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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Martin kernel and Martin representation is what you are after. Positive solutions are: $$ u(x) = \int_{\partial B_r} \frac{(1 - |x|^2)^s}{|x - y|^N} \, \mu(dy) $$ for any positive measure $\mu$. Signed solution can also be of that form with signed $\mu$, but there are other (more singular) solutions, too.
In other words: take a classical harmonic function $v$ and multiply it by $(1 - |x|^2)^{s-1}$ to get a $(2s)$-harmonic function $u$.
This result is due to Hmissi:
- F. Hmissi, Fonctions harmoniques pour les potentiels de Riesz sur la boule unité, Expo. Math. 12(3) (1994), 281–288.
answered Feb 17, 2021 at 8:53
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$\begingroup$ Thank you. If $s=1$, and we assume that $u$ is radially symmetric we get $u(x) = C_1\frac{r}{N} + C_2 \frac{1}{r^{N-1}}$. Is there an analogue of this result for $s \in (0,1)$? In particular, it looks that $u(x)$ from your answer has always a singularity, while this does not for $C_2 = 0$ $\endgroup$ Commented Feb 17, 2021 at 8:57
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$\begingroup$ Radial solutions for $s = 1$ are just constants, and $u(x) = C_1 + C_2 |x|^{2-N}$ are solutions in the punctured ball $B_r \setminus \{0\}$. Analogous result is available for the fractional Laplace operator, $u(x) = C_1 |x|^{s-1} + C_2 G_B(x/r,0)$, where $G_B$ is the Green function for $(-\Delta)^s)$ in the unit ball. You may like to have a look at my survey: M. Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Vol. 1: Basic Theory, De Gruyter, Berlin, 2019, it lists many results for the ball. $\endgroup$ Commented Feb 17, 2021 at 9:05
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$\begingroup$ Thank you very much for the reference! What if we look for solutions in the form $u(x) = v(\rho)\omega$, with $\omega \in {S}^{N-1}$ and $\rho \in (0,r)$? For $s=1$, we have $v(\rho) = \frac{c_1}{N}\rho + \frac{c_2}{\rho^{N-1}}$. What is the fractional analogue of this expression? $\endgroup$ Commented Feb 17, 2021 at 9:19
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$\begingroup$ If I understand correctly — this is again very similar, by Bochner's relation. I discuss it briefly in section 3.4 of the survey linked above, and in more detail in a joint paper with Bartłomiej Dyda and Alexey Kunzetsov, Fractional Laplace operator and Meijer G-function, Constructive Approx. 45(3) (2017): 427–448, doi.org/10.1007/s00365-016-9336-4. The solution should be $v(\rho)=C_1 \rho(1-\rho)^{s-1} + C_2 \rho g(\rho)$, where $g(\rho)$ is the radial profile of $G_B(x,0)$ in dimension $N + 2$. $\endgroup$ Commented Feb 17, 2021 at 10:41
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$\begingroup$ Thank you! Could you add some more explanation about what is the Bochner's relation and how it produces the solution $v(\rho) = C_1 \rho(1-\rho)^{s-1} + C_2\rho g(\rho)$. By the way, is it correct to say that, if $v$ is also s-harmonic in $B_r$ then $C_2 \equiv 0$? $\endgroup$ Commented Feb 22, 2021 at 22:07