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In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energiesIαI_{\alpha}that describe the interaction of particles confined in an elliptic subset of the plane. The caseα=0\alpha=0corresponds to purely Coulomb interactions, while the caseα=1\alpha=1describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter  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 caseα=0\alpha=0of purely Coulomb interactions, which is given by an explicit measure supported on the boundary of the elliptic confining domain. Our result does not seem to agree with the mechanical conjecture that positive edge dislocations 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.

Equilibrium measure for a nonlocal dislocation energy with physical confinement