To Do List

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.

• drift-diffusion equations.

• 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.

• 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.

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.