  We study the regularity of minimizers of the functional $\mathcal E(u):=
[u]_{H^s(\Omega)}^2 +\int_\Omega fu$. This corresponds to understanding
solutions for the regional fractional Laplacian in $\Omega\subset\mathbb R^N$.
More precisely, we are interested on the global (up to the boundary) regularity
of solutions, both in the case of free minimizers in $H^s(\Omega)$ (i.e.,
Neumann problem), or in the case of Dirichlet condition $u\in H^s_0(\Omega)$
when $s>\frac12$.
  Our main result establishes the sharp regularity of solutions in both cases:
$u\in C^{2s+\alpha}(\overline\Omega)$ in the Neumann case, and
$u/\delta^{2s-1}\in C^{1+\alpha}(\overline\Omega)$ in the Dirichlet case. Here,
$\delta$ is the distance to $\partial\Omega$, and $\alpha<\alpha_s$, with
$\alpha_s\in (0,1-s)$ and $2s+\alpha_s>1$. We also show the optimality of our
result: these estimates fail for $\alpha>\alpha_s$, even when $f$ and
$\partial\Omega$ are $C^\infty$.
