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Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels. (English) Zbl 1258.47064
In this paper, the authors consider the regularity of the viscosity solutions of integro-differential operators with possibly nonsymmetric kernel: \[ \mathcal L u(x) = p.v. \int_{R*N}\mu(u,x,y)K(y) \, dy, \] where \(\mu(u,x,y) = u(x+y) - u(x) - (\nabla u(x)\cdot y) \chi{B_1}(y),\) which describes the infinitesimal generator of a given purely jump processes. The authors are able to extend suitable versions of the Alexandroff-Backelman-Pucci estimate corresponding to the full class \(\mathcal S^{\mathfrak {L}_0}\) of uniformly elliptic nonlinear equations with \(1 < \sigma < 2\) (subcritical case) and to their subclass \(\mathcal S_\eta^{\mathfrak {L}_0}\) with \(0 < \sigma \leq 1\). The authors show that \(\mathcal S_\eta^{\mathfrak {L}_0}\) is meaningful because it includes a large number of nonlinear operators as well as linear operators. Lastly, they show a Harnack inequality, Hölder regularity, and \(C ^{1,\alpha }\)-regularity of the solutions by obtaining decay estimates of their level sets.

MSC:
47G20 Integro-differential operators
45K05 Integro-partial differential equations
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35D40 Viscosity solutions to PDEs
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