Timeline for Harnack inequality for the minimal surface equation
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 2 at 15:04 | comment | added | Aidan Backus | On further inspection, a bunch of the comments here seem to have been deleted, so that example is now rather contextless.... but the point of it really is that the OP's conjecture is false. | |
| Dec 2 at 14:40 | comment | added | Aidan Backus | @gaoqiang The Harnack inequality definitely has applications for minimal hypersurfaces, provided that you have an a priori bound on their extrinsic curvatures, so I would not call it "almost useless". But as I already said, the OP's original conjecture that its constant only depends on n is false -- that was the whole point of including that example. | |
| Dec 2 at 12:42 | comment | added | gaoqiang | Almost useless, since the constant of Harnack constant depends on the elliptic constant, and in your arguments, it depends on the function u itself. | |
| Aug 18, 2023 at 0:35 | vote | accept | user88544 | ||
| Aug 18, 2023 at 0:34 | comment | added | user88544 | Sorry, I misunderstood your example. You are right! | |
| Aug 17, 2023 at 15:47 | comment | added | Aidan Backus | I think by looking at pictures of half-catenoids it is pretty easy to geometrically see why this is a counterexample. As $\varepsilon \to 0$, the neck gets thinner and thinner, while the wider part gets fatter and fatter. The Harnack constant measures the ratio of the fatness to the thinness, and as $\varepsilon \to 0$, the neck shrinks to a point while the wide part becomes infinitely wide. | |
| Aug 17, 2023 at 15:45 | comment | added | Aidan Backus | @user88544 A catenary curve is given by the graph of cosh, not inverse cosh -- I don't even know how inverse cosh would come up in this situation. | |
| Aug 17, 2023 at 1:04 | comment | added | Aidan Backus | @user88544 The point I was trying to make is that on the linear level, Harnack's inequality depends on the ellipticity, so on the quasilinear level, the same thing had better be true. However, this was kind of hazy and worded confusingly by me, so I am sorry for that. I have added a more explicit counterexample. The Harnack inequalities for the catenoids blow up as $\varepsilon \to 0$. | |
| Aug 17, 2023 at 1:03 | history | edited | Aidan Backus | CC BY-SA 4.0 |
Adding an explicit counterexample
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| Aug 14, 2023 at 20:47 | history | edited | Aidan Backus | CC BY-SA 4.0 |
fix a typo
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| Aug 14, 2023 at 20:36 | history | answered | Aidan Backus | CC BY-SA 4.0 |