Timeline for Equivalence of alternative definitions of 'viscosity solution'
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 28, 2016 at 21:24 | vote | accept | CommunityBot | ||
| Oct 10, 2016 at 19:52 | comment | added | parsiad | Here's a reference for the local iff global argument: mathoverflow.net/questions/154542/… | |
| Oct 5, 2016 at 19:27 | comment | added | parsiad | Oh sorry. I misread your question. The answer handles "global max" $\iff$ "strict global max". I believe the idea is similar for "local max" $\iff$ "global max". Start with a test function in which a local max is admitted and try to turn that local max into a global max while maintaining all of the local properties. As for the density argument, start with a test function in $C^1$ and approximate it by $C^k$ functions via mollification. | |
| Oct 5, 2016 at 19:17 | comment | added | user99249 | Thanks for your reply. 1. Why is the global case already handled? 2. What kind of density argument are you thinking of? | |
| Oct 4, 2016 at 16:08 | comment | added | parsiad | @Kei: The global case is already handled by my answer in Q1. To answer your second question, intuitively, $\zeta$ is used to mollify the value of $\psi$ on $\partial B_{1}$ (where it is equal to $\varphi$) with its value on $\partial B_{2}$ (where it is equal to zero). The $C^k$ replacement can be obtained by a density argument. | |
| Oct 4, 2016 at 12:05 | comment | added | user99249 | Also, what is the motivation behind defining $\zeta$ that way? | |
| Oct 4, 2016 at 12:03 | comment | added | user99249 | Thank you. So this answer leaves open only the cases "global maximum" (or "minimum") and replacing $C^1$ with $C^k$, $1 < k \le \infty$. | |
| Oct 4, 2016 at 2:02 | history | answered | parsiad | CC BY-SA 3.0 |