\mathcal{A}(p) generators for H^*V and Singer's homological transfer

We list explicitly a minimal set of generators for the cohomology of an elementary abelian p-group, V, of rank 1 or 2, as a module over the mod p Steenrod algebra, for an odd prime p. Following Singer, we then construct a transfer map to the vector space spanned by such generators, where V now has arbitrary rank, from the homology of the Steenrod algebra. We show that this map takes images in the subspace of GL(V)-invariants and that it is an isomorphism for V having rank 1 or 2. Received June 11, 1996; in final form June 9, 1997     

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

We show that a solution of the Cauchy problem for the KdV equation, has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for () data satisfying the condition the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator . Received 22 March 1999     

Full regularity for a class of degenerated parabolic systems in two spatial variables

The authors consider quasilinear parabolic systems

Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity

Summary. In this paper we consider the numerical solutions of the nonlinear time-dependent Ginzburg-Landau model which describes the phase transitions taking place in superconducting films. We propose a semi-implicit finite element scheme which is based on a linear finite element approximation of the order parameter and a mixed finite element discretization for the equation involving the magnetic potential A. The error estimates of the scheme are derived. Received September 5, 1994 / Revised version received April 23, 1995     

Global Existence and Asymptotic Limits of Weak Solutions of the Bipolar Hydrodynamic Model for Semiconductors

For the case of the adiabatic exponents being larger than , we establish the global existence of entropy weak solutions of the Cauchy problem to the bipolar hydrodynamic model for semiconductors. Using the theory of compensated compactness, we hence give finally a complete answer on the related existence problems with the -law pressure relation. A new kind of singular limit of the modified entropy weak solution is discussed. To some extent, the limit of this sort can provide some information about the uniform boundedness of the scaled solution sequences. The quasineutral-relaxation limit of the entropy weak solutions is also investigated.     

Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals

We consider parametric variational double integrals with elliptic Lagrangians F depending on the surface normal and prove a compactness theorem for -critical immersions. As a key ingredient for the relevant a priori estimates we use F. Sauvigny's F-conformal parameters adapted to the parametric integrand F. As a by-product of our analysis we obtain an isoperimetric inequality for -critical immersions generalizing the classical isoperimetric inequality for minimal surfaces. Received November 19, 1999 / Accepted February 4, 2000 / Published online July 20, 2000     

Scaling identity for crossing Brownian motion in a Poissonian potential

We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|^ξ, whereas the distance fluctuation is of order |y|^χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for this conjecture. Received: 31 December 1997 / Revised version: 14 April 1998     

Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory

We study a perturbed semilinear problem with Neumann boundary condition where is a bounded smooth domain of , , , if or if and is the unit outward normal at the boundary of . We show that for any fixed positive integer K any “suitable” critical point of the function generates a family of multiple interior spike solutions, whose local maximum points tend to as tends to zero. Received March 7, 1999 / Accepted October 1, 1999 / Published online April 6, 2000     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

High temperature regime for a multidimensional Sherrington–Kirkpatrick model of spin glass

Comets and Neveu have initiated in [5] a method to prove convergence of the partition function of disordered systems to a log-normal random variable in the high temperature regime by means of stochastic calculus. We generalize their approach to a multidimensional Sherrington-Kirkpatrick model with an application to the Heisenberg model of uniform spins on a sphere of ℝ ^d , see [9]. The main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical energy path. Received: 30 October 1996 / In revised form: 13 October 1997     

Analysis of a part design procedure

Summary. In this paper we analyze and illustrate a new "ab initio" part design procedure, in which, given a cost function which reflects performance, materials, and manufacturing considerations, the topology and the geometry of the part are automatically produced. The analysis is based on demonstration of, first, the compactness of the metric space over which the cost function is defined, and, second, lower semi-continuity of the cost function. Examples include beams and elastic supports. Received November 15, 1993     

Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993