Fida El Chami

• Professor
• Professor at Lebanese University

Given a compact Riemannian manifold (M n , g) with boundary $\partial$M , we give an estimate for the quotient $\partial$M f d$\mu$ g M f d$\mu$ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a differential inequality for f...

We extend the buckling and clamped-plate problems to the context of differential forms on compact Riemannian manifolds with smooth boundary. We characterize their smallest eigenvalues and prove that, in the case of Euclidean domains, their spectra on forms coincide with the spectra of the corresponding problems. We obtain various estimates involvin...

We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the first eigenvalue, some of which involve eigen-values of other problems such as the Dirichlet, Neumann, Robin and S...

On a Riemannian manifold (M, g) endowed with a Riemannian flow, we study in this paper the curvature term in the Bochner–Weitzenböck formula of the basic Laplacian. We prove that this term splits into two parts; a first part that depends on the curvature operator of the manifold M and a second part that can be expressed in terms of the O’Neill tens...

Given a compact Riemannian manifold $(M^n,g)$ with boundary $\partial M$, we give an estimate for the quotient $\frac{\int_{\partial M} fd\mu_g}{\int_M fd\mu_g}$, where $f$ is a smooth positive function defined on $M$ that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established by Alessandro Savo in A mean valu...

In this paper, we consider a Riemannian manifold (M, g) endowed with a Riemannian flow and we study the curvature term in the Bochner-Weitzenb{\"o}ck formula of the basic Laplacian on M. We prove that this term splits into two parts. The first part depends mainly on the curvature operator of the underlying manifold M and the second part is expresse...

In [4], we gave a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. In this paper, we extend this result to the case of the first eigenvalue on basic $p$-forms for $p>1$. As in [4], the limiting case allows to characterize the m...

Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O’Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a doma...

Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a doma...

In [4], we gave a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. In this paper, we extend this result to the case of the first eigenvalue on basic $p$-forms for $p>1$. As in [4], the limiting case allows to characterize the m...

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and the boundary is $\eta$-umbili...

In this paper, we consider a Riemannian foliation whose normal bundle carries a parallel or harmonic basic form. We estimate the norm of the O'Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show th...

In this article, we study the Stokes problem with some nonstandard boundary conditions. The variational formulation decouples into a system for the velocity and a Poisson equation for the pressure. The corresponding discrete system do not need an inf-sup condition. Hence, the velocity is approximated with “curl” conforming finite elements and the p...

In this paper, we give a branching law from the group Sp(n) to the subgroup Sp(q) × Sp(n-q). We propose an application of this result to compute the Laplace spectrum on the forms of the manifold Sp(n)/Sp(q)×Sp(n-q), using the “identification” of the Laplace operator with the Casimir operator in symmetric spaces.

In this note, we first give a quick presentation of the supergeometry underlying supergravity theories, using an intrinsic differential geometric language. For this, we adopt the point of view of Cartan geometries, and rely as well on the work of John Lott, who has found a unified geometrical interpretation of the torsion constraints for many super...

In this paper, we compute the Laplace spectrum on the forms of the manifold Sp(n)/Sp(q) XSp(n-q). The method is based on the representation theory of compact Lie groups and the ”identification” of the Laplace operator with the Casimir operator in symmetric spaces.

This work is devoted to the optimal and a posteriori error estimates of the Stokes problem with some non-standard boundary conditions in three dimensions. The variational formulation is decoupled into a system for the velocity and a Poisson equation for the pressure. The velocity is approximated with curl conforming finite elements and the pressure...

In this paper we study the time dependent Stokes problem with some different boundary conditions. We establish a decoupled variational formulation into a system of velocity and a Poisson equation for the pressure. Hence, the velocity is approximated with ${\bf curl}$ conforming finite elements in space and Euler scheme in time and the pressure with...

In this paper we study the Coupling Darcy-Stokes Systems. We establish a coupled variational formulation with the velocity and the pressure. The velocity is approximated with ${\bf curl}$ conforming finite elements and the pressure with standard continuous elements. We establish optimal a priori and a posteriori error estimates.

We are interested in the relation between the Laplace spectrum on the differential forms and the geometric spectrum. We consider examples of isospectral non-isometric spherical space forms constructed by A. Ikeda: lens spaces isospectral up to a fixed order and not above, and spherical space forms with fundamental groups of type 1 isospectral on th...

In this paper we study the Stokes problem with some different boundary conditions. We establish a decoupled variational formulation into a system of velocity and a Poisson equation for the pressure. The continuous and corresponding discrete system do not need an inf-sup condition. Hence, the velocity is approximated with curl conforming finite elem...