Timeline for The first eigenfunction of fractional laplacian
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Dec 1, 2020 at 18:03 | history | bounty ended | CommunityBot | ||
| S Dec 1, 2020 at 18:03 | history | notice removed | CommunityBot | ||
| S Nov 23, 2020 at 16:46 | history | bounty started | inoc | ||
| S Nov 23, 2020 at 16:46 | history | notice added | inoc | Authoritative reference needed | |
| Nov 5, 2020 at 22:15 | vote | accept | inoc | ||
| Nov 24, 2020 at 18:22 | |||||
| Nov 3, 2020 at 9:15 | answer | added | Mateusz Kwaśnicki | timeline score: 2 | |
| Nov 3, 2020 at 8:05 | comment | added | inoc | There does not seem to be any proof in the article by Bañuelos and Kulczycki that eigenfunctions of $(-\Delta)^s$ are $C^\infty$. Can you suggest some articles where i can find this proof please? | |
| Nov 1, 2020 at 21:47 | comment | added | Mateusz Kwaśnicki | If I remember correctly, in the article by Bañuelos and Kulczycki it is proved that they are even $C^\infty$. | |
| Nov 1, 2020 at 7:28 | comment | added | inoc | Do you have some reference that proves that the eigenfunction of $(-\Delta)^s$ are $C^{2s+\epsilon}$? | |
| Oct 30, 2020 at 0:54 | comment | added | Mateusz Kwaśnicki | If using something more advanced (e.g. some regularity theory) is forbidden, I do not see a simple, direct proof. Even in order to write $(-\Delta)^s e_1(x)$ one needs $e_1$ to be at least, say, $C^{2s+\epsilon}$ near $x$. | |
| Oct 29, 2020 at 11:30 | comment | added | inoc | These articles are more than I need, and in my classroom note i have that only $e_1\in C(\mathbb{R}^n)$, can you give a me sketch of proof that $(-\Delta)^se_1(x)=\lambda_1e_1(x), \forall x\in\Omega$ holds? Please. | |
| Oct 29, 2020 at 11:13 | comment | added | Mateusz Kwaśnicki | Regarding the other question, you might like to have a look at my Ten equivalent definitions of the fractional Laplace operator, DOI:10.1515/fca-2017-0002. This is about definitions in all of $\mathbb R^n$, but still hopefully related. | |
| Oct 29, 2020 at 11:11 | comment | added | Mateusz Kwaśnicki | The eigenfunctions are known to be smooth in $\Omega$ (in fact, with no regularity conditions on the open set $\Omega$), and equation $(-\Delta)^s e_n(x) = \lambda_{n,s} e_n(x)$ indeed holds pointwise. If you just need a reference, I think the article The Cauchy process and the Steklov problem by Rodrigo Bañuelos and Tadeusz Kulczycki (JFA 2004, DOI:10.1016/j.jfa.2004.02.005) is a good source. | |
| Oct 29, 2020 at 10:07 | history | asked | inoc | CC BY-SA 4.0 |