  The problem of minimizing an integral functional of a vector-valued
Lagrangian on a set of admissible arcs with given endpoints is considered. The
problem is tackled by embedding it into a set-optimization problem such that
the image space becomes a complete lattice. This procedure allows to define the
concepts of minimizer and of infimizer, as two completely different notions. An
infimizer is proved to contain minimizers or at least minimizing sequences for
the linearly scalarized problems, with the scalarizing parameters being
elements in the dual of the ordering cone. In this way set-valued
Euler-Lagrange equations are obtained and weak solutions are defined for these
equations. Following the guidelines of the classical results, under suitable
convexity and coercivity hypotheses an existence result for an infimizer is
proved. In addition to the unconstrained problem, Euler-Lagrange equations are
provided for the case with isoperimetric constraints. Finally, the results are
applied to the optimization of the shape of energy-saving buildings.
