  This paper is devoted to the Laplacian operator of fractional order $s\in
(0,1)$ in several dimensions. We first establish a representation formula for
the partial derivatives of the solutions of the homogeneous Dirichlet problem.
Along the way, we obtain a Pohozaev-type identity for the fractional Green
function and of the fractional Robin function. The latter extends to the
fractional setting a formula obtained by Br\'ezis and Peletier, see
\cite{Bresiz}, in the classical case of the Laplacian. As an application we
consider the particle system extending the classical point vortex system to the
case of a fractional Laplacian. We observe that, for a single particle in a
bounded domain, the properties of the fractional Robin function are crucial for
the study of the steady states. We also extend the classical Hadamard
variational formula to the fractional Green function as well as to the shape
derivative of weak solution to the homogeneous Dirichlet problem. Finally we
turn to the in homogeneous Dirichlet problem and extend a formula by J.L.
Lions, see \cite{Lions}, regarding the kernel of the reproducing kernel Hilbert
space of harmonic functions to the case of $s$-harmonic functions. We observe
that, despite the order of the operator is not $2$, this formula looks like the
Hadamard variational formula, answering in a negative way to an open question
raised in \cite{ELPL}.
