[Paper Review] The Neumann problem for the fractional Laplacian: regularity up to the boundary

Alessandro Audrito, Juan-Carlos Felipe-Navarro+1 more|arXiv (Cornell University)|Jun 17, 2020

Nonlinear Partial Differential Equations4 citations

TL;DR

This paper establishes the first boundary regularity results for the Neumann problem of the fractional Laplacian, proving that weak solutions are H"older continuous up to the boundary and, for $ s > \frac{1}{2} $, belong to the H"older space $ C^{2s-1+\alpha}(\overline{\Omega}) $. The authors develop a novel boundary Moser iteration with logarithmic corrections to overcome the challenges of nonlocality and lack of reflection symmetry.

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 $(-Δ)^s u=f$ in $Ω$, $\mathcal N_s u=0$ in $Ω^c$, then $u$ is $C^α$ up tp the boundary for some $α>0$. Moreover, in case $s>\frac12$, we then show that $u\in C^{2s-1+α}(\overlineΩ)$. 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.

Motivation & Objective

• To establish the first boundary regularity theory for the Neumann problem of the fractional Laplacian, where continuity and H"older regularity up to the boundary were previously unknown.
• To extend the regularity analysis to the regional fractional Laplacian, which models censored stochastic processes and has a different underlying energy structure.
• To overcome the fundamental difficulty that classical reflection techniques fail in the nonlocal setting due to the absence of odd/even symmetry.
• To develop a new boundary Moser iteration scheme incorporating logarithmic corrections to control oscillations near the boundary in nonlocal equations.
• To prove that $ C^{2s-1+\alpha} $ regularity is optimal, as demonstrated by the model solution $ |x_N|^{2s-1} $ in the half-space.

Proposed method

• Adapt the method of Moser iteration to the nonlocal Neumann setting, introducing logarithmic corrections in the boundary estimates to handle the singular kernel and nonlocal interactions.
• Use a variational formulation based on the energy functional $ \mathcal{E}(u) = \frac{c_{N,s}}{4} \iint_{\mathbb{R}^{2N} \setminus (\Omega^c)^2} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} \, dx\,dy - \int_\Omega f u $, which characterizes weak solutions.
• Define the nonlocal Neumann operator $ \mathcal{N}_s u(x) = c_{N,s} \int_\Omega \frac{u(x)-u(y)}{|x-y|^{N+2s}} \, dy $ for $ x \in \Omega^c $, ensuring conservation of mass and variational consistency.
• Establish a duality between the full and regional fractional Laplacians by proving that solutions to the Neumann problem for the regional operator satisfy the same regularity estimates.
• Prove that the boundary regularity threshold $ 2s-1 $ is sharp by constructing a model solution $ |x_N|^{2s-1} $ that solves the equation pointwise but is not a weak solution.
• Use integration by parts identities and kernel decomposition to relate the nonlocal Neumann problem to divergence-form equations with bounded measurable coefficients, enabling the application of advanced regularity techniques.

Experimental results

Research questions

• RQ1 What is the optimal boundary regularity for weak solutions to the Neumann problem $ (-\Delta)^s u = f $ in $ \Omega $, $ \mathcal{N}_s u = 0 $ in $ \Omega^c $, when $ f \in L^q(\Omega) $ with $ q > \frac{N}{2s} $?
• RQ2 Can the boundary regularity theory for the fractional Laplacian Neumann problem be extended to the regional fractional Laplacian, which arises from censored processes?
• RQ3 Why does classical reflection-based regularity theory fail in the nonlocal setting, and what new analytical tools are required to achieve boundary regularity?
• RQ4 Is the regularity threshold $ C^{2s-1+\alpha} $ optimal, and how does it relate to explicit solutions like $ |x_N|^{2s-1} $?
• RQ5 How can Moser iteration be adapted to the nonlocal boundary setting, and what role do logarithmic corrections play in controlling boundary oscillations?

Key findings

• For any bounded Lipschitz domain $ \Omega $, weak solutions $ u $ to the Neumann problem satisfy $ \|u\|_{C^{\alpha}(\overline{\Omega})} \leq C\left(\|f\|_{L^q(\Omega)} + \|u\|_{L^2(\Omega)}\right) $ for some $ \alpha > 0 $, provided $ f \in L^q(\Omega) $ with $ q > \frac{N}{2s} $ and $ \int_\Omega f = 0 $.
• When $ s > \frac{1}{2} $, $ q > N $, and $ \Omega $ is $ C^1 $, the solution belongs to $ C^{2s-1+\alpha}(\overline{\Omega}) $, with the norm controlled by $ \|f\|_{L^q(\Omega)} + \|u\|_{L^2(\Omega)} $.
• The regularity threshold $ 2s-1 $ is sharp, as demonstrated by the function $ |x_N|^{2s-1} $, which solves the equation pointwise in the half-space but is not a weak solution.
• The boundary Moser iteration technique is adapted to the nonlocal setting with logarithmic corrections, enabling control of oscillations near $ \partial\Omega $ despite the lack of reflection symmetry.
• The same regularity results hold for the Neumann problem associated with the regional fractional Laplacian, confirming the robustness of the method across different nonlocal operators.
• The nonlocal Neumann operator $ \mathcal{N}_s u $ satisfies a natural integration by parts formula and ensures conservation of mass, aligning with the classical Neumann problem in the limit $ s \to 1^- $.