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. (CITATIONS still needed)

• 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.

• Keller-Segel equation.

• Active scalar equations.

• Ising Models With Nonlocal Interactions

• Nonlocal Interaction Energies

• 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...)

• Boundary Harnack inequality it would be good to possibly include this in the description of what is different/same regarding local vs. nonlocal equations.

• Describe, with references, the connection between holder estimates of viscosity solutions of stationary and or parabolic equations and the existence and uniqueness of solutions to the relevant Martingale Problem

• Neumann Boundary Conditions

• Jump processes via Subordination

• Processes in the "limit" $\alpha=0$, Zero Order Operators

• Discussion of the Barlow-Bass-Chen-Kassmann Counterexample to Regularity of $L$-Harmonic functions.

• Discussion of Jean Picard's paper "ON THE EXISTENCE OF SMOOTH DENSITIES FOR JUMP PROCESSES" (regularity for pure jump processes with non-degenerate, translation invariant generators)

• Add Courrège's Structure theorem for operators and other structure theorems (generators of Feller semigroups, Hunt processes... etc)

(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.

• The surface quasi-geostrophic equation.

• A page about semilinear equations.

• 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.

• It is convenient to have a mini second order elliptic wiki inside this wiki.

• open problems.

• 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.