Timeline for The first eigenfunction of fractional laplacian
Current License: CC BY-SA 4.0
8 events
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| Feb 11, 2021 at 22:54 | comment | added | Mateusz Kwaśnicki | (To clarify: in the above comment $\mathcal E$ is the quadratic form corresponding to $(-\Delta)^s$.) | |
| Feb 11, 2021 at 22:53 | comment | added | Mateusz Kwaśnicki | @mathqestion: I think that in the end step 2 is about weak (rather than pointwise) eigenfunctions. The Green operator can be defined in the weak sense by $\mathcal E(G_B u, v) = \langle u, v\rangle$ for all $u, v \in H_0^s(B)$ (with $G_B u$ itself in $H_0^s(B)$), and the harmonic reduction operator $P_B$ satisfies $\mathcal E(P_B u, v) = 0$ for all $u \in H^s$ and $v \in H_0^s(B)$ (with $P_B u$ equal to $u$ in the complement of $B$, and $P_B u$ in $H^s$). And $e_n$ is in $H^s$ and it is a weak eigenfunction in $\Omega$. This should be enough to get the initial identity in step 2, I think. | |
| Feb 11, 2021 at 13:56 | comment | added | mathqestion | May I ask a clarifying question? It looks like Steps 2 and 4 justify that any "pointwise-sense" eigenfunciton is a "weak-sense" eigenfunction. But how to show the converse (the exact OP's question)? In particular, can the displayed Green's formula be directly applied to a "weak-sense" eigenfunction? | |
| Nov 28, 2020 at 20:59 | comment | added | Mateusz Kwaśnicki | And once we know that $\int (-\Delta)^s e_n(x) \phi(x) dx = \lambda_n \int e_n(x) \phi(x) dx$ for all smooth $\phi$ compactly supported in $\Omega$, we conclude that $(-\Delta)^s e_n = \lambda_n e_n$ almost everywhere in $\Omega$. | |
| Nov 28, 2020 at 20:57 | comment | added | Mateusz Kwaśnicki | Regarding using Fubini's theorem to rearrange the integrals: I think I already did, in a greater generality, in an answer to a different question. | |
| Nov 24, 2020 at 18:22 | comment | added | inoc | can you give me the details of the point 4 in your answer please? | |
| Nov 5, 2020 at 22:15 | vote | accept | inoc | ||
| Nov 24, 2020 at 18:22 | |||||
| Nov 3, 2020 at 9:15 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |