Noether Theorems for Lagrangians involving fractional Laplacians

Abstract

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 $\Omega$ 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.

126 pages

References in corpus (2)

• Conformally Invariant Variational Problems
• Free boundary minimal surfaces: a nonlocal approach