Regularity and structure of non-planar $p$-elasticae

We prove regularity and structure results for $p$-elasticae in ${\mathbb{R}}^n$, with arbitrary $p\in (1,\mathrm{\infty })$ and $n\ge 2$. Planar $p$-elasticae are already classified and known to lose regularity. In this paper, we show that every non-planar $p$-elastica is analytic and three-dimensional, with the only exception of flat-core solutions of arbitrary dimensions. Subsequently, we classify pinned $p$-elasticae in ${\mathbb{R}}^n$ and, as an application, establish a Li-Yau type inequality for the $p$-bending energy of closed curves in ${\mathbb{R}}^n$. This extends previous works for $p=2$ and $n\ge 2$ as well as for $p\in (1,\mathrm{\infty })$ and $n=2$.