Inserted: 23 jul 2020
Last Updated: 23 jul 2020
In this paper we characterize the equilibrium measure for a family of nonlocal and anisotropic energies $I_\alpha$ that describe the interaction of particles confined in an elliptic subset of the plane. The case $\alpha=0$ corresponds to purely Coulomb interactions, while the case $\alpha=1$ describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter $\alpha$ and favors the alignment of particles. We show that the equilibrium measure is completely unaffected by the anisotropy and always coincides with the optimal distribution in the case $\alpha=0$ of purely Coulomb interactions, which is given by an explicit measure supported on the boundary of the elliptic confining domain. Our result seems to be in constrast with the mechanical conjecture that positive edge dislocation at equilibrium tend to arrange themselves along "wall-like" structures. Moreover, this is one of the very few examples of explicit characterization of the equilibrium measure for nonlocal interaction energies outside the radially symmetric case.