Variational field theory from the point of view of direct methods

In this paper we show that the classical field theory of Weierstrass–Hilbert can be strengthen on applying direct methods. Concretely, given a field of extremals and an extremal that is an element of the field, we can show that the latter gives minimum in the class of Lipschitz functions with the same boundary data and with the graphs in the set covered by the field. We suggest the two proofs: a modern one (exploiting Tonelli's Theorem on lower semicontinuity of integral functionals with respect to the weak convergence of admissible functions in $W^{1,1}$) and the one based only on arguments available already in the 19th century.