The Neumann problem for the fractional Laplacian: regularity up to the boundary

Abstract

We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if u is a weak solution of (−∆) s u = f in Ω, Nsu = 0 in Ω c , then u is C α up tp the boundary for some α > 0. Moreover, in case s > 1 2 , we then show that u ∈ C 2s−1+α (Ω). To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections.Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result.Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.

Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement

“…In Section V we have discussed the nonlocal Neumann condition proposal of Ref. [15][16][17], presenting an explicit construction of the related random walk, in conjunction with the asymptotic pdf data. By theory of [15], the pertinent pdf should be uniform in the interval.…”

“…In the present section, we shall outline rudiments of another "reflecting" framework for Lévy processes in the interval, with the fractional heat (Fokker-Planck) dynamics leading asymptotically to the uniform distribution. Its major ingredient is the so-called nonlocal Neumann condition [15][16][17]20]. We note that there are other Neumann condition proposals in existence, [13,14] and [12], but transparent probabilistic pictures, amenable to a computer-assisted (path-wise) verification, appear to be lacking.…”

“…Conditions for the existence of the regional Laplacian for all x ∈ D, need to be carefully set. For 1 ≤ α < 2, the existence of the regional Laplacian for all x ∈ ∂D, is granted if and only if a derivative (a non-conventional Neumann condition, that is adapted to the nonocal setting) of a each function in the domain in the inward direction vanishes, [13,14,16].…”

Finite element approximation of fractional Neumann problems

“…Indeed, a crucial aspect in the numerical analysis of differential equations is the regularity of solutions. Reference [10] studies the Hölder regularity of solutions to (1.1) whenever α = 0 and g ≡ 0. However, to the best of our knowledge, there are no Sobolev regularity estimates for Neumann problems involving the integral fractional Laplacian in the literature.…”

Nonlocal equations with fractional order dependence

“…The main difficulty in the proof of Theorem 3.1.3 is the lack of boundedness and regularity estimates for the renormalized regional fractional Laplacian D s Ω which are uniform in s ∈ (0, 1). In fact, even for fixed s ∈ (0, 1), the elliptic boundary regularity theory for this operator has only been developed very recently with regularity estimates containing s-dependent constants, see [14,73,76]. For the proof of Theorem 3.1.3, we need to consider uniform L ∞ -estimates related to the operator family D s Ω , s ∈ [0, 1) first.…”