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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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References:
[1] Awatif, Équations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité, Comm Partial Differential Equations 16 (6) pp 1057– (1991)
[2] Barles, Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann Inst H Poincaré Anal Non Linéaire 25 (3) pp 567– (2008)
[3] Bass, Harnack inequalities for non-local operators of variable order, Trans Amer Math Soc 357 (2) pp 837– (2005) · Zbl 1052.60060
[4] Bass, Hölder continuity of harmonic functions with respect to operators of variable order, Comm Partial Differential Equations 30 (7) pp 1249– (2005) · Zbl 1087.45004
[5] Bass, Harnack inequalities for jump processes, Potential Anal 17 (4) pp 375– (2002) · Zbl 0997.60089
[6] Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann of Math (2) 130 (1) pp 189– (1989) · Zbl 0692.35017
[7] Caffarelli, Fully nonlinear elliptic equations 43 (1995) · doi:10.1090/coll/043
[8] Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm Pure Appl Math 42 (1) pp 15– (1989) · Zbl 0645.35025
[9] Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch Rational Mech Anal 101 (1) pp 1– (1988) · Zbl 0708.35019
[10] Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ Math J 55 (3) pp 1155– (2006) · Zbl 1101.45004
[11] Soner, Optimal control with state-space constraint. II, SIAM J Control Optim 24 (6) pp 1110– (1986) · Zbl 0597.49023
[12] Song, Harnack inequality for some classes of Markov processes, Math Z 246 (1) pp 177– (2004) · Zbl 1052.60064
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