On W 1 , p - solvability for special vectorial Hamilton – Jacobi systems

We study the solvability of special vectorial Hamilton–Jacobi systems of the form F(Du(x))= 0 in a Sobolev space. In this paper we establish the general existence theorems for certain Dirichlet problems using suitable approximation schemes called W1,p-reduction principles that generalize the similar reduction principle for Lipschitz solutions. Our approach, to a large extent, unifies the existing methods for the existence results of the special Hamilton–Jacobi systems under study. The method relies on a new Baire’s category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for W1,p-reduction are also given along with certain generalization of some known results and a specific application to the boundary value problem for special weakly quasiregular mappings.  2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. MSC: 35F30; 35A25; 49J10