Regularity theory for tangent-point energies: The non-degenerate sub-critical case

Abstract

AbstractIn this article we introduce and investigate a new two-parameter family of knot energies ${\operatorname{TP}^{(p,\,q)}}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy ${\operatorname{TP}^{(p,\,q)}}$ in the sub-critical range p ∈ (q+2,2q+1) and see that those are all injective and regular curves in the Sobolev–Slobodeckiĭ space ${W^{\scriptstyle (p-1)/q,q}(\mathbb {R}/\mathbb {Z},\mathbb {R}^n)}$. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case q = 2: a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of $\operatorname{TP}^{(p,2)}$ + λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth.