We analyze the Ginzburg-Landau equation for a superconductor in the case of a 2-dimensional model: a cylindrical conductor with a magnetic field parallel to the axis. This amounts to find the extrema of the free energywhere Ω is a bounded domain with smooth boundary in IR 2 , A = (A ί9 A 2 ) the vector potential, B A = d 1 A 2 -d 2 A ί the magnetic field, Φ Ά complex field. We describe the connected components of the maximal configuration space, i.e. of the set of all {A, Φ) with components in the Sobolev space H ί (Ω) and such that | Φ\ = 1 on the boundary, modulo the action of the gauge group. In the critical case K = 1 we give a complete description of the minimal configurations in each component.

The paper deals with an initial boundary value problem for the system of equations describing suspension motion in the case of specular reflecting boundary of the domain. A definition of a global generalized solution of Hopf class is given and its existence proved. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

The theory of infinite dimensional oscillatory integrals by finite dimensional approximations is shown to provide new information on the trace formula for Schrodinger operators. In particular, the explicit computation of contributions given by constant and non constant periodic orbits, for potentials which are quadratic plus a bounded nonlinear part, is provided. The heat semigroup as well as the Schrodinger group are discussed and it is shown in particular that their singular supports are contained in an explicit countable set independent of the bounded part of the potential.

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An algebraic formalism is presented which simplifies and makes natural various arguments in the theories where some notion of ‘‘connected operator’’ appears. As a first example, in this paper the case of N-body systems is considered.