An explicit solution to a system of implicit di¤erential equations

In this article we give an explicit solution to the vectorial di¤erential inclusion where we have to …nd a map u : R ! R with u = 0 on @ and Du (x) 2 O (2) ; a.e. x 2 : Titre: Une solution explicite d’un système implicite d’équations différentielles Résumé: Nous construisons une solution explicite d’une inclusion di¤érentielle où on cherche une application u : R ! R avec u = 0 sur @ et Du (x) 2 O (2) ; p.p. x 2 : 1 Introduction In the last few years methods of …nding almost everywhere solutions of partial di¤erential equations of implicit type have been developed; the theory can be applied also to nonlinear systems of pdes, for which the notion of viscosity solution seem not to be appropriate, mostly because of the lack of maximum principle. Some references in this …eld are the article [7] and the monograph [8] by Dacorogna and Marcellini, where the Baire category method is a crucial step in the proof, and the method of convex integration by Gromov as in Müller and Sverak [14]. These methods are not constructive, i.e., they give existence of solutions but they do not give a rule to compute them. Cellina introduced the use of the Baire category method in the study of di¤erential equations (see [3], in the simple but pioneering context of one single ode). Cellina in [4], [5] and Friesecke [11] gave in the scalar case, for a single pde, an explicit solution, which in the monograph we named pyramid (see Section 2.3.1 in [8]). Cellina and Perrotta [6] also considered a genuine 3 3 system of pdes of implicit type and proposed an explicit solution for the associated Dirichlet problem. The solution is described through an iterative scheme: the set is divided into a countable number of cells and the value of the vector-valued function u at x 2 is de…ned taking into account values of u previously de…ned