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In this wiki we collect several result 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.

We keep a list of open problems and also upcoming events.

Contents

Why nonlocal equations

All partial differential equations are a limit case of nonlocal equations. One could even go further and boldly say that in nature all equations are nonlocal, and PDEs are a simplification. 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

Existence and uniqueness results

For a variety of nonlinear elliptic and parabolic equations, the existence of viscosity solutions can be obtained using Perron's method. The uniqueness of solutions is a consequence of the comparison principle.

There are some equations for which this general framework does not work, for example the surface quasi-geostrophic equation. One could say that the underlying reason is that the equation is not purely parabolic, but it has one hyperbolic term.

Regularity results

The regularity tools used for nonlocal equations vary depending on the type of equation.

Nonlinear equations

The starting point to study the regularity of solutions to a nonlinear elliptic or parabolic equation are the Holder estimates which hold under very weak assumptions and rough coefficients. They are related to the Harnack inequality.

For some fully nonlinear integro-differential equation with continuous coefficients, we can prove $C^{1,\alpha}$ estimates.

For the Bellman equation, the solutions are classical due to the nonlocal version of Evans-Krylov theorem.

Semilinear equations

There are several interesting models that are semilinear equations. Those equations consists of either the fractional Laplacian or fractional heat equation plus a nonlinear term. There are challenging regularity questions especially when the Laplacian interacts with gradient terms in Drift-diffusion equations.

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