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Regularity results for nonlocal equations by approximation. (English) Zbl 1231.35284
This paper is concerned with existence and regularity results for nonlocal elliptic equations which include applications in stochastic control problems and stochastic games. By using compactness and perturbative methods and previous results for the translation-invariant case, the authors obtain Cordes-Nirenberg type ($$C^{1,\alpha}$$) estimates for nonlocal elliptic equations that are not necessarily translation-invariant. Applications of the abstract theorem are provided in the paper, which include linear and nonlinear equations with variable coefficients and nonlinear equations with constant coefficients but non-differentiable kernels.

##### MSC:
 35R09 Integral partial differential equations 35B65 Smoothness and regularity of solutions to PDEs 35D40 Viscosity solutions to PDEs 91A15 Stochastic games, stochastic differential games
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##### References:
 [1] Barles, G., Chasseigne, E., Imbert, C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (to appear) · Zbl 1207.35277 [2] Caffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009) · Zbl 1170.45006 · doi:10.1002/cpa.20274 [3] Caffarelli L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. of Math. (2) 130(1), 189–213 (1989) · Zbl 0692.35017 · doi:10.2307/1971480 [4] Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, Vol. 43. American Mathematical Society, Providence, RI, 1995 · Zbl 0834.35002 [5] Cordes H.O.: Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen. Math. Ann. 131, 278–312 (1956) · Zbl 0070.09604 · doi:10.1007/BF01342965 [6] Kassmann M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differ. Equ. 34(1), 1–21 (2009) · Zbl 1158.35019 · doi:10.1007/s00526-008-0173-6 [7] Mikulyavichyus R., Pragarauskas G.: Nonlinear potentials of the Cauchy–Dirichlet problem for the Bellman integro-differential equation. Liet. Mat. Rink. 36(2), 178–218 (1996) [8] Nirenberg, L.: On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations. In Contributions to the Theory of Partial Differential Equations. Annals of Mathematics Studies, Vol. 33. Princeton University Press, Princeton, 95–100, 1954 · Zbl 0057.08604 [9] Silvestre L.: Hölder estimates for solutions of integro-differential equations like the fractional laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006) · Zbl 1101.45004 · doi:10.1512/iumj.2006.55.2706
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