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We study the regularity of minimizers of the functional $\mathcal E(u):=[u]_{H^s(\Omega)}^2 +\int_\Omega fu$. This corresponds to understandingsolutions for the regional fractional Laplacian in $\Omega\subset\mathbb R^N$.More precisely, we are interested on the global (up to the boundary) regularityof 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 ourresult: these estimates fail for $\alpha>\alpha_s$, even when $f$ and$\partial\Omega$ are $C^\infty$.

Global Schauder theory for minimizers of the $H^s(\Omega)$ energy