In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies I_alpha defined on probability measures in R^2. The purely nonlocal term in I_alpha is of convolution type, and is isotropic for alpha = 0 and anisotropic otherwise. The cases alpha = 0 and alpha = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of I_alpha 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 different approach, that we believe can be applied to more general energies.
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