Published Paper

Inserted: 13 oct 2022
Last Updated: 6 oct 2023

Journal: Advances in Mathematics
Year: 2023
Links: Published Paper (Open Access)

Abstract:

In this paper we consider a general class of anisotropic energies in three dimensions and give a complete characterisation of their minimisers. We show that, depending on the Fourier transform of the interaction potential, the minimiser is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional ellipse. In particular, it is always an ellipsoid if the transform is strictly positive, while when the Fourier transform is degenerate both cases can occur. Finally, we show an explicit example where loss of dimensionality of the minimiser does occur.

Keywords: Potential theory, Nonlocal energy, anisotropic interaction

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