Introduction to nonlocal equations

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Description

Traditional partial differential equations are relations between the values of an unknown function and its derivatives of different orders. In order to check whether a partial differential equation holds at a particular point, one needs to known only the values of the function in an arbitrarily small neighborhood, so that all derivatives can be computed. A nonlocal equation is a relation for which the opposite happens. In order to check whether a nonlocal equations holds at a point, information about the values of the function far from that point is needed. Most of the times, this is because the equation involves integral operators.

When dealing with nonlocal equation, one can still talk about elliptic, parabolic and hyperbolic ones. This classification is vague, as much as it is for usual partial differential equations. An equation is elliptic when it fulfills some characteristics that are common to elliptic equations. Some characteristic properties of elliptic equations, for example, are the maximum principle and the interior regularity results. These equations, being local or nonlocal, describe a function which does not evolve with time. The equation describes a situation in which the values of the function at every point equal some form of weighted average of other values of the function. There are several settings in which that kind of equations occur. An important example is the problems in stochastic control, which motivate the study of fully nonlinear integro-differential equations. The Laplace equation is the prime example of a classical elliptic equation. Likewise, the equations involving the fractional Laplacian are the prime example of nonlocal elliptic equations.

The parabolic equations are those which describe the evolution of a function which tries to converge to its equilibrium because of the effect of dissipation (in a broad sense). Much of the theory developed for elliptic equations can be extended to parabolic equations, but one often nontrivial difficulties in the process. In these equations, we also find some form of maximum principle and regularization results.

[...] To be continued.

which long range as well as short range behavior play significant roles in the equation describing the unknown function. This can happen in increasing levels of nonlocal strength from a zero order term as a convolution with a nice kernel (citation/link), to a a gradient of a convolution (citation/link), and finally an integro-differential term involving a weighted average of the differences of the unknown, u, against a kernel which is singular at the origin and the equation requires regularity of u(x+y)−u(x) to even make sense of the equation (citation/link). Sometimes these equations obey comparison principles between subsolutions and supersolutions which puts them in the realm of viscosity solutions (citation/link), and sometimes these equations do not have a comparison principle (citation/link, bio-agg, nonlocal cahn hilliard, etc...).

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.

Under certain hypothesis, the nonlocal Bellman equation from optimal stochastic control has classical solutions 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. A simple method that has been successful in proving the well posedness of some semilinear equations with drift terms in the critical case (when both terms have the same scaling properties) is the conserved modulus of continuity approach, often called "nonlocal maximum principle method".