Timeline for A harmonic function degenerate in one direction
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 28, 2022 at 23:26 | vote | accept | Leo Moos | ||
| Nov 25, 2022 at 10:00 | history | edited | Leo Moos | CC BY-SA 4.0 |
removed the example and posted it as an answer
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| Nov 25, 2022 at 9:59 | answer | added | Leo Moos | timeline score: 3 | |
| Nov 24, 2022 at 21:26 | comment | added | Alexandre Eremenko | @Leo Moos: My second comment was not correct, and I deleted it. Anyway, you solved the problem yourself, probably inspired by the first (trivial) comment. | |
| Nov 24, 2022 at 20:07 | comment | added | Leo Moos | @AlexandreEremenko By the way, if you wanted to post your comments as an answer, I'd be happy to accept it! | |
| Nov 24, 2022 at 16:32 | comment | added | Leo Moos | @AlexandreEremenko Never mind, I think you can just take a linear combination of two homogeneous polynomials $P_m + \delta Q_n$ and take $\delta > 0$ small enough that no additional singular points are created. I worked out an example, see the most recent edit. | |
| Nov 24, 2022 at 16:30 | history | edited | Leo Moos | CC BY-SA 4.0 |
gave answer according to suggestions in the comments
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| Nov 24, 2022 at 15:52 | history | edited | Leo Moos | CC BY-SA 4.0 |
clarified question by adding equation
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| Nov 24, 2022 at 15:49 | comment | added | Leo Moos | @GiorgioMetafune I guess you can basically think of it as the lowest-order (non-zero) homogeneous harmonic polynomial $p$ so that $u = p + o(\lvert x \rvert^m)$, where $m$ is the degree of $p$. If I am not mistaken the tricky point is to show that this polynomial exists at all, which you do via monotonicity of the frequency functional. (This whole viewpoint makes Alexandre's comment all the more natural.) I'll edit the question to make this point a bit less opaque. | |
| Nov 24, 2022 at 15:42 | comment | added | Leo Moos | @GiorgioMetafune Here I meant the limit as $r \to 0$ of the functions $x \in B_1 \mapsto \lvert u (r \cdot) \rvert_{L^2(\partial B_1)}^{-1} u(rx)$, so the rescaling by the $L^2$-norm. Sorry it took me so long to reply to your comment - as I was typing it, Alexandre's comment popped up. | |
| Nov 24, 2022 at 15:37 | comment | added | Leo Moos | @AlexandreEremenko Of course, how embarrassing that I missed that - thanks for pointing this out! | |
| Nov 24, 2022 at 15:33 | comment | added | Alexandre Eremenko | The answer to the first question is negative: a counterexample is $u(x,y,x)=P_m(x,y)+Q_n(x,y,z),$ where $P_m,P_n$ are homogeneous harmonic polynomials of degrees $n>m$, and $P_n$ depends on all three variables. | |
| Nov 24, 2022 at 15:06 | comment | added | Giorgio Metafune | Could you define the homogenuous harmonic blow-up? | |
| Nov 24, 2022 at 10:14 | history | asked | Leo Moos | CC BY-SA 4.0 |