English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this work we derive Noether Theorems for energies of the form\begin{equation*} E(u)=\int_\OmegaL\left(x,u(x),(-\Delta)^\frac{1}{4}u(x)\right)dx \end{equation*} forLagrangians exhibiting invariance under a group of transformations actingeither on the target or on the domain of the admissible functions $u$, in termsof fractional gradients and fractional divergences. Here $\Omega$ stays eitherfor an Euclidean space $\mathbb{R}^n$ or for the circle $\mathbb{S}^1$. We thendiscuss some applications of these results and related techniques to the studyof nonlocal geometric equations and to the study of stationary points of thehalf Dirichlet energy on $\mathbb{S}^1$. In particular we introduce the$\frac{1}{2}$-fractional Hopf differential as a simple tool to characterizestationary point of the half Dirichlet energy in$H^\frac{1}{2}(\mathbb{S}^1,\mathbb{R}^m)$ and study their properties. Finallywe show how the invariance properties of the half Dirichlet energy on$\mathbb{R}$ can be used to obtain Pohozaev identities.

Noether Theorems for Lagrangians involving fractional Laplacians