Stationary points of O'Hara's knot energies

In this article we study the regularity of stationary points of the knot energies E ((alpha)) introduced by O'Hara (Topology 30(2):241-247, 1991; Topol Appl 48(2):147-161, 1992; Topol Appl 56(1):45-61, 1994) in the range . In a first step we prove that E ((alpha)) is C (1) on the set of all regular embedded curves belonging to and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E ((alpha)) plus a positive multiple of the length. We show that stationary points of finite energy are of class C (a)-so especially all local minimizers of E ((alpha)) among curves with fixed length are smooth.