# Regularity results for nonlocal equations

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## Regularity results for linear equations with constant coefficients

## Regularity results for linear equations with smooth coefficients

## Regularity results for linear equations with rough coefficients

## Regularity results for nonlinear equations

- For general fully nonlinear integro-differential equations, interior $C^{1,\alpha}$ estimates can be proved in a variety of situations. The simplest assumption would be for a translation invariant uniformly elliptic equations with respect to the class of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 1
^{[1]}. There are several other $C^{1,\alpha}$ estimates for variations of this situation (smooth coefficients, kernels close to the smoothness class, etc...)^{[2]}.

- A nonlocal version of Evans-Krylov theorem says that for the Bellman equation, for a family of kernels that are uniformly elliptic of order $s$ and in the smoothness class of order 2, the solutions are $C^{s+\alpha}$
^{[3]}. This is enough regularity for the solutions to be classical.

## References

- ↑ Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations",
*Communications on Pure and Applied Mathematics***62**(5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 - ↑ Caffarelli, Luis; Silvestre, Luis (2009), "Regularity results for nonlocal equations by approximation",
*Archive for Rational Mechanics and Analysis*(Berlin, New York: Springer-Verlag): 1–30, ISSN 0003-9527 - ↑ Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations",
*Annals of Mathematics*, ISSN 0003-486X