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

By Francesca Da Lio, T. Rivière. 2010

109 citations recorded by the source corpus.

Abstract We consider non-local linear Schrodinger-type critical systems of the type (1) Δ 1 / 4 v = Ω v in R , where Ω is antisymmetric potential in L 2 ( R , so ( m ) ) , v is an R m valued map and Ωv denotes the matrix multiplication. We show that every solution v ∈ L 2 ( R , R m ) of (1) is in fact in L loc p ( R , R 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 loc 0 , α regularity of weak 1/2-harmonic maps into C 2 compact sub-manifolds without boundary.

Subfields: Applied Mathematics

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In Applied Mathematics

Citations in the corpus

Blow-up analysis of a nonlocal Liouville-type equation (2015, unknown): “(47) Equation (47) says that φ is a 12 -harmonic map into 0 (see [Da Lio and Rivière 2011]).”
Fractional div-curl quantities and applications to nonlocal geometric equations (2017, methodology): “With this, one obtains a decay estimate following the typical procedure for critical nonlocal equations, see [11, 10, 35].”
On a new class of fractional partial differential equations (2014, methodology): “For example, in recent work of Da Lio and Rivière [7], Da Lio and Schikorra [8], and Schikorra [26, 27, 28] we find alternative use of the two in establishing Hölder continuity for critical points of integral functionals of the fractional Laplacian where the underlying fields take values in a mani”