当前位置: X-MOL 学术Calc. Var. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
p -Harmonic maps to $$S^1$$ S 1 and stationary varifolds of codimension two
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-10-10 , DOI: 10.1007/s00526-020-01859-6
Daniel Stern

We study the limiting behavior as \(p\uparrow 2\) of the singular sets \(Sing(u_p)\) and p-energy measures \(\mu _p:=(2-p)|du_p|^pdvol\) for families of stationary p-harmonic maps \(u_p\in W^{1,p}(M,S^1)\) from a closed, oriented manifold M to the circle. When the measures \(\mu _p\) have uniformly bounded mass, we show that—up to subsequences—the singular sets \(Sing(u_p)\) converge in the Hausdorff sense to the support of a stationary, rectifiable varifold V of codimension 2, and the measures \(\mu _p\) converge weakly in \((C^0(M))^*\) to a limit of the form

$$\begin{aligned} \mu =\Vert V\Vert +|h|^2dvol, \end{aligned}$$

where h is a harmonic one-form. For solutions on two-dimensional domains, we show moreover that the density of V takes values in \(2\pi {\mathbb {N}}\). Finally, we observe that nontrivial families \(u_p\) of such maps arise naturally on any closed Riemannian manifold of dimension \(n\ge 2\), via variational methods.



中文翻译:

p-调和映照到$$ S ^ 1 $$ S 1和余维二的平稳变量

我们以奇异集\(Sing(u_p)\)p-能量测度\(\ mu _p:=(2-p)| du_p | ^ pdvol \)的极限行为作为(p \ uparrow 2 \)用于从闭合定向歧管M到圆的平稳p调和映射\(u_p \ in W ^ {1,p}(M,S ^ 1)\)的族。当测度\(\ mu _p \)具有一致的边界质量时,我们证明,直到子序列为止,奇异集\(Sing(u_p)\)在Hausdorff方向上收敛,得到了一个固定的,可校正的V变量V的支持。余维2,并且测度\(\ mu _p \)\((C ^ 0(M))^ * \)中弱收敛 到形式的极限

$$ \ begin {aligned} \ mu = \ Vert V \ Vert + | h | ^ 2dvol,\ end {aligned} $$

其中h是谐波单形式。对于二维域上的解,我们进一步证明V的密度取\(2 \ pi {\ mathbb {N}} \)中的值。最后,我们观察到,这些图的非平凡族\(u_p \)通过变分方法自然出现在任何维\(n \ ge 2 \)的闭合黎曼流形上。

更新日期:2020-10-11
down
wechat
bug