| Summary: | In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies <em>I</em><sub><em>α</em></sub> defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in <em>I</em><sub><em>α</em></sub> is of convolution type, and is isotropic for <em>α</em> = 0 and anisotropic otherwise. The cases <em>α</em> = 0 and <em>α</em> = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of <em>I</em><sub><em>α</em></sub> 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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