Optimal sets for a geometric oscillation energy

TL;DR

This paper studies a nonlocal geometric energy related to oscillations of normals and tangents, establishing optimal inequalities, existence, and shape characterization of extremal sets under constraints.

Contribution

It introduces a variational framework for the energy, determines optimal constants, and characterizes extremal shapes depending on the parameter p.

Findings

Optimal constants c(n,p) and C(n,p) are identified.

Existence of extremal sets under perimeter and volume constraints is proven.

Shapes of extremal sets are characterized depending on p.

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.