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).     

Estimation de la constante de Bernoulli dans le problème intérieur à frontière libre pour le p-laplacien

In this Note, considering the p-Laplacian operator, we first establish an existence and regularity result for an optimisation problem of form. From a monotony result we show the existence of a solution to the interior problem with a free surface for a family of Bernoulli constants; we also give an optimal estimation for the upper bound for the Bernoulli constant. To cite this article: I. Ly, D. Seck, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Nouveau noyau de Green associé au problème de Poisson–Dirichlet sur un rectangle

This work is focused on the study of a ‘discretization’ method for the Laplacian operator, in the two-dimensional Poisson problem on a rectangle, with Dirichlet boundary conditions. The Laplacian operator is approximated by a block Toeplitz matrix, the blocks of which are Toeplitz matrices again, and a formula of the inverse matrix blocks is given. Then an asymptotic development of the inverse matrix trace and the Toeplitz matrix determinant are obtained. Finally, the continuum expression of the Laplacian operator is found by calculating the ergodic limit of the inverse matrix. A new asymptotic formula for the well known Green function for the Poisson problem that we obtain converges more rapidly than the usual one. To cite this article: J. Chanzy, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

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.     

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).     

Solutions variationnelles du problème stationnaire des fluides de grade troisVariational solutions of stationary problem of third grade fluids

We show the existence of solutions and the uniqueness of the stationary problem of third grade fluids, for a boundary of class C^2,1 and small data, by a method of energy estimates. The resolution of the problem of third grade fluids in three dimensions requires a condition on the parameters occuring in the equation, which allows us to show that the differential operator of the equation verifies properties of ellipticity. To cite this article: J.-M. Bernard, E.H. Ouazar, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 609–613.     

A generation theorem for kinetic equations with non-contractive boundary operatorsUn résultat de génération pour des équations cinétiques soumises à des conditions frontières non contractives

In this Note, we present some c₀-semigroup generation results in L_p-spaces for the advection operator submitted to non-contractive boundary conditions covering in particular the classical Maxwell-type boundary conditions. To cite this article: B. Lods, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 655–660.     

Diffraction d'une onde électromagnétique par un dièdre infini absorbant

We study the diffraction of electromagnetic waves by an infinite wedge of dielectric material. For this aim we consider the stationary coupled vacuum-dielectric Maxwell equations with an outgoing condition in the vacuum. We show the equivalence of the latter problem to a Caldéron boundary operator system for particular classes of incoming data. We study the solutions of this system in the case of traces of monochromatic plane waves. In particular, we give the asymptotics of the diffracted signal in high frequency regime at a given point with fixed distance to the boundary and away from some incident directions. To cite this article: J.-M. Caron, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions

In this Note, we discuss the numerical solution of a system of Eikonal equations with Dirichlet boundary conditions. Since the problem under consideration has infinitely many solutions, we look for those solutions which are nonnegative and of maximal (or nearly maximal) L¹-norm. The computational methodology combines penalty, biharmonic regularization, operator splitting, and finite element approximations. Its practical implementation requires essentially the solution of cubic equations in one variable and of discrete linear elliptic problems of the Poisson and Helmholtz type. As expected, when the spatial domain is a square whose sides are parallel to the coordinate axes, and when the Dirichlet data vanishes at the boundary, the computed solutions show a fractal behavior near the boundary, and particularly, close to the corners. To cite this article: B. Dacorogna et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

Existence of solutions to nonlinear Hammerstein integral equations and applications

In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L²(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.     

Periodic unfolding and Robin problems in perforated domains

The periodic unfolding method was introduced in 2002 by D. Cioranescu et al. for the study of classical periodic homogenization. In this Note, we extend this method to perforated domains introducing also a boundary unfolding operator. As an application, we study the homogenization of some elliptic problems with Robin condition on the boundary of the holes. To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On an integral equation approach for the exterior Robin problem for the Helmholtz equation

A boundary integral equation for the exterior Robin problem for Helmholtz's equation is analyzed in this paper. This integral operator is not compact. A proof based on a suitable regularization of this integral operator and the Fredholm alternative for the regularized compact operator was given by other authors. In this paper, we will give a direct existence and uniqueness proof for the boundary non-compact integral equation in the space settings C^1,λ(S) and C^0,λ(S), where S is a closed bounded smooth surface.     

A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition

In the Hilbert space , we consider nonselfadjoint singular Sturm-Liouville boundary value problem (with two singular end points a and b) in limit-circle cases at a and b, and with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Sturm-Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and Sturm-Liouville boundary value problem.     

From classical to semiclassical non-trapping behaviour

For the semiclassical Schrödinger operator with smooth long-range potential, we prove in a new way, making use of semiclassical measures, that the boundary values of its resolvent at non-trapping energies are bounded by O(1/h), h being the semiclassical parameter. To cite this article: T. Jecko, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Solutions entropiques des problèmes non locauxEntropy solutions of nonlocal problems

We introduce a notion of entropy solution for the nonlocal problem $\text{C}\text{f+f=ψ}$ on $\text{?Ω}$, where $\text{ψ∈}\text{L}{}^1\text{(?Ω)}$ and $\text{C}$ is a nonlinear capacity operator. We prove its existence and uniqueness. This notion of solution allows also to solve a general elliptic problem with nonlinear boundary conditions. To cite this article: K. Ammar, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 751–756.     

New characterization of the kernel of the n-dimensional Laplace operator in exterior domains

In this Note, we study the characterization of the kernel of the Laplace operator with Dirichlet boundary conditions in exterior domains. We consider data in weighted Sobolev spaces. To cite this article: C. Amrouche, Huy Hoang Nguyen, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

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.     

Remarques sur le spectre de l'opérateur de Dirac

We describe a new family of examples of hypersurfaces in the sphere satisfying the limiting-case in C. Bär's extrinsic upper bound for the smallest eigenvalue of the Dirac operator. To cite this article: N. Ginoux, C. R. Acad. Sci. Paris, Ser. I 337 (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).