Showing papers by "Jean-Pierre Eckmann published in 1996"

Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

Abstract: In this paper we describe invariant geometrical ~structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider's recent proof that these states are nonlinearly stable.

Zeta functions with Dirichlet and Neumann boundary conditions for exterior domains

Abstract: We generalize earlier studies on the Laplacian for a bounded open domain $\Omega\in \real^2$ with connected complement and piecewise smooth boundary We compare it with the quantum mechanical scattering operator for the exterior of this same domain Using single layer and double layer potentials we can prove a number of new relations which hold when one chooses {\em independently} Dirichlet or Neumann boundary conditions for the interior and exterior problem This relation is provided by a very simple set of $\zeta$-functions, which involve the single and double layer potentials We also provide Krein spectral formulas for all the cases considered and give a numerical algorithm to compute the $\zeta$-function