Showing papers by "University of Montpellier published in 1971"

Numerical computations in the inverse-scattering problem at fixed energy.

Abstract: Constructing potentials from the phase shifts at a given energy yields an infinity of equivalent solutions. The deviations of these solutions from each other can, however, be analyzed according to a priori limitations on the derivatives and other features of "acceptable" potentials. A sketch of this analysis is given together with a numerical comparison of usual potential forms with the equivalent potentials obtained through Newton's method. The observed deviation gives an appraisal of the deviations from each other of all the equivalent potentials with similar bounds on the derivatives. The deviation is small when there are many phase shifts available, all of them definitely smaller than $\frac{\ensuremath{\pi}}{2}$. For a static potential these conditions can be met for high energies.

Weak and Strong Solutions of Dual Problems

Abstract: Publisher Summary This chapter discusses weak and strong solutions of dual problems. Many problems arising from various domains of physics consist the investigation of functions defined, for instance, on a subset S of IR n , fulfilling certain requirements at each interior point of S and other requirements at each boundary point. Such conditions, expressing physical laws of local character, generally involve continuity of the functions and their partial derivatives up to a certain order. At this stage of problem, physicists frequently characterize the solution, if it exists, by variational properties or even extremal properties. It is that element of a certain class of functions where a certain functional attains its minimum. These variational characterizations of solutions have suggested, to mathematicians, the idea of shifting from the strong formulation of problems, that is, the naive formulation provided by physics to milder systems of requirements, yielding solutions denoted as weak and whose existence is more easily established. The chapter presents a rather general model involving triplets of equivalent ways of characterizing elements in a function space, which is a priori not complete: two minimization properties, said to be dual to each other, and a decomposition property. This is done for nonlinear problems, without differentiability being considered for the functionals in question, but under certain, convexity hypotheses.