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Regularity theory for fully nonlinear integro-differential equations. (English) Zbl 1170.45006
The authors study regularity properties of solutions $$u(x)$$ to the equation
$I\, u(x) = 0,\;\; \forall x\in \Omega \subset {\mathbb R}^n,$ where the operator $$I$$ is a nonlinear integro-differential operator that arises in stochastic control problems, of the form $I\,u(x) = \inf_{\beta}\sup_{\alpha}L_{\alpha \beta} u(x),$ with $Lu(x) = \text{PV}\,\int_{{\mathbb R}^n} (u(x+y)-u(x))K(y)\,dy,$ for some positive, symmetric kernel $$K$$, satisfying a standard Lévy-Khintchine condition.
The paper is planned to be the first one in a series of papers that extend the existing theory for nonlinear second-order elliptic equations to the case of discontinuous processes. The results in this paper are: a comparison principle for a general nonlinear integro-differential equation, a nonlocal version of the Aleksandrov-Bakel’man-Pucci estimate, the Harnack inequality and Hölder regularity result for integro-differential equations with fractional Laplacian type discontinuous kernels and a $$C^{1,\,\alpha}$$ regularity result for a large class of nonlinear integro-differential equations.

##### MSC:
 45K05 Integro-partial differential equations 45G10 Other nonlinear integral equations 93E03 Stochastic systems in control theory (general)
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