Communications in Contemporary Mathematics

In this paper, we study the Ginzburg–Landau equations for a two dimensional domain which has small size. We prove that if the domain is small, then the solution has no zero, that is no vortex. More precisely, we show that the order parameter Ψ is almost constant. Additionnally, we obtain that if the domain is a disc of small radius, then any non normal solution is symmetric and unique. Then, in the case of a slab, that is a one dimensional domain, we use the same method to derive that solutions are symmetric. The proofs use a priori estimates and the Poincaré inequality.

This paper achieves, among other things, the following:

• It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations.

• It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an -Hilbert module of finite type ( a finite von Neumann algebra). The result of [3] corresponds to .

• It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M.

These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).

We determine the decomposition of into a sum of irreducible T-modules for general l where D_l is the root lattice of type D_l and T is the tensor product of l+1 Virasoro vertex operator algebras with central charges c₁=1/2, c₂=7/10, c₃=4/5, and c_i=1 for 4≤i≤l+1.

We study the global Cauchy problem for the equation iu_t+Δu+λ|u|^αu=0 in ℝ^N. Using generalized Strichartz' inequalities we show that, under some restrictions on α, if the initial value is sufficiently small in some weak L^p space then there exists a global solution. This result provides a common framework to the "classical" H^s solutions and to the self-similar solutions, thereby extending previous results by Planchon.