Some Remarks on Pohozaev-Type Identities

The aim of this note is to discuss in more detail the Pohozaev-type identities that have been recently obtained by the author, Paul Laurain and Tristan Rivi\`ere in the framework of half-harmonic maps defined either on $R$ or on the sphere $S^1$ with values into a closed manifold $N^n\subset R^m$. Weak half-harmonic maps are critical points of the following nonlocal energy $$\int_{R}|(-\Delta)^{1/4}u|^2 dx~~\mbox{or}~~\int_{S^1}|(-\Delta)^{1/4}u|^2\ d\theta.$$ If $u$ is a sufficiently smooth critical point of the above energy then it satisfies the following equation of stationarity \frac{du}{dx}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $R$}~~\mbox{or}~~\frac{\partial u}{\partial \theta}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $S^1$.} By using the invariance of the equation of stationarity in $S^1$ with respect to the trace of the M\"obius transformations of the $2$ dimensional disk we derive a countable family of relations involving the Fourier coefficients of weak half-harmonic maps $u\colon S^1\to N^n.$ In the same spirit we also provide as many Pohozaev-type identities in $2$-D for stationary harmonic maps as conformal vector fields in $R^2$ generated by holomorphic functions.