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.
- 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.
- A reminder for later (to make a full article out of this): it has become clear (only now, although in hindsight its trivial) that what physicists call "strong interactions" is the same as "very non-local". Namely, strong interactions correspond to forces that approach infinity much, much faster than say Coulomb's force as the distance between two particles goes to zero, mathematically, this means the interaction kernel $k(x,y)$ goes like $|x-y|^{-\gamma}$ for a very big positive number $\gamma$.