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We establish a comparison principle providing accurate upper bounds for themodulus of vector valued minimizers of an energy functional, associated whenthe potential is smooth, to elliptic gradient systems. Our assumptions are verymild: we assume that the potential is lower semicontinuous, and satisfies amonotonicity condition in a neighborhood of its minimum. As a consequence, wegive a sufficient condition for the existence of dead core regions, where theminimizer is equal to one of the minima of the potential.

A comparison principle for vector valued minimizers of semilinear elliptic energy, with application to dead cores