اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدMinimizing 1/2-harmonic maps into spheres
100
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $mgeq 3$, 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 $m=2$, we prove that, up to an orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous minimizing $1/2$-harmonic map from the plane into the circle $mathbb{S}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d domain into $mathbb{S}^1$ has a topological charge equal to $pm1$.
قيم البحث
اقرأ أيضاً
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 operator related to the square root Laplacian as defined in Fourie
r space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. The proof relies on the representation of the square root Laplacian as a Dirichlet-to-Neumann operator in one more dimension, and on the analysis of a boundary version of the Ginzburg-Landau equation. Besides the analysis of the fractional Ginzburg-Landau equation, we also give a general partial regularity result for stationary 1/2-harmonic maps in arbitrary dimension.
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $sin(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^infty$ away from a small closed singular
set. The Hausdorff dimension of the singular set is also estimated in terms of $sin(0,1)$ and the stationarity/minimality assumption.
We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n geq 4$. For minimizing harmonic maps $uin W^{1,2}(Omega,mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension
of Almgren and Liebs linear law, namely [mathcal{H}^{n-3}(textrm{sing} u) le C int_{partial Omega} | abla_T u|^{n-1} ,dmathcal{H}^{n-1};] (2) An extension of Hardt and Lins stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.
We consider minimizing harmonic maps $u$ from $Omega subset mathbb{R}^n$ into a closed Riemannian manifold $mathcal{N}$ and prove: (1) an extension to $n geq 4$ of Almgren and Liebs linear law. That is, if the fundamental group of the target manifo
ld $mathcal{N}$ is finite, we have [ mathcal{H}^{n-3}(textrm{sing } u) le C int_{partial Omega} | abla_T u|^{n-1} ,d mathcal{H}^{n-1}; ] (2) an extension of Hardt and Lins stability theorem. Namely, assuming that the target manifold is $mathcal{N}=mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s in (frac{1}{2},1]$ and $p in [2,infty)$ satisfying $sp geq 2$. We also discuss sharpness.
This article addresses the regularity issue for minimizing fractional harmonic maps of order $sin(0,1/2)$ from an interval into a smooth manifold. Holder continuity away from a locally finite set is established for a general target. If the target is
the standard sphere, then Holder continuity holds everywhere.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
