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 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 related topics that will be necessary to add (and that I will write about soon) include Geometric Scattering Theory, Ambient Metric Construction, GJMS Operators and Singular Yamabe Problem. -Nestor.)

• 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.

• Luis C. would be very disappointed if we did not even mention topics from good old potential theory. Very importantly, Riesz potentials and the Poisson kernel for Fractional harmonic functions. Someone better start dusting off their copy of Landkof.

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.