Weak and Strong Solutions of Dual Problems

Abstract

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 IRn, 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.