Timeline for The Existence of PDE by Banach vs Leray-Schauder fixed point
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2022 at 17:59 | answer | added | Daniele Tampieri | timeline score: 2 | |
| Feb 9, 2022 at 13:14 | comment | added | mlk | Why use a screwdriver when a hammer is a much more intuitive tool? There are simply too many problems that cannot be written as a contraction and are thus inaccessible to Banach. On the other hand, there are enough problems where you are interested in uniqueness, so Leray-Schauder is no help. In some sense PDE-theory is all about having the right tool for the right job and knowing when to use it. Also, judging the importance of a tool by the amount of content in a book is never a good idea. You could equally argue that if Banach is easier to use, then it should take less pages to explain. | |
| Feb 9, 2022 at 11:21 | comment | added | Ben McKay | @Hannes: does the Banach fixed point theorem perhaps also allow you to work with fixed points? | |
| Feb 9, 2022 at 9:38 | comment | added | Hannes | Welcome to MO. My personal impression is that the LS theorem is popular because its usual formulation allows you to work with fixed points, so, in the PDE context, solutions to the problem at hand. This allows to use all sorts of trickery with (weak) formulations, clever testing, and the like, whose principles are very well established but still quite an art. (Usually this is summarized under a priori estimates, see also this nice explanation in a related question.) | |
| S Feb 9, 2022 at 6:40 | review | First questions | |||
| Feb 9, 2022 at 8:39 | |||||
| S Feb 9, 2022 at 6:40 | history | asked | Math The Novice | CC BY-SA 4.0 |