Timeline for Concentration compactness. Can this concept be stated in a theorem?
Current License: CC BY-SA 3.0
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| Aug 26, 2017 at 2:59 | comment | added | Terry Tao | See also Klainerman's essay "PDE as a unified subject", explaining how partial differential equations are best approached not from the perspective of maximum generality, but rather from studying families of PDE clustered around fundamental examples (e.g. heat equation, wave equation, etc..). The same is often true for the tools used in the analysis of PDE. web.math.princeton.edu/~seri/homepage/papers/telaviv.pdf | |
| Aug 26, 2017 at 2:53 | comment | added | Terry Tao | I think you may find Gowers' "two cultures" distinction to be useful here: dpmms.cam.ac.uk/~wtg10/2cultures.pdf . Like most other useful things in the analysis of PDE, concentration compactness is definitely a concept coming from the "problem solving" culture of mathematics rather than the "theory building" culture. | |
| Aug 25, 2017 at 21:27 | comment | added | Zinkin | thank you, I did not want to bring heat into this debate, I was just curious to understand how far developed this concept is. | |
| Aug 25, 2017 at 12:56 | history | edited | Dirk | CC BY-SA 3.0 |
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| Aug 25, 2017 at 12:38 | comment | added | Dirk | But such a discussion is exactly what Tao does right before the $l^1$ example starts. If this is not the direction you want, I have no idea what you are looking for. My guess is that the answer to "To which spaces/sets can this concept be applied so that everything works out?" is something like: "You have to check in every case and it may get cumbersome…" | |
| Aug 25, 2017 at 12:08 | comment | added | Zinkin | My question is: To which spaces/sets can this concept be applied so that everything works out? Neither Tao's $l^1$ discussion nor your quote seem to provide an answer to that question. Could you please elaborate on that point? All I want is a discussion of this concept that is not an example. | |
| Aug 25, 2017 at 11:30 | history | answered | Dirk | CC BY-SA 3.0 |