# Minimizing 1/2-harmonic maps into spheres

@article{Millot2019Minimizing1M,
title={Minimizing 1/2-harmonic maps into spheres},
author={Vincent Millot and Marc Pegon},
journal={Calculus of Variations and Partial Differential Equations},
year={2019},
volume={59},
pages={1-37}
}
• Published 2019
• Mathematics, Physics
• Calculus of Variations and Partial Differential Equations
In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of Millot and Sire (Arch Ration Mech Anal 215:125–210, 2015), Moser( J Geom Anal 21:588–598, 2011) in the case where the target manifold is the $$(m-1)$$ ( m - 1 ) -dimensional sphere. For $$m\geqslant 3$$ m ⩾ 3 , we show that minimizing 1/2-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For $$m=2$$ m = 2 , we prove that, up to an… Expand
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#### References

SHOWING 1-10 OF 65 REFERENCES
Partial regularity for stationary harmonic maps into spheres
In an interesting recent paper [12], F. HI~LEIN has shown that any weakly harmonic mapping from a two-dimensional surface into a sphere is smooth. I present here a kind of generalization to higherExpand
On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres
• Mathematics
• 2015
This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg–Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operatorExpand
Relaxed energies for Hl 12-maps with values into the circle and measurable weights
• Mathematics
• 2009
We consider, for maps f ∈ H 1/2 (R 2 ;S 1 ), an energy E(f) related to a seminorm equivalent to the standard one. This seminorm is associated to a measurable matrix field in the half space. UnderExpand
Compactness and Bubbles Analysis for 1/2-harmonic Maps
In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps $u_k\colon\R\to {\cal{S}}^{m-1}$ such that $|u_k|_{\dot H^{1/2}(\R,{\cal{S}}^{m-1})}\le C.$ MoreExpand
Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds
• Mathematics
• 1988
We study maps qp from domains Q in R3 to S2 which have prescribed boundary value functions 4 and which minimize Dirichlet's integral energy. Such (p's can have isolated singular points, and ourExpand
Free boundary minimal surfaces: a nonlocal approach
• Physics, Mathematics
• 2017
Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we considerExpand
An optimal estimate for the singular set of a harmonic map in the free boundary.
• Mathematics
• 1989
We consider the following Situation: The parameter domain M for our maps is a Riemannian manifold of dimension m ^ 3 and the free boundary Σ is a non-empty subset of the boundary d M. As targetExpand
Minimizing fractional harmonic maps on the real line in the supercritical regime
• Mathematics
• 2017
This article addresses the regularity issue for minimizing fractional harmonic maps of order $s\in(0,1/2)$ from an interval into a smooth manifold. H\"older continuity away from a locally finite setExpand
A variational problem with lack of compactness for H1/2 (S1;S1) maps of prescribed degree
• Mathematics
• 2004
Abstract We consider, for maps in H 1/2 ( S 1 ; S 1 ), a family of (semi)norms equivalent to the standard one. We ask whether, for such a norm, there is some map in H 1/2 ( S 1 ; S 1 ) of prescribedExpand
Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps
• Mathematics
• 2010
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 andExpand