Composition of weak diffeomorphisms (Q1358205)

The paper is devoted to a systematic analysis of the chain rule in classes of weakly differentiable functions. The main two problems under investigation are: 1) the differentiability properties of \(f\circ u\), where \(f\) is a Sobolev map and \(u\) is, in a weak sense, a change of the independent variable; 2) the stability of classes of weak diffeomorphisms under composition. In both problems, first the approximate differentiability properties of the composite function and the summability of the approximate differential are investigated. Then, under additional assumptions it is proved that the approximate differential provides also a derivative in the sense of distributions. In addition, suitable nonlinear integration by parts formulas involving the minors of the Jacobian matrix are satisfied.