Some Remarks on Pohozaev-Type Identities

Abstract

In this note we present some Pohozaev-type identities that have been recently established in a joint work with Paul Laurain and Tristan Rivière in the framework of half-harmonic maps defined either on the real line or on the unit circle with values into a closed n-dimensional manifold. Weak half-harmonic maps are defined as critical points of the so-called half Dirichlet energy.

By using the invariance of the half Dirichlet energy with respect to the trace of the Möbius transformations we derive a countable family of relations involving the Fourier coefficients of weak half-harmonic maps. We also present a short overview of Pohozaev formulas in 2-D in connection with Noether's theorem.