Optimal sets for a geometric oscillation energy
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.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Point processes and geometric inequalities
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 26 February 2026
- Language
- EN
- Document type
- Preprint