论文标题
涉及分数laplacians的Lagrangians的Noether定理
Noether Theorems for Lagrangians involving fractional Laplacians
论文作者
论文摘要
在这项工作中，我们为形式的能量{等式*} e（u）= \int_Ωl\ left（x，x，u（x），（ - δ）^\ frac {1} {1} {4} {4} u（x）u（x）u（x）u（x）在分数梯度和分数差异方面，可允许的功能$ U $。在这里，$ω$要么用于欧几里得空间$ \ mathbb {r}^n $或circle $ \ mathbb {s}^1 $。然后，我们讨论了这些结果和相关技术的一些应用，以研究非局部几何方程的研究，以及研究$ \ mathbb {s}^1 $上半dirichlet能量的固定点的研究。特别是，我们介绍了$ \ frac {1} {2} $ - 分数型差速器作为一个简单的工具，以表征$ h^\ frac {1} {1} {1} {2} {2}（\ mathbb {s}^1，\ s}^1，\ mathbb {r}^m）$的固定点。最后，我们展示了如何使用$ \ mathbb {r} $上的半Dirichlet Energy的不变性属性来获得Pohozaev身份。
In this work we derive Noether Theorems for energies of the form \begin{equation*} E(u)=\int_ΩL\left(x,u(x),(-Δ)^\frac{1}{4}u(x)\right)dx \end{equation*} for Lagrangians exhibiting invariance under a group of transformations acting either on the target or on the domain of the admissible functions $u$, in terms of fractional gradients and fractional divergences. Here $Ω$ stays either for an Euclidean space $\mathbb{R}^n$ or for the circle $\mathbb{S}^1$. We then discuss some applications of these results and related techniques to the study of nonlocal geometric equations and to the study of stationary points of the half Dirichlet energy on $\mathbb{S}^1$. In particular we introduce the $\frac{1}{2}$-fractional Hopf differential as a simple tool to characterize stationary point of the half Dirichlet energy in $H^\frac{1}{2}(\mathbb{S}^1,\mathbb{R}^m)$ and study their properties. Finally we show how the invariance properties of the half Dirichlet energy on $\mathbb{R}$ can be used to obtain Pohozaev identities.
