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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.

45K05 Integro-partial differential equations
45G10 Other nonlinear integral equations
93E03 Stochastic systems in control theory (general)
Full Text: DOI arXiv
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