On \(W^{1,p}\)-solvability for special vectorial Hamilton--Jacobi systems. (Q1408355)

In this paper a rather weak notion of almost everywhere solution in a Sobolev space to the Dirichlet problem \[ F(Du(x))= 0,\quad x\in\Omega,\qquad u(x)= \varphi(x),\quad x\in\partial\Omega, \] is studied, where \(\Omega\) is a bounded open set in \(\mathbb{R}^n\) and \(u: \Omega\to\mathbb{R}^m\) is a unknown vector field. The general existence theorems for certain Dirichlet problems using suitable approximation schemes called \(W^{1,p}\)-reduction principles that generalize the similar reduction principle for Lipschitz solutions are established. The method relies on a new Baire's category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for \(W^{1,p}\)-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.