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.