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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 (mostly semilinear).
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).
We should give priority in this wiki to the topics that we consider most important. A basic guidance of importance is fully nonlinear $\approx$ quasilinear > semilinear > linear. Unless a linear result has applications to nonlinear equations (for example Holder estimate with measurable coefficients), in which case linear > semilinear. Of course this is only a guideline (Navier-Stokes is semilinear).
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.
what to do?
Eventually we will need to organize all result in categories and also write an introduction 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.
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" might be true, but it would be inappropriate.
- Avoid using words like outstanding, remarkable, groundbreaking or magnum opus when describing a result.
- If you think that an article is a triviality or is wrong, do not include it in the citations. It is better to ignore than to criticize.
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.