The Neumann problem for the fractional Laplacian: optimal regularity via the Mellin transform

TL;DR

This work resolves the optimal boundary regularity for the nonlocal Neumann problem for the fractional Laplacian $(-\Delta)^s$ by marrying 1D Liouville theory with Mellin-transform techniques and transferring the information to higher dimensions through compactness and carefully constructed boundary corrections. A central contribution is a meta-theorem: the classification of 1D solutions to homogeneous equations $Lu=0$ is governed by the zeros of an explicit meromorphic symbol $f$, which yields precise regularity exponents $B_0$ that determine the best possible $C^{\cdot}$-regularity in $\Omega$ (and the corresponding boundary condition $\partial_{\nu} u=0$ when applicable). The authors then develop a distributional Mellin framework to rigorously justify the 1D–multi-dimensional reduction, prove Liouville theorems for both single and paired operators, and apply these results to obtain the optimal boundary Schauder-type estimates for the Neumann problem. In particular, for $s\le 1/2$ all solutions are $C^{2s+\alpha}$ up to the boundary, while for $s\ge 1/2$ they are $C^{s+1/2+\alpha}$, with $\partial_{\nu}u=0$ on $\partial\Omega$ when $2s+\alpha>1$, and these exponents are sharp. The analysis also uncovers highly oscillatory 1D solutions of the form $u(x)=x^{a}\cos(b\log x)$, reflecting rich structure in the nonlocal Neumann problem that is absent in the Dirichlet setting.

Abstract

We establish the optimal regularity of solutions to the Neumann problem for the fractional Laplacian, $(-Δ)^s u=h$ in $Ω$, with the external condition $\mathcal N^s u=0$ in $Ω^c$. For this, a key point is to establish a 1D Liouville theorem for functions with growth, which we prove by using complex analysis and the Mellin transform. More precisely, we prove a ``meta-theorem'' relating the classification of 1D solutions to general linear homogeneous equations of the type $Lu=0$ in $(0,\infty)$ to the (complex) roots of an explicit meromorphic function $f(z)$ that depends on $L$. In case of the fractional Laplacian with Neumann conditions, we show that all solutions are $C^{2s+α}$ when $s\leq 1/2$, and $C^{s+\frac12+α}$ when $s\geq1/2$. Moreover, quite surprisingly, we prove that even in 1D there exist highly oscillating solutions of the type $u(x)=x^{a} \cos(b \log x)$ for $x>0$, with $a>0$ and $b>0$ that depend on $s$, and $a<2s$ for $s\sim1$.

Theorems & Definitions (136)

• theorem 1
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