We prove that if a curve γ ∈ H 3/2 (R/Z, R n ) parametrized by arc length is a stationary point of the Möbius energy introduced by Jun O'Hara in [O'H91], then γ is smooth. Our methods only rely on purely analytical arguments, entirely without using Möbius invariance. Furthermore, they are not fundamentally restricted to one-dimensional domains, but are generalizable to arbitrary dimensions. 2The price we pay is that, instead of the very appealing geometric argument in [FHW94], we have to adapt some sophisticated techniques originally developed by Tristan Rivière and Francesca Da Lio [DLR11a, DLR11b, DL11] and the third author [Sch12, Sch11] to deal with n 2 -harmonic maps into manifolds. The first task in order to prove this result, is to derive the Euler-Lagrange equation for such stationary points. In [FHW94], it was shown that for simple closed curves γ ∈ C 1,1 (R/Z, R n ) and h ∈ C 1,1 (R/Z, R n ) we have δE (2) (γ; h) := lim τց0

In this paper, we will give a necessary and sufficient condition under which O'Hara's Ej,p-energies are bounded. We show that a regular curve has bounded Ej,p-energy if and only if it is injective and belongs to a certain Sobolev–Slobodeckij space.

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The natural setting for the Lane-Emden equation −Δu= |u| p−2 u on a domain Ω ⊂ R n , n≥ 3, for supercritical exponents p > 2 * = 2n/(n − 2) is identified as the space of func-with finite scale-invariant Morrey norms. We show that this Morrey regularity is propagated by the heat flow associated with this equation, and we study the blow-up profiles.