To Do List

Things that need to be done

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.

• Some discussion on Dirichlet forms, and maybe some models from 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.

• 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.

• nonlocal image processing

• Aggregation 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.)

• Definition of viscosity solutions for nonlocal equations. Also a discussion on existence using Perron's method and uniqueness through the comparison principle.

• 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.

• drift-diffusion equations.

• Nonlocal minimal surfaces and Nonlocal mean curvature flow.

• There is also plenty of work on Dislocation dynamics that we ought to add later on.

• 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...

• Having a list of equations may make it easier to navigate the wiki.

Other ideas

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

• Nestor has started a Literature on Nonlocal Equations to dump there all papers we want to reference or are already referencing on the wiki.