2025/10/15 by Serena Dipierro, Dipierro, Serena, Xavier Ros‐Oton +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2510.13340
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 Ns u=0 in Ωc. 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,∞) 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 C2s+α when s≤ 1/2, and Cs+\frac12+α when s≥1/2. Moreover, quite surprisingly, we prove that even in 1D there exist highly oscillating solutions of the type u(x)=xa cos(b log x) for x>0, with a>0 and b>0 that depend on s, and a<2s for s∼1.