On the Euler-Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in Lloc() to the Euler\u2013Lagrange equation for the variational problem infu+W01()(ID(u)+g(u))dx(01) with D convex closed subset of Rn with non empty interior. By means of a disintegration theorem, we next show that the Euler\u2013Lagrange equation can be reduced to an ODE along characteristics, and we deduce that there exists a solution to Euler\u2013Lagrange different from 0 a.e. and satisfies a uniqueness property. These results prove a conjecture on the existence of variations on vector fields stated in Bertone and Cellina (On the existence of variations)