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

2021/06/14 by Mouhamed Moustapha Fall, Xavier Ros‐Oton, Fall, Mouhamed Moustapha +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2106.07593

openalex publication_date 2021/06/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We study the regularity of minimizers of the functional \mathcal E(u):= [u]_H^s(Ω)² +∫_Ωfu. This corresponds to understanding solutions for the regional fractional Laplacian in Ω⊂\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(Ω) (i.e., Neumann problem), or in the case of Dirichlet condition u∈ H^s₀(Ω) when s>\frac12. Our main result establishes the sharp regularity of solutions in both cases: u∈ C^2s+α(Ω) in the Neumann case, and u/δ^2s-1∈ C^1+α(Ω) in the Dirichlet case. Here, δ is the distance to ∂Ω, and α1. We also show the optimality of our result: these estimates fail for α>α_s, even when f and ∂Ω are C^∞.

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