretin's groups and Kazhdan's property (T)

We consider Neretin's group of boundary extensions of piecewise isometries of a homogeneous simplicial tree. We prove that if a subgroup Γ of this group has Kazhdan's property (T), then a finite index subgroup of Γ stabilizes a finite collection of balls of the boundary of the tree, acting isometrically on each of them. To cite this article: A. Navas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 789–792.     

On the ground state energy for a magnetic Schrödinger operator and the effect of the de Gennes boundary condition

Motivated by the Ginzburg–Landau theory of superconductivity, we estimate the ground state energy of a magnetic Schrödinger operator with de Gennes boundary condition in the semi-classical limit and we study the localization of the corresponding ground states. We exhibit cases when the de Gennes boundary condition has a strong effect on this localization. To cite this article: A. Kachmar, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Homogénéisation de frontière en élasticité linéaire

We describe the asymptotic behaviour of the solution of a linear elastic problem posed in a domain of R³, with homogeneous Dirichlet boundary conditions imposed on small zones of size less than ɛ distributed on some part of the boundary of this domain, when the parameter ɛ tends to 0. We use epi-convergence arguments in order to establish this asymptotic behaviour.     

Une conduite cylindrique n'est pas optimale pour minimiser l'énergie dissipée par un fluide

We consider an incompressible fluid in a three-dimensional cylindrical pipe, following the Navier–Stokes system with classical boundary conditions on the boundary of the cylinder. We are interested in the following question: is the cylinder the optimal shape for the criterion “energy dissipated by the fluid”? We prove that it is not the case. For that purpose, we explicit the first order optimality condition, thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system. To cite this article: A. Henrot, Y. Privat, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Stabilisation d'ondes électromagnétiques dans un domaine extérieur 2D

We consider the stabilization of Maxwell equations in two dimensional on the exterior of an arbitrary bounded obstacle. Using some results concerning of wave equations with Neumann boundary condition in the exterior domain. We deduce under the ‘exterior geometric control condition’ the behavior of the solution for large time. To cite this article: A. Moulahi, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Inégalité de Kato et inégalité de Kato jusqu'au bord

Haïm Brezis and Augusto Ponce introduced and studied in their works several extensions of Kato's inequality, in particular Kato's inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a function. Using Potential theoretic methods we answer here some questions raised in Brezis and Ponce (2008). To cite this article: A. Ancona, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron

As representatives of a larger class of elliptic boundary value problems of mathematical physics, we study the Dirichlet problem for the Laplace operator and the electric boundary problem for the Maxwell operator. We state regularity results in two families of weighted Sobolev spaces: A classical isotropic family, and a new anisotropic family, where the hypoellipticity along an edge of a polyhedral domain is taken into account. To cite this article: A. Buffa et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Obstacle problem for Arithmetic Asian options

We prove existence, regularity and a Feynman–Ka? representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average. To cite this article: L. Monti, A. Pascucci, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Transparent boundary conditions for a class of boundary value problems in some ramified domains with a fractal boundary

We consider some boundary value problems in self-similar ramified domains, with Laplace and Helmholtz equations. We discuss transparent boundary conditions. These conditions permit computing the restriction of the solutions to domains obtained by stopping the geometric construction after a finite number of steps. To cite this article: Y. Achdou et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Conditions d'optimalité du second ordre nécessaires ou suffisantes pour les problèmes de commande optimale avec une contrainte sur l'état et une commande scalaires

This Note deals with optimal control problems with only one control variable and one state constraint, of arbitrary order. We consider the case of finitely many boundary arcs and touch times. We obtain a no-gap theory of second-order conditions, allowing us to characterize second-order quadratic growth. To cite this article: J.F. Bonnans, A. Hermant, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Solutions concentrating at curves for some singularly perturbed elliptic problems

We study positive solutions of the equation ?ε²Δu+u=u^p, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain $\text{Ω}$. We prove the existence of solutions concentrating along some closed curve on $\text{?Ω}$. To cite this article: A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary

This note deals with some boundary value problems in a self-similar ramified domain of ${\mathbb{R}}^2$, with a fractal boundary. The partial differential equation is Laplace's equation, and there are nonhomogeneous generalized Neumann boundary conditions on the fractal boundary. We propose a multiscale strategy for approximating the restriction of the solutions to simple subdomains. This strategy is based on transparent boundary conditions and on a wavelet expansion of the Neumann datum. To cite this article: Y. Achdou, N. Tchou, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L:?=?P ^???T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional adjoint operator P ^??? of P is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators P and B. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.     

The modified KdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann–Hilbert problem in the complex k-plane. This Riemann–Hilbert problem has explicit (x,t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition q(x,0)=q₀(x), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Numerical analysis of the electric field formulation of an eddy current problem

In this paper we analyze a finite element method for the numerical solution of an eddy current problem in a bounded conducting domain. We use a weak formulation in terms of the electric field and impose non-local non-standard boundary conditions. The unique data are the input current intensities which are imposed by means of some special curves lying on the boundary of the domain. Optimal error estimates are shown and implementation issues are discussed. To cite this article: A. Bermúdez et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A three field stabilized finite element method for the Stokes equationsUne méthode d'éléments finis stabilisée à trois champs pour les équations de Stokes

We consider in this work the boundary value problem for Stokes equations on a two dimensional domain in cases where non-standard boundary conditions are given. We study the cases where pressure and normal or tangential components of the velocity are given in different parts of the boundary and solve the problem with a minimal regularity. We introduce the problem and its variational formulation which is a mixed one. The principal unknowns are the pressure and the vorticity, the multiplier is the velocity. We present the numerical discretization which needs some stabilization. We prove the convergence and the behavior of the a priori error estimates. Some numerical tests are also presented. To cite this article: M. Amara et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 603–608.     

Analysis of curvature influence on effective boundary conditionsAnalyse de l'influence de la courbure sur les conditions aux limites équivalentes

The aim of this work is the construction of effective boundary conditions (wall laws) for elliptic problems defined in domains with curved, rough boundaries with periodic wrinkles. We present error estimates for first and second order approximations, and a numerical test. To cite this article: A. Madureira, F. Valentin, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 499–504.     

Missing boundary data reconstruction by the factorization method

We consider the data completion problem for the Laplace equation in a cylindrical domain. The Neumann and Dirichlet boundary conditions are given on one face of the cylinder while there is no condition on the other face. This Cauchy problem has been known since Hadamard (1953) to be ill-posed. Here it is set as an optimal control problem with a regularized cost function. We use the factorization method for elliptic boundary value problems. For each set of Cauchy data, to obtain the estimate of the missing data one has to solve a parabolic Cauchy problem in the cylinder and a linear equation. The operator appearing in these problems satisfy a Riccati equation that does not depend on the data. To cite this article: A. Ben Abda et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Odd and even hamming spheres also have minimum boundary

Combinatorial problems with a geometric flavor arise if the set of all binary sequences of a fixed length n, is provided with the Hamming distance. The Hamming distance of any two binary sequences is the number of positions in which they differ. The (outer) boundary of a set A of binary sequences is the set of all sequences outside A that are at distance 1 from some sequence in A. Harper [6] proved that among all the sets of a prescribed volume, the ‘sphere’ has minimum boundary.We show that among all the sets in which no pair of sequences have distance 1, the set of all the sequences with an even (odd) number of 1's in a Hamming ‘sphere’ has the same minimizing property. Some related results are obtained. Sets with more general extremal properties of this kind yield good error-correcting codes for multi-terminal channels.