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user173196
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What is the expression of the (non $u \equiv 0$) solutions to \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} where $$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy, $$$$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}} dy, $$ ($0<s<1$) is the fractional Laplacian?

What is the expression of the (non $u \equiv 0$) solutions to \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} where $$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy, $$ ($0<s<1$) is the fractional Laplacian?

What is the expression of the (non $u \equiv 0$) solutions to \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} where $$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}} dy, $$ ($0<s<1$) is the fractional Laplacian?

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user173196
user173196
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Solution of the fractional Laplace equation on a ball

What is the expression of the (non $u \equiv 0$) solutions to \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} where $$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy, $$ ($0<s<1$) is the fractional Laplacian?