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== Things that need to be done ==
To see the contents of this wiki, start on [[starting page|this page]].


There is also a list of topics in the [[Mwiki:Current events|"to do" list]].
We need to come up with some organization for the articles.
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The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki.


* We better start thinking hard about writing the [[Introduction to nonlocal equations]]. We gotta start somewhere, so any random idea or small thing you want to write should go in there. This is high priority, since if someone reads one page of this wiki, it will likely be this one.


== Purpose ==
* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].


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.
* Fractional curvatures in conformal geometry.  


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.
* We need  to explain further the  [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geometric Scattering Theory]], [[Ambient Metric Construction]], [[GJMS Operators]] and [[Singular Yamabe Problem]]..)


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 [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page Dispersive wiki].
* It would be wise (once the wiki is more mature) to add pages about the [[Boltzmann equation]], since it is one of the more "classical" and better known integro-differential equations.


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).
* Pages about [[Homogenization]] (local and nonlocal) should appear here too.


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.
* Limit of $\sigma$-order operators as $\sigma \to 2^-$, maybe this should go on a page on [[Linear integro-differential operators]].


We should make an effort to negate the following '''myths''':
* Given the recent works of Osher/Gilboa and Bertozzi/Flenner on Ginzburg-Landau  on graphs we should have an article on the natural similarities between non-local operators and [[elliptic operators on graphs]].
# 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? ==
* [[nonlocal image processing]]
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 on this wiki? ==
* [[Aggregation equation]]
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.
== (partially) Completed tasks ==


== what to do? ==
* A sort of [[Starting page]] that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.)


Eventually we will need to organize all result in categories and also write an [[intro to nonlocal equations]] as a starting point for dummies.
* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]].


Right now, in [[Mwiki:Current events|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.
* Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].


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.
* A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]].


if you don't know how to start, you can use the pages that are already written as a sample.
* Some discussion on models involving [[Levy processes]] and [[stochastic control]].


Use the website http://zeteo.info/ to generate the references.
* Some references to equations from fluids including the [[surface quasi-geostrophic equation]].


== Writing guidelines ==
* A page about [[semilinear equations]] including the [[surface quasi-geostrophic equation]] and also some form of KPP.


Here is some advice to take into account when writing an article.
* [[Nonlocal porous medium equation]].
# '''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.
* [[drift-diffusion equations]].


When writing an article, also keep the following priorities in mind.
* [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]].
# 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.
* There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on.


Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.
* Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
 
* We may want to include a [[mini second order elliptic wiki]] inside this wiki.
 
* We may want to have a section on [[open problems]], for example integral ABP for general kernels, harnack for kernels with a
less restrictive bound below, $C^{1,\alpha}$ estimate in bounded domains with a nonsmooth kernel, supercritical quasi-geostrophic,
classification of nonlocal minimal cones, etc...
 
* Having a [[list of equations]] may make it easier to navigate the wiki.
 
== Other ideas ==
 
 
* Fill up the list of [[upcoming events]] such as conferences, workshops, summer schools.
 
* [[User:Nestor|Nestor]] has started a [[Literature on Nonlocal Equations]] to dump there all papers we want to reference or are already referencing on the wiki.

Revision as of 15:51, 12 March 2012

Things that need to be done

We need to come up with some organization for the articles.

The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki.

  • We better start thinking hard about writing the Introduction to nonlocal equations. We gotta start somewhere, so any random idea or small thing you want to write should go in there. This is high priority, since if someone reads one page of this wiki, it will likely be this one.
  • Fractional curvatures in conformal geometry.
  • It would be wise (once the wiki is more mature) to add pages about the Boltzmann equation, since it is one of the more "classical" and better known integro-differential equations.
  • Pages about Homogenization (local and nonlocal) should appear here too.
  • Given the recent works of Osher/Gilboa and Bertozzi/Flenner on Ginzburg-Landau on graphs we should have an article on the natural similarities between non-local operators and elliptic operators on graphs.

(partially) Completed tasks

  • A sort of Starting page that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.)
  • Phase transitions involving non-local interactions, in particular, pages about Particle Systems, discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
  • We may want to have a section on open problems, for example integral ABP for general kernels, harnack for kernels with a

less restrictive bound below, $C^{1,\alpha}$ estimate in bounded domains with a nonsmooth kernel, supercritical quasi-geostrophic, classification of nonlocal minimal cones, etc...

Other ideas

  • Fill up the list of upcoming events such as conferences, workshops, summer schools.
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