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The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy

Filippo Gaia

Abstract

We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional analogous to the Hopf differential. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.

The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy

Abstract

We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional analogous to the Hopf differential. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.
Paper Structure (5 sections, 18 theorems, 118 equations)

This paper contains 5 sections, 18 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. The first inner variation
  3. The fractional Hopf differential
  4. A Noether theorem for the half Dirichlet energy
  5. Commutator estimates and fractional divergences

Key Result

Lemma 1.1

$u\in H^\frac{1}{2}(\partial D^2)$ is a stationary point of the half Dirichlet energy if and only if $\mathscr{H}_\frac{1}{2}(u)=0$.

Theorems & Definitions (37)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Definition 3.1
  • ...and 27 more