On a constrained Dirichlet problem

We consider a Dirichlet minimum problem with a pointwise constraint on the gradient, i.e., ||▽u(x)|| ≤ 1 a.e., or, equivalently, an unconstrained minimum problem with an extended-valued integrand. Since the subdifferential of this integrand is defined on the whole effective domain, the problem of the validity of the Euler - Lagrange equation (or, equivalently, of the Pontryagin maximum principle) for a solution w can be posed. To show that this equation is verified along a solution, the equivalence of the problem considered here and of a problem with obstacles is proved, and a generalization of Stampacchia's bounded slope condition result is presented.