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).     

Approximation par éléments finis de problèmes elliptiques d'optimisation de forme

We consider a problem of elliptic optimal design. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n=2, S?veràk proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. The proof (J. Math. Pures Appl. 72 (1993) 537–551) is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet Laplacian in H¹ with respect to it. In this Note we consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one as the mesh-size tends to zero. The key point of the proof is that finite-element approximations of the solution of the Dirichlet Laplacian converge in H¹ whenever the polygonal domains converge in the sense of that topology. To cite this article: D. Chenais, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equationsSolutions visqueuses du problème avec une dérivée oblique dégénérée pour une classe d'équations complètement non linéaires

In this paper we prove a comparison principle between the semicontinuous viscosity sub- and supersolutions of the tangential oblique derivative problem and the mixed Dirichlet–Neumann problem for fully nonlinear elliptic equations. By means of the comparison principle, the existence of a unique viscosity solution is obtained. To cite this article: P. Popivanov, N. Kutev, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 661–666.     

Application of global Carleman estimates with rotated weights to an inverse problem for the wave equation

We establish geometrical conditions for the inverse problem of determining a stationary potential in the wave equation with Dirichlet data from a Neumann measurement on a suitable part of the boundary. We present the stability results when we measure on a part of the boundary satisfying a rotated exit condition. The proofs rely on global Carleman estimates with angle type dependence in the weight functions. To cite this article: A. Doubova, A. Osses, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray-type solutions towards a vector field which satisfies the usual 2D Navier–Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat–type equation elsewhere. The method of proof uses weak compactness arguments. To cite this article: I. Gallagher, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Explosion of Jump-Type Symmetric Dirichlet Forms on ℝ d

We give a sufficient condition for a class of jump-type symmetric Dirichlet forms on ? ^d to be conservative in terms of the jump kernel and the associated measure. Our condition allows the coefficients dominating big jumps to be unbounded. We derive the conservativeness for Dirichlet forms related to symmetric stable processes. We also show that our criterion is sharp by using time changed Dirichlet forms. We finally remark that our approach is applicable to jump-diffusion type symmetric Dirichlet forms on ? ^d .     

Factorialité de l'anneau des séries de Dirichlet analytiques

We study arithmetical properties of the ring of analytic Dirichlet series. In particular, we prove a theorem of division by several series and we deduce from it that the ring is factorial. To cite this article: F. Bayart, A. Mouze, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Hölder continuity of solutions to a basic problem in the calculus of variations

For the basic problem in the calculus of variations where the Lagrangian is convex and depends only on the gradient, we establish the continuity of the solutions when the Dirichlet boundary condition is defined by a continuous function ϕ. When ϕ is Lipschitz continuous, then the solutions are Hölder continuous. To cite this article: P. Bousquet et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Solutions sous-analytiques globales de certaines équations d'Hamilton–Jacobi

In the 1980 Crandall and Lions introduced the concept of viscosity solution in order to get existence and/or unicity results for Hamilton–Jacobi equations. In this Note we focus on the Dirichlet problem for Hamilton–Jacobi equations stemming from calculus of variations, and assert that if the data are analytic then the viscosity solution is moreover subanalytic. We extend this result to generalized eikonal equations, stemming from sub-Riemannian geometry problems. To cite this article: E. Trélat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions

We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of O(h ² ln h ⁻¹), where h is the step of a cubic grid.     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

Monotonicity in integrodifferential equations

We study the behavior of positive solutions of the Dirichlet problem Lu=f(u) in $\text{Ω}$ with $\text{Ω=(a,+∞)}$, where a can be ?∞, and L is an abstract operator which is non-increasing under translation and satisfies a strong maximum principle property. This covers the case of many integral operators. Under some assumptions on f (e.g., bistable, monostable), we show that any solution exhibits a monotone behavior. To cite this article: J. Coville, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

Abstract theory of universal series and an application to Dirichlet series

We present an abstract theory of universal series; in particular, we give a necessary and sufficient condition for the existence of universal series of a certain type. Most of the known results can be proved or strengthened by using this condition. We also obtain new results, for example, related to universal Dirichlet series. To cite this article: V. Nestoridis, C. Papadimitropoulos, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Un algorithme de type Robin pour des problèmes de contact unilatéral

In this Note, we propose a Robin domain decomposition algorithm to approximate a frictionless unilateral problem between two elastic bodies. Indeed, this algorithm combines on the contact zone the Dirichlet and Neumann boundaries conditions (Robin boundary condition). The primary feature of this algorithm is the resolution of the same variational inequality on each sub-domain. To cite this article: M. Ipopa, T. Sassi, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Beurling–Deny formula of semi-Dirichlet forms

In this Note we announce a structure result for non-symmetric Dirichlet forms and semi-Dirichlet forms. Our result is regarded as an extension of the celebrated Beurling–Deny formula which is up to now available only for symmetric Dirichlet forms. The result can also be regarded as an extension of Lévy–Khinchine formula or more generally, an extension of Courrège's Theorem in the semi-Dirichlet forms setting. To cite this article: Z.-C. Hu, Z.-M. Ma, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Algorithme de Neumann–Dirichlet pour des problèmes de contact unilatéral : Résultat de convergenceNeumann–Dirichlet algorithm for unilateral contact problems: Convergence results

In this Note, we propose and we prove the convergence of a Neumann–Dirichlet algorithm in order to approximate a Signorini problem between two elastic bodies. The idea is to retain the natural interface between the two bodies as numerical interface for the domain decomposition and to replace the Dirichlet problem in [4] by a variational inequality. To cite this article: G. Bayada et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 381–386.     

Bornes sur la densité pour un problème de Navier–Stokes compressible à frontière variable avec conditions aux limites de Dirichlet

We extend a compactness result shown by P.-L. Lions in 1998 to an isentropic compressible Navier–Stokes problem ($\gamma ?1$) defined on a time dependent domain with Dirichlet boundary conditions. This result can be useful for the study of some fluid–structure interaction problems, for the analysis of some pollution water problems (shallow water equations with free boundary: $\gamma =1$) or for the modelling of a river level. To cite this article: F. Flori, B. Giudicelli, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A reaction-diffusion system of a competitor-competitor-mutualist model

We investigate the homogeneous Dirichlet problem and Neumann problem to a reaction-diffusion system of a competitor-competitor-mutualist model. The existence, uniqueness, and boundedness of the solutions are established by means of the comparison principle and the monotonicity method. For the Dirichlet problem, we study the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities. For the Neumann problem, we analyze the contant equilibrium solutions and their stabilities. The main method used in studying of the stabilities is the spectral analysis to the linearized operators. The O.D.E. problem to the same model was proposed and studied by B. Rai, H. I. Freedman, and J. F. Addicott (Math. Biosci. 65 (1983), 13–50).     

On the horizontal distribution of zeros of linear combinations of Euler products

We investigate the behavior of the zero counting function of certain natural Dirichlet series with functional equation in the immediate vicinity of the critical line $\text{{}\text{Re}\text{(s)=}\text{1}\text{2}\text{}}$. To cite this article: D.A. Hejhal, C. R. Acad. Sci. Paris, Ser. I 338 (2004).