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Purpose

The purpose of this wiki is to create a clear reference for the methods, difficulties, open problems, and applications of elliptic and parabolic nonlocal equations.

Especially since this is a relatively new subject, it is important to clearly point out where the fundamental mathematical challenges are, and what are the applications to the equations studied.

We would like to write a reference that is easy to read and navigate. It should be convenient to easily find the precise assumptions for which several theorems are proved (for example Harnack inequality or $C^{1,\alpha}$ estimates). It would also be desirable to have some indication of the methods used in the proofs, which may be a non rigorous idea, or an analogy with a classical theorem.

The emphasis of this wiki will be on nonlocal nonlinear (nonbananas?) equations of elliptic and parabolic type. There are several hyperbolic nonlocal equations that are described in the Dispersive wiki.

The ultimate goal of this wiki would be to prove for all equations that they are either well posed in the classical sense, or a weak solution and their possible singularities are well understood. The emphasis should then be on regularity results. We may also include some topics only indirectly related to regularity estimates (for example homogenization).

We should make an effort to negate the myths about nonlocal equations