Finding a solutions for an equation

fa.practical evaluation – Fractional Laplacian equation on a ball and categorical options retort

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fa.practical evaluation – Fractional Laplacian equation on a ball and categorical options

Let us assume
commence{align*}
(-Delta)^s u &= 0 && x in B_r(0)
u&=0 && x in mathbb R^N setminus B_r(0),
stop{align*}

the place $$
(-Delta)^s u(x) = int_{mathbb{R}^N} frac{u(x)-u(y)}{|x-y|^{N+2s}} dy,
$$
($0<s<1$)
is the fractional Laplacian. How can I discover options of this drawback within the design $v(r)omega$, with $omega in S^{N-1}$?

In a remark on Solution of the fractional Laplace equation on a ball, it was stated that the answer in $B_r(0) setminus {0}$ is
$$v(rho) = C_1rho(1-rho)^{s-1} + C_2 rho g(rho),$$ the place $g(rho)$ is the radial silhouette of the Green’s responsibility $G(x,0)$ in dimension $N+2$ and that to carry an answer of the issue on $B_r(0)$, we too necessity $C_2 equiv 0$. Is this rectify? How assassinate you show it?

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