A unified fictitious domain model for general embedded boundary conditions

This Note addresses the analysis of a new fictitious domain method for elliptic problems in order to handle general embedded boundary conditions (E.B.C.): Fourier, Neumann and Dirichlet conditions on an immersed interface. Our method is based on a recent model of fracture combining flux and solution jumps on the interface Σ separating the original domain $\stackrel{?}{\Omega }$ from the auxiliary exterior domain ${\Omega }_e$. A class of methods is derived within the same unified formulation with either no penalty or exterior control in ${\Omega }_e$, or surface penalty on Σ, volume $H^1$ or $L^2$ penalty in ${\Omega }_e$, or both. The consistency (no penalty) or optimal error estimates with respect to the penalty parameter are proved for such methods. To cite this article: Ph. Angot, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity

We study positive solutions of the equation ${\mathrm{?}}^2\mathrm{\Delta }u?u+u^{\frac{n+2}{n?2}}=0$, where $n=3,4,5$ and $\mathrm{?}>0$ is small, with Neumann boundary condition in a unit ball B. We prove the existence of solutions with an interior bubble at the center and a boundary layer at the boundary ?B. To cite this article: J. Wei, S. Yan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

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).     

Borne sur l'advection et la hauteur d'eau du problème de shallow-water avec conditions aux limites de Dirichlet

Using some properties of the Hardy spaces, we give a regularity result on the advection term of the shallow-water equations and we show a $L^2$ bound holding up to the boundary on the water height with Dirichlet boundary conditions. To cite this article: F. Flori, P. Orenga, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The Helmholtz equation with impedance in a half-space

In this Note we obtain existence and uniqueness results for the Helmholtz equation in the half-space ${\mathbb{R}}_{+}^3$ with an impedance or Robin boundary condition. Basically, we follow the procedure we have already used in the bi-dimensional case (the half-plane). Thus, we compute the associated Green's function with the help of a double Fourier transform and we analyze its far field in order to obtain radiation conditions that allow us to prove the uniqueness of an outgoing solution. Again, these radiation conditions are somewhat unusual due to the appearance of a surface wave guided by the boundary. An integral representation of the solution is presented by means of the Green's function and the boundary data. To cite this article: M. Durán et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

L2-flow of elastic curves with clamped boundary conditions

In this paper we investigate the $L^2$-flow of elastic curves with clamped boundary conditions in n-dimensional Euclidean spaces. The $L^2$-flow corresponds to a fourth-order parabolic equation. In the case of closed curves, the long-time existence of solutions of this evolution equation has been derived in the literature. We extend this result to the case of open (i.e., non-closed) curves with clamped boundary conditions.     

Braces and symmetric groups with special conditions

We study left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like Raut and lri on the properties of the left brace and its associated solution of the Yang–Baxter equation (YBE). We show that the solution $(G,r_G)$ of the YBE associated to the structure group $G=G(X,r)$ (with the natural structure of a left brace) of a nontrivial solution $(X,r)$ of the YBE has multipermutation level 2 if and only if G satisfies lri. It is known that every (left) brace with lri satisfies condition Raut. We prove that for a graded Jacobson radical ring with no elements of additive order two the conditions lri and Raut are equivalent. We construct a finite two-sided brace with condition Raut which does not satisfy lri. We show that a finitely generated two-sided brace which satisfies lri has a finite multipermutation level which is bounded by the number of its generators.     

The Camassa–Holm equation on the half-line

We study the initial-boundary-value problem for the Camassa–Holm equation on the half-line by associating to it a matrix Riemann–Hilbert problem in the complex k-plane; the jump matrix is determined in terms of the spectral functions corresponding to the initial and boundary values. We prove that if the boundary values $u(0,t)$ are ?0 for all t then the corresponding initial-boundary-value problem has a unique solution, which can be expressed in terms of the solution of the associated RH problem. In the case $u(0,t)<0$, the compatibility of the initial and boundary data is explicitly expressed in terms of an algebraic relation to be satisfied by the spectral functions. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The asymptotically sharp Korn interpolation and second inequalities for shells

We consider shells in three-dimensional Euclidean space that have bounded principal curvatures. We prove Korn's interpolation (or the so-called first and a half¹) and the second inequalities on that kind of shells for $\mathit{u}\in W^{1,2}$ vector fields, imposing no boundary or normalization conditions on u. The constants in the estimates are optimal in terms of the asymptotics in the shell thickness h, having the scalings h or $O(1)$. The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to simply proving a Poincaré-type estimate with the symmetrized gradient on the right-hand side. In particular, this applies to linear geometric rigidity estimates for shells, i.e. Korn's fist inequality without boundary conditions.     

Uniform asymptotic formulae for Green's kernels in regularly and singularly perturbed domains

Asymptotic formulae for Green's kernels $G_{\epsilon }(\mathbf{x},\mathbf{y})$ of various boundary value problems for the Laplace operator are obtained in regularly perturbed domains and certain domains with small singular perturbations of the boundary, as $\epsilon \to 0$. The main new feature of these asymptotic formulae is their uniformity with respect to the independent variables x and y. To cite this article: V. Maz'ya, A. Movchan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Solutions to the 1-harmonic flow with values into a hyper-octant of the N-sphere

We announce existence results for the 1-harmonic flow from a domain of ${\mathbb{R}}^m$ into the first hyper-octant of the $N$-dimensional unit sphere, under homogeneous Neumann boundary conditions. The arguments rely on a notion of “geodesic representative” of a BV-vector field on its jump set.     

Lower bounds for the first eigenvalue of the magnetic Laplacian

We consider a Riemannian cylinder Ω endowed with a closed potential 1-form A and study the magnetic Laplacian ${\mathrm{\Delta }}_A$ with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.     

Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains

We study the qualitative properties of sign changing solutions of the Dirichlet problem $\mathrm{\Delta }u+f(u)=0$ in Ω, $u=0$ on ?Ω, where Ω is a ball or an annulus and f is a $C^1$ function with $f(0)?0$. We prove that any radial sign changing solution has a Morse index bigger or equal to $N+1$ and give sufficient conditions for the nodal surface of a solution to intersect the boundary. In particular, we prove that any least energy nodal solution is non radial and its nodal surface touches the boundary. To cite this article: A. Aftalion, F. Pacella, C. R. Acad. Sci. Paris, Ser. I 339 (2004).