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

Abstract

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 $C^\infty$.