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

2010/12/11 by Francesca Da Lio, Da Lio, Francesca, Tristan Riviere +1 · 1 citation
Mathematics · #35B65 #35J20 #35J60 #35S99 #58E20 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B65 #msc:35J20 #msc:35J60 #msc:35S99 #msc:58E20

paper · pdf · doi:10.48550/arxiv.1012.2438

52 pages

arxiv created 2010/12/11 · arxiv updated 2010/12/14

Abstract

We consider nonlocal linear Schrödinger-type critical systems of the type Δ^1/4 v=Ω v~~~in \R . where Ω is antisymmetric potential in L²(\R,so(m)), v is a \R^m valued map and Ω v denotes the matrix multiplication. We show that every solution v∈ L²(\R,\R^m) of \receqabstr is in fact in L^p_loc(\R,\R^m), for every 2≤ p<+∞, in other words, we prove that the system (\refeqabstr) which is a-priori only critical in L² 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 C² compact manifold without boundary.

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