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

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 $Omegasubsetmathbb R^N$. 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 $uin H^s_0(Omega)$ when $s>frac12$. Our main result establishes the sharp regularity of solutions in both cases: $uin C^{2s+alpha}(overlineOmega)$ in the Neumann case, and $u/delta^{2s-1}in C^{1+alpha}(overlineOmega)$ in the Dirichlet case. Here, $delta$ is the distance to $partialOmega$, and $alpha<alpha_s$, with $alpha_sin (0,1-s)$ and $2s+alpha_s>1$. We also show the optimality of our result: these estimates fail for $alpha>alpha_s$, even when $f$ and $partialOmega$ are $C^infty$.