Timeline for Solution of the fractional Laplace equation on a ball
Current License: CC BY-SA 4.0
11 events
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| Sep 6, 2021 at 5:56 | comment | added | Mateusz Kwaśnicki | I am not aware of any explicit results for annular regions. Regarding the punctured ball, Bochner's relation and the Green function of a ball are discussed in both papers that I mentioned, what else do you need? | |
| Feb 23, 2021 at 18:00 | vote | accept | user173196 | ||
| Feb 23, 2021 at 18:00 | comment | added | user173196 | I put this additional question in another post. Let me know if you have any remarks: mathoverflow.net/questions/384757/… Thanks again for your help! | |
| Feb 22, 2021 at 22:42 | comment | added | user173196 | Thanks! Yes, in particular, I'd like to know how you use it to compute $v(\rho)$ if it seems to be about problems in the whole space | |
| Feb 22, 2021 at 22:30 | comment | added | Mateusz Kwaśnicki | Is there anything particular that you would like to ask about? Explaining Bochner's relation would not fit into this comment, I think. Regarding the other part of your comment: yes, if $C_2 \ne 0$, then $v$ is not harmonic near the origin. | |
| Feb 22, 2021 at 22:07 | comment | added | user173196 | 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 $C_2 \equiv 0$? | |
| Feb 17, 2021 at 10:41 | comment | added | Mateusz Kwaśnicki | 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 $N + 2$. | |
| Feb 17, 2021 at 9:19 | comment | added | user173196 | 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? | |
| Feb 17, 2021 at 9:05 | comment | added | Mateusz Kwaśnicki | 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. | |
| Feb 17, 2021 at 8:57 | comment | added | user173196 | 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 $C_2 = 0$ | |
| Feb 17, 2021 at 8:53 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |