A maximum-principle approach to the minimisation of a nonlocal dislocation energy

In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies defined on probability measures in two dimensions. The purely nonlocal term in the energy is of convolution type, and is isotropic for the parameter alpha equal to 0 and anisotropic otherwise. The cases when alpha is equal to 0 and 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of I have been characterised by the same authors in an earlier paper, by exploiting some formal similarities with the Euler equation, and by means of complex-analysis techniques. We here propose a dierent approach, that we believe can be applied to more general energies.