Timeline for Extremals versus minima for variational problems
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 14, 2012 at 1:04 | vote | accept | Colin McLarty | ||
| Apr 11, 2012 at 13:37 | answer | added | Robert Bryant | timeline score: 6 | |
| Apr 10, 2012 at 22:00 | comment | added | Deane Yang | Yes, I agree about the need to assume sufficient regularity for the solution. | |
| Apr 10, 2012 at 21:36 | comment | added | Rbega | I think one also needs to be careful about the regularity of the class of solutions. For instance if $\Sigma$ is a branched minimal surface in $\Real^3$ then $\Sigma$ locally minimizes area away from the branch set but cannot minimize area in any neighborhood of a branch point. | |
| Apr 10, 2012 at 19:26 | comment | added | Deane Yang | On the other hand, I think if the energy functional satisfies an appropriate ellipticity or convexity condition, then any critical point is "locally minimal". I think the same argument used for geodesics can be used more generally. | |
| Apr 10, 2012 at 18:54 | comment | added | alvarezpaiva | I don't think this "local minimality" is really general. I'm thinking of the variational problem that defines Reeb orbits for a given contact form: it takes the form $\gamma \mapsto \int_\gamma \alpha$, where $\alpha$ is the contact $1$-form and so pieces of $\gamma$ don't seem to be maxima nor minima. For $1$-dimensional problems at least convexity/ellipticity of the integrand seems to play a role. Note: I'm writing this off the top of my head so please check. | |
| Apr 10, 2012 at 18:00 | history | asked | Colin McLarty | CC BY-SA 3.0 |