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

Abstract

We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions are very mild: we assume that the potential is lower semicontinuous, and satisfies a monotonicity condition in a neighbourhood of its minimum. 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. Our results extend and provide variational versions of several classical theorems, well-known for solutions of scalar semilinear elliptic PDE.