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| == Things that need to be done == | | <table {{#if:{{{id|}}}|id="{{{id|}}}"}} class="plainlinks fmbox |
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| 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.
| | | system <!-- system = default --> |
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| * 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.
| | }} {{{class|}}}" style="{{{style|}}}"> |
| | | <tr> |
| * Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].
| | {{#ifeq:{{{image|}}}|none |
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| * Fractional curvatures in conformal geometry.
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| * 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]]..)
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| * 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.
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| * Pages about [[Homogenization]] (local and nonlocal) should appear here too. (CITATIONS still needed)
| | | system <!-- system = default --> |
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| * 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]].
| | }}|40x40px|link=|alt=]] |
| | | }}</td> |
| * [[nonlocal image processing]].
| | }} |
| | | <td class="mbox-text" style="{{{textstyle|}}}"> {{{text}}} </td> |
| * [[Aggregation equation]].
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| | | | <td class="mbox-imageright"> {{{imageright}}} </td> |
| * [[Keller-Segel equation]].
| | }} |
| | | </tr> |
| * [[Active scalar equations]].
| | </table><!-- |
| | | Detect and report usage with faulty "type" parameter: |
| * [[Ising Models With Nonlocal Interactions]]
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| * [[Nonlocal Interaction Energies]]
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| * [[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...)
| | | system = <!-- Do nothing, valid "type" --> |
| | | | #default = <div style="text-align: center;">This message box is using an invalid "type={{{type|}}}" parameter and needs fixing.</div>[[Category:Wikipedia message box parameter needs fixing|{{main other|Main:}}{{FULLPAGENAME}}]]<!-- Sort on namespace --> |
| * [[Boundary Harnack Inequality]] it would be good to possibly include this in the description of what is different/same regarding local vs. nonlocal equations.
| | }}<noinclude> |
| | | {{documentation}} |
| * describe, with references, the connection between [[H\"older Regularity]] of viscosity solutions of stationary and or parabolic equations and the existence and uniqueness of solutions to the relevant [[Martingale Problem]]
| | <!-- Add categories and interwikis to the /doc subpage, not here! --> |
| | | </noinclude> |
| == (partially) Completed tasks ==
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| * 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]].
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| * Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].
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| * A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]].
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| * Some discussion on models involving [[Levy processes]] and [[stochastic control]].
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| * The [[surface quasi-geostrophic equation]].
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| * A page about [[semilinear equations]].
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| * [[Nonlocal porous medium equation]].
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| * [[drift-diffusion equations]].
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| * [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]].
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| * There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on.
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| * Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
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| * It is convenient to have a [[mini second order elliptic wiki]] inside this wiki.
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| * [[open problems]].
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| * Having a [[list of equations]] may make it easier to navigate the wiki.
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| == Other ideas ==
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| * Fill up the list of [[upcoming events]] such as conferences, workshops, summer schools.
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| * [[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.
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