Optimal sets for a geometric oscillation energy

Matteo Novaga; Fumihiko Onoue; Emanuele Paolini

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arXiv · 2602.22910

Optimal sets for a geometric oscillation energy

Matteo Novaga·Fumihiko Onoue·Emanuele Paolini

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Abstract

We investigate the nonlocal energy corresponding to the $p$-oscillation of the unit normal vector for hypersurfaces, or the unit tangent vector for curves. The energy satisfies geometric inequalities with optimal constants $c(n,p)$ and $C(n,p)$ which are determined by a variational problem over the probability measures on the sphere. The extremal measures for such problem depend critically on the value of $p$. We prove existence of optimal sets for this energy under perimeter and volume constraint, and characterize their shape.

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Matteo Novaga, Fumihiko Onoue, Emanuele Paolini. 2026-02-26. Optimal sets for a geometric oscillation energy. https://arxiv.org/abs/2602.22910

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