Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps

Abstract

We consider nonlocal linear Schrödinger-type critical systems of the type(1)where Ω is antisymmetric potential in L 2 (IR, so(m)), v is a IR m valued map and Ω v denotes the matrix multiplication. We show that every solution v ∈ L 2 (IR, IR m ) of (1) is in fact in L p loc (IR, IR m ), for every 2 ≤ p < +∞, in other words, we prove that the system (1) which is a-priori only critical in L 2 happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the C 0,α loc regularity of weak 1/2-harmonic maps into C 2 compact manifold without boundary.

Nonlinear commutators for the fractional p-Laplacian and applications

“…e.g. [8,7,5,3,21,19,20], and we will not repeat them in detail. We will also assume that κ 1 > κ 2 > κ 3 .…”

n/p-harmonic maps: Regularity for the sphere case

“…Compensation phenomena for 3-term commutators were first observed by the first author and Rivière in [DLR11b] and [DLR11a] in the context of half-harmonic maps by using the so-called para-products. Such compensation phenomena can be formulated in different ways, for instance as an expansion of lower order derivatives.…”

“…Our work is motivated by recent results [DLR11b], [DLR11a], [Sch12], [DL10], [Sch11] which proved regularity for critical points of the energy F n acting on maps v : R n → R N ,…”

“…In two dimensions, these facts were observed in Rivière's celebrated [Riv07] for all conformally invariant variational functionals (of which the Dirichlet energy is a prototype). We refer the interested reader to the introductions of [DLR11b], [DLR11a] for more on this.…”

“…To this end, in [DLS12] we will treat this case of general manifolds, but with the condition p ≤ 2. The proof relies on a suitable adaption of the arguments in [DLR11b], [DLR11a], [Sch12], [DL10], [Sch11], the details of which we will explain in the next section: The Euler-Lagrange equations of a critical point, see [DLR11b], [Sch12], imply that…”