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Purpose
The purpose of this wiki is to stimulate research in elliptic and parabolic nonlocal equations, and to advertise the most interesting open problems in the area.
We would like to write a reference that is easy to read and navigate. It should be convenient to easily find the precise assumptions for which several theorems are proved (for example Harnack inequality or $C^{1,\alpha}$ estimates). It would also be desirable to have some indication of the methods used in the proofs, which may be a non rigorous idea, or an analogy with a classical theorem.
The emphasis of this wiki will be on nonlocal nonlinear (nonbananas?) equations of elliptic and parabolic type. There are several hyperbolic nonlocal equations that are described in the Dispersive wiki.
The ultimate goal of this wiki would be to prove for all equations that they are either well posed in the classical sense, or a weak solution and their possible singularities are well understood. The emphasis should then be on regularity results. We may also include some topics only indirectly related to regularity estimates (for example homogenization).
It would be desirable if the most important topics get a central role in the wiki. This is hard to do for many reasons, including that there is not always a consensus on what the most important results are. Depending on how things go, the wiki might have to take a position some day about what the priority topics are.
We should make an effort to negate the following myths:
- There are no new difficulties in nonlocal equations and everything is proved analogously as in the classical case.
- Nonlocal equations is a field in which one replaces the Laplacian by the fractional Laplacian in whatever equation and writes a paper.
- Nonlocal equations are bizarre and unnatural objects.
- Most equations in nature are local.
- All statements and proofs in nonlocal equations involve gigantic formulas.
Why a wiki?
If this wiki project works well, it may become a massive reference which is always up to date. It can potentially be better than a book. For that we need several people involved and willing to edit the articles.
Why would I spend time writing in this wiki?
If you are a mathematician who has done some research in the area, you definitely want people to know about your results. If you write an easy to read reference in this wiki, that would help more people know about your work and how it is related with other results in the area. Just be careful not to overplay the importance of your own results (or you will be banned from editing again). The appropriate thing to do is to write about all the related results by other people as much as you write about yours. Also remember to follow the rules below in the #Writing guidelines.
If you are a student learning the subject, writing in this wiki may help you understand the topics better (especially if someone comes after you to correct you). Moreover, if you are learning the subject, you probably appreciate the existence of this wiki more than others and are willing to contribute back.
what to do?
Eventually we will need to organize all result in categories and also write an intro to nonlocal equations as a starting point for dummies.
Right now, in Current Events there is a to do list. Click on the links and edit the pages. The red links denote that there is a page needed that was not even started.
There are several pages already. But the wiki is still in a very premature state. Most pages need some more work. The idea of having a wiki is that no version of a page will ever be a final version. However, right now they make that very apparent.
if you don't know how to start, you can use the pages that are already written as a sample.
Use the website http://zeteo.info/ to generate the references.
Writing guidelines
Here is some advice to take into account when writing an article.
- Do not offend anybody. A sentence like: "the last time Fourier analysis proved an important result in parabolic PDEs was when Fourier used it to solve the heat equation" or anything in that tone, would be totally unacceptable.
- Avoid using words like outstanding, remarkable, groundbreaking or tour de force when describing a result.
- If you think that an article is a triviality or is wrong, it is better not to include it in the citations.
Making a contribution to the wiki is fairly simple and it can take an arbitrarily small amount of time. Most of the articles are currently not perfect. You can add a paragraph here and there if you have little time. Or you can add a new article which just states a result and hope that someone will pick up the rest.
When writing an article, also keep the following priorities in mind.
- It has to be easy to read. This is the top priority.
- It should be clear what is proved and what is not. But see comment below.
- Avoid too much technicalities. If the assumptions of a general result are too complicated, it is ok to just list the major examples.
- Give references to the papers where theorems are proved.
- Explain the ideas of the proofs when appropriate.
- If a result is a nonlocal version of a classical theorem, mention it.
If you disagree with something that you find in an article, just edit it. If you think that your edition might be controversial, it would be convenient to write a note in the corresponding discussion page before.
Consult the User's Guide for information on using the wiki software.