Global Schauder theory for minimizers of the Hs(Ω) energy

Abstract

We study the regularity of minimizers of the functional E(u):=[u]Hs(Ω)2+∫Ωfu. This corresponds to understanding solutions for the regional fractional Laplacian in Ω⊂RN. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in Hs(Ω) (i.e., Neumann problem), or in the case of Dirichlet condition u∈H0s(Ω) when [Formula presented]. Our main result establishes the existence of a constant αs∈(0,1−s) satisfying 2s+αs>1 such that for all α∈(0,αs) the solution u∈C2s+α(Ω‾) in the Neumann case, and u/δ2s−1∈C1+α(Ω‾) in the Dirichlet case. Here, δ is the distance to ∂Ω. We also show the optimality of our result: these estimates fail for α>αs, even when f and ∂Ω are C∞. © 2022 Elsevier Inc.