To Do List: Difference between revisions

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We need to come up with some organization for the articles.
== Things that need to be done ==


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.
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.
* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].
 
* Fractional curvatures in conformal geometry.
 
* Create a page on [[Non-local Lagrangian | Non-local Lagrangians]] and mention relevant examples, note Non-local Lagrangian is a term used by particle physicists and it is not necessarily the same as a [[Dirichlet form]].
 
* 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]]..)
 
* It would be good to have an article on the natural similarities between non-local operators and [[elliptic operators on graphs]].
 
* [[nonlocal image processing]].
 
* [[Aggregation equation]].
 
* [[Keller-Segel equation]].
 
* Improve the page on [[Active scalar equations]].
 
* [[Ising Models With Nonlocal Interactions]]
 
* [[Nonlocal Interaction Energies]]
 
* [[Nonlocal Phase Field Equations]] this will include many types of evolution equations such as nonlocal reaction-diffusion, models arising from nonlocal ising models, nonlocal cahn-hilliard, nonlocal biological aggregation, sharp interface limits (giacomin-lebowitz, bates-et-al, de massi-pressutti, etc...)
 
* [[Boundary Harnack inequality]] it would be good to possibly include this in the description of what is different/same regarding local vs. nonlocal equations.
 
* Describe, with references, the connection between [[holder estimates]] of viscosity solutions of stationary and or parabolic equations and the existence and uniqueness of solutions to the relevant [[Martingale Problem]]
 
* [[Neumann Boundary Conditions]]
 
* Processes in the "limit" $\alpha=0$, [[Zero Order Operators]]
 
* Discussion of the [[Barlow-Bass-Chen-Kassmann Counterexample]] to Regularity of $L$-Harmonic functions.
 
* Discussion of Jean Picard's paper (regularity for pure jump processes with non-degenerate, translation invariant generators)
 
* Add Courrège's Structure theorem for operators and other structure theorems (generators of Feller semigroups, Hunt processes... etc)
 
* A page on the nonlocal analogue of the thin film equation
 
== (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.)
* 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.)
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* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]].
* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]].


* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].
* Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Aleksandrov-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].
 
* Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].


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


* Some discussion on [[models]] involving [[Levy processes]] and [[stochastic control]].
* Some discussion on models involving [[Levy processes]] and [[stochastic control]].


* Some references to equations from fluids including the [[surface quasi-geostrophic equation]].
* The [[surface quasi-geostrophic equation]].


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


* [[Nonlocal porous medium equation]] and its more regular sister, the [[Aggregation equation]].
* [[Nonlocal porous medium equation]].


* [[drift-diffusion equations]].
* [[drift-diffusion equations]].


* [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]] (specially in connection to [[nonlocal image processing]]).
* [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]].


* Fractional curvatures in conformal geometry.  
* There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on.


* 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]]..)
* Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
 
* 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.
 
* There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on.


* Pages about [[Homogenization]] (local and nonlocal) should appear here too.
* It is convenient to have a [[mini second order elliptic wiki]] inside this wiki.


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


* 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]].
* Having a [[list of equations]] may make it easier to navigate the wiki.


* Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
* Jump processes via [[Subordination]]


* We may want to include a [[mini second order elliptic wiki]] inside this wiki.
== Wiki Maintenance ==


* We may want to have a section on [[open problems]], for example integral ABP for general kernels, harnack for kernels with a
* Get access to LocalSettings.php (ask Maorong for access maybe?) so we can change the page name of the Main Page and the Starting Page (specially since those are the names shown by a google search)
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.
* 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.
* [[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.

Latest revision as of 15:11, 25 January 2016

Things that need to be done

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.

  • Fractional curvatures in conformal geometry.
  • Create a page on Non-local Lagrangians and mention relevant examples, note Non-local Lagrangian is a term used by particle physicists and it is not necessarily the same as a Dirichlet form.
  • Nonlocal Phase Field Equations this will include many types of evolution equations such as nonlocal reaction-diffusion, models arising from nonlocal ising models, nonlocal cahn-hilliard, nonlocal biological aggregation, sharp interface limits (giacomin-lebowitz, bates-et-al, de massi-pressutti, etc...)
  • Boundary Harnack inequality it would be good to possibly include this in the description of what is different/same regarding local vs. nonlocal equations.
  • Describe, with references, the connection between holder estimates of viscosity solutions of stationary and or parabolic equations and the existence and uniqueness of solutions to the relevant Martingale Problem
  • Discussion of Jean Picard's paper (regularity for pure jump processes with non-degenerate, translation invariant generators)
  • Add Courrège's Structure theorem for operators and other structure theorems (generators of Feller semigroups, Hunt processes... etc)
  • A page on the nonlocal analogue of the thin film equation

(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.

Wiki Maintenance

  • Get access to LocalSettings.php (ask Maorong for access maybe?) so we can change the page name of the Main Page and the Starting Page (specially since those are the names shown by a google search)
  • Fill up the list of upcoming events such as conferences, workshops, summer schools.
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