The Neumann problem for the fractional Laplacian: optimal regularity via the Mellin transform
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| Format: | Preprint |
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2025
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| author | Dipierro, Serena Ros-Oton, Xavier Valdinoci, Enrico Weidner, Marvin |
| author_facet | Dipierro, Serena Ros-Oton, Xavier Valdinoci, Enrico Weidner, Marvin |
| contents | 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 $s\sim1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_13340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Neumann problem for the fractional Laplacian: optimal regularity via the Mellin transform Dipierro, Serena Ros-Oton, Xavier Valdinoci, Enrico Weidner, Marvin Analysis of PDEs 47G20, 35B65 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 $s\sim1$. |
| title | The Neumann problem for the fractional Laplacian: optimal regularity via the Mellin transform |
| topic | Analysis of PDEs 47G20, 35B65 |
| url | https://arxiv.org/abs/2510.13340 |