This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential. 2000Academic Press

We consider a three-dimensional viscous incompressible fluid governed by the Navier-Stokes equations, interacting with an elastic plate located on one part of the fluid boundary. We do not neglect the deformation of the fluid domain which consequently depends on the displacement of the structure. The purpose of this work is to study the solutions of this unsteady fluid-structure interaction problem, as the coefficient modeling the viscoelasticity (resp. the rotatory inertia) of the plate tends to zero. As a consequence, we obtain the existence of at least one weak solution for the limit problem (Navier-Stokes equation coupled with a plate in flexion) as long as the structure does not touch the bottom of the fluid cavity.

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In this paper we prove new Hardy-like inequalities with optimal potential singularities for functions in W 1,p (Ω), where Ω is either bounded or the whole space R n and also some extensions to arbitrary Riemannian manifolds. We also study the spectrum of perturbed Schrödinger operators for perturbations which are just below the optimality threshold for the corresponding Hardy inequality.2000 Mathematics Subject Classification: 35R45, 35J10, 35J85, 35P05, 81Q10.

The purpose of this note is to derive compactness properties for both incompressible and col~lpressible viscous flows in a boullded donlaill interacting with a finite number of rigid bodies. We prove the global existence of weak solutions away from collisions.

In this paper we prove the existence of stationary solutions of some nonlinear Dirac equations. We do it by using a general variational technique. This enables us to consider nonlinearities which are not necessarily compatible with symmetry reductions.

SynopsisIn this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 of -Au=/(u) in (1, u = 0 on Sil, u(x) i " 5 " > 0 for any smooth / satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

No abstract