To Do List: Difference between revisions
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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.) | * 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.) | ||
* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron 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 discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]]. | ||
Revision as of 17:35, 6 July 2011
Current Projects / To do list
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.
- 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.)
- 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 forms, and maybe some models from nonlocal image processing.
- 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.
- Some discussion on models involving Levy processes and stochastic control.
- Some references to equations from fluids including the surface quasi-geostrophic equation.
- A page about semilinear equations including the surface quasi-geostrophic equation and also some form of KPP.
- Nonlocal porous medium equation and its more regular sister, the Aggregation equation.
- Nonlocal minimal surfaces and Nonlocal mean curvature flow (specially in connection to nonlocal image processing).
- Fractional curvatures in conformal geometry.
- 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 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.
- Limit of $\sigma$-order operators as $\sigma \to 2^-$, maybe this should go on a page on Linear integro-differential operators.
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.