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'''MediaWiki has been successfully installed.'''
We need to come up with some organization for the articles.


Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.
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.


== Getting started ==
* 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.
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
 
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* 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.)
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]
 
<!-- some LaTeX macros we want to use: -->
* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]].
$
 
  \newcommand{\Re}{\mathrm{Re}\,}
* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].
  \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
 
$
* Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].
   
 
We consider, for various values of $s$, the $n$-dimensional integral
* A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]].
\begin{align}
 
  \label{def:Wns}
* Some discussion on models involving [[Levy processes]] and [[stochastic control]].
  W_n (s)
 
  &:=
* Some references to equations from fluids including the [[surface quasi-geostrophic equation]].
  \int_{[0, 1]^n}
 
    \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
* A page about [[semilinear equations]] including the [[surface quasi-geostrophic equation]] and also some form of KPP.
\end{align}
 
which occurs in the theory of uniform random walk integrals in the plane,
* [[Nonlocal porous medium equation]] and its more regular sister, the [[Aggregation equation]].
where at each step a unit-step is taken in a random direction.  As such,  
 
the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
* [[drift-diffusion equations]].
to the origin after $n$ steps.
 
* [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]] (specially in connection to [[nonlocal image processing]]).
By experimentation and some sketchy arguments we quickly conjectured and
 
strongly believed that, for $k$ a nonnegative integer
* Fractional curvatures in conformal geometry.
\begin{align}
 
  \label{eq:W3k}
* 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]]..)
  W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
 
\end{align}
* 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.
Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.  
 
The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained
* There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on.
at the end of the paper.
 
* Pages about [[Homogenization]] (local and nonlocal) should appear here too.
 
* Limit of $\sigma$-order operators as $\sigma \to 2^-$, maybe this should go on a page on [[Linear integro-differential operators]].
 
* 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]].
 
* 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...
 
*Fill up the list of [[upcoming events]] such as conferences, workshops, summer schools.
 
* I have started a [[Literature on Nonlocal Equations]] to dump there all papers we want to reference or are already referencing on the wiki.
 
* Having a [[list of equations]] may make it easier to navigate the wiki.

Revision as of 17:45, 4 March 2012

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.
  • 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.)
  • 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.
  • 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...

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