additional difficulties with respect to the extension problem corresponding to the lower order fractional case $s\in(0,1)$ and consisting in a single equation associated with a Neumann boundary condition. In particular the proof of a monotonicity formula for is made quite delicate by the appearance in the derivative of the frequency $\mathcal N$ of a term of the type $$-r\int_{\partial B_r'}huv\,dS'+2\int_{B_r'}hu\,x\cdot
  \nabla_x v\,dx,$$ see Lemma \[mono\]. While in the lower order case we have only one component $u=v$ so that an integration by parts allows rewriting the above sum as an integral over $B_r'$, in the case of two components $u,v$ this is no more possible and an estimate of the integral over “the boundary of the boundary” $\int_{\partial B_r'}huv\,dS'$ is required. The method developed here to overcome this difficulty is based on estimates in terms of boundary integrals (see Lemma \[l:STIMA\_DI\_nu\_2\]) and represents one of the main technical novelty of the present paper in the context of monotonicity formulas; we think that this procedure could have future applications in the extension of some of the results of [@FF] to rough potentials, since it could avoid the integration by parts needed to write the above sum as an integral over $B_r'$, 