The equilibrium measure for an anisotropic nonlocal energy

Abstract In this paper we characterise the minimisers of a one-parameter family nonlocal and anisotropic energies $$I_\alpha $$ I α defined on probability measures in $${\mathbb {R}}^n$$ R n , with $$n\ge 3$$ ≥ 3 . The energy consists purely term convolution type, whose interaction kernel reduces to Coulomb potential for $$\alpha =0$$ = 0 is otherwise, quadratic confinement. two-dimensional case arises study defects metals has been solved by authors means complex-analysis techniques. We prove that \in (-1, n-2]$$ ∈ ( - 1 , 2 ] minimiser unique (normalised) characteristic function spheroid. This result paradigmatic example role anisotropy shape minimisers. particular, phenomenon loss dimensionality, observed dimension $$n=2$$ does not occur higher at value =n-2$$ corresponding sign change Fourier transform potential.