Minimizing 1/2-harmonic maps into spheres

  title={Minimizing 1/2-harmonic maps into spheres},
  author={Vincent Millot and Marc Pegon},
  journal={Calculus of Variations and Partial Differential Equations},
  • V. Millot, Marc Pegon
  • 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
Partial regularity for fractional harmonic maps into spheres
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order s ∈ (0, 1) in arbitrary dimensions. It is shown that such fractional harmonicExpand
Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes
Let $\Sigma$ a closed $n$-dimensional manifold, $\mathcal{N} \subset \mathbb{R}^M$ be a closed manifold, and $u \in W^{s,\frac ns}(\Sigma,\mathcal{N})$ for $s\in(0,1)$. We extend the monumental workExpand
Approximation of fractional harmonic maps
Weak compactness results for critical points of the fractional Dirichlet energy on unit-length vector fields imply the convergence of numerical approximations in the approximation of fractional harmonic maps. Expand


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
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
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
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
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.
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
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
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
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