Timeline for Can we get rid of this test function?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2017 at 4:26 | comment | added | M. Rahmat | I have a question: will this $\phi$ be between 0 and 1? | |
| Sep 18, 2017 at 4:42 | comment | added | M. Rahmat | Just pick any test function that satisfies the conditions. Thanks. | |
| Sep 18, 2017 at 1:14 | comment | added | Johannes Hahn | Again: Is $\phi$ a test function or any test function? (The former meaning "you know exactly which one and we are not allowed to pick another", the latter meaning "it doesn't matter which one, just pick any test function that satisfies the conditions"). If it is the latter, than @MateuszKwaśnicki constructs the function like this: Fix $\phi_0:\mathbb{R}\to\mathbb{R}$ with support [-2,+2] and plateau [-1,+1]. Then use a scaled, rotationally symmetric version of $\phi_0$ as your $\phi$. Scaling with $\lambda$ introduces the factor $\lambda^{-2}$ in 2nd derivatives. Done. | |
| Sep 18, 2017 at 0:45 | history | edited | M. Rahmat | CC BY-SA 3.0 |
added 67 characters in body
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| Sep 18, 2017 at 0:43 | comment | added | M. Rahmat | $ \phi $ is a non-negative test function with compact support in $B$ that takes the value of 1 on $\overline{B(x,r)}$, or on any other set containing $\overline{B(x,r)}$ and contained in $B$. I didn't get the function $\phi$ you are talking about. Can you please explain? | |
| Sep 17, 2017 at 19:19 | comment | added | Mateusz Kwaśnicki | Is $\phi$ arbitrary? If yes, then you can simply choose $\phi$ in such a way that $|\Delta \phi| \leqslant C_1 (\operatorname{dist}(B^c, B(x, R)))^{-2}$ to get $|f(x)| \leqslant C_2 M (\operatorname{dist}(x, B^c))^{-2}$. | |
| Sep 17, 2017 at 18:58 | history | asked | M. Rahmat | CC BY-SA 3.0 |