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 $u\in H_0^s(\Omega )$ when $s>\frac{1}{2}$. 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 $C^{\infty }$.