Timeline for Ratio of measure of level region for harmonic functions
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2023 at 16:22 | comment | added | Alexandre Eremenko | The conjecture is that if both { z:u(z)>0} and { z: u(z)<0} are connected then your assertion about areas must be true. But if only one of them is connected, probably not. Both statements look plausible but difficult to prove. | |
| May 1, 2023 at 14:52 | comment | added | Re_0 | Yes! I have been wondering if there is any proof or counterexample of this case ( the positive part and negative part are both connected) | |
| Apr 28, 2023 at 12:05 | comment | added | Alexandre Eremenko | @Sam and Jim: do you also want $\{ z:u(z)<0\}$ to be connected? | |
| Apr 28, 2023 at 0:38 | vote | accept | Re_0 | ||
| Apr 28, 2023 at 0:27 | comment | added | Re_0 | Oh, you are right and I am afraid I forgot to assume $\{z: u(z)>0\}$ is also a region. | |
| Apr 27, 2023 at 18:12 | comment | added | Alexandre Eremenko | I repeat that the set $\{ z:u(z)>0\}$ is ALWAYS simply connected. | |
| Apr 27, 2023 at 16:03 | comment | added | Re_0 | Then the positive part is not simply connected in this example, which does not satisfy the condition. | |
| Apr 27, 2023 at 10:56 | comment | added | Alexandre Eremenko | Each component of the set $\{ z:u(z)>0\}$ for any harmonic function is simply connected; this followd from the Maximum/Minimum Principle. But the number of components can be large. | |
| Apr 27, 2023 at 4:16 | comment | added | Re_0 | But I'm wondering if the positive part $\{x\in B_1(0): u(x)>0\}$ of such a harmonic function $u(z)$ is always simply connected? | |
| Apr 26, 2023 at 13:00 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 104 characters in body
|
| Apr 26, 2023 at 12:51 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |