  We establish a general comparison principle for vector valued minimizers of a
functional, associated when the potential is smooth, to elliptic gradient
systems. As a consequence, we give a sufficient condition for the existence of
dead core regions, where the minimizer is equal to one of the minima of the
potential. In the scalar case, this condition is also necessary, due to the
availability of a strong maximum principle. Our results provide variational
versions of several classical theorems well-known for solutions of scalar
semilinear elliptic PDE.
