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Consider the spectral fractional laplacian in a unit ball \begin{equation} \label{a1} \left\{\begin{aligned} (-\Delta)^{s} u &=f(r) &&\text{in } B, \\ u &=0 &&\text{on } \partial B \end{aligned} \right. \end{equation} Is it possible to write a radial solution of $u(r)$ in the closed form.

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  • $\begingroup$ You have for $s\in (0,1)$, $$ (-\Delta) u=(-\Delta)^{1-s}(-\Delta)^{s} u=(-\Delta)^{1-s} f(r)=f_s(r), $$ where $f_s$ is also radial (check the Fourier transform). You can find $u$ radial such that $(-\Delta) u=f_s(r)$ with $u$ vanishing on the unit sphere: it is a good candidate. $\endgroup$ – Bazin May 16 '18 at 19:54
  • $\begingroup$ I do not think there is an explicit expression neither for the kernel of $(-\Delta_B)^s$ nor for its Green operator $(-\Delta_B)^{-s}$. Even in dimension one the expression for both objects involves polylogarithms, if I am not mistaken. $\endgroup$ – Mateusz Kwaśnicki May 16 '18 at 20:30

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