In this wiki we collect several results about nonlocal elliptic and parabolic equations. If you want to know what a nonlocal equation refers to, a good starting point would be the Intro to nonlocal equations.
The wiki has an assumed bias towards regularity results and consequently to equations for which some regularization occurs. But we also include some topics which are tangentially related, or even completely unrelated, to regularity.
Why nonlocal equations?
All partial differential equations are a limit case of nonlocal equations. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are
In financial mathematics it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the obstacle problem.
Nonlocal electrostatics is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.