Regularity theory for fully nonlinear integro‐differential equations

By Luis Caffarelli, Luis Silvestre. 2007

714 citations recorded by the source corpus.

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C1, α regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/81c874b12fbb64364923ecbe40e850b2cd809888

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Citations in the corpus

On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, background): “Moreover, in this analysis, the regularity of the principal eigenfunction and a Harnack type inequality [16, 17, 61] for some non negative solution of (1.”
Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations (2010, methodology): “In [18], there is a version of M− which covers many more operators than the one appearing in (1.”
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space (2015, neutral): “6 in [6] for integro differential equation).”
Global H\"older regularity for the fractional $p$-Laplacian (2014, background): “For non-local, fully non-linear, uniformly elliptic equation, higher interior regularity (up to C) is proved in [5], [6], [21] when the kernel satisfies additional structural and regularity assumption, but no such result is known for the s-fractional p-Laplacian.”
Integro-Differential equations with nonlinear directional dependence (2014, methodology): “These results extend those in [9] and are also uniform as the order of operators approaches 2.”
Dirichlet heat kernel estimates for rotationally symmetric Lévy processes (2013, background): “Recently, quite a few people in PDE are interested in problems related to non-local operators; see, for example, [5, 6, 7, 24] and the references therein.”
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary (2012, background): “Some other results dealing with various aspects concerning the Dirichlet problem are the following: estimates for the heat kernel (of the parabolic version of this problem) and for the Green function, e.g., [3, 10]; an explicit expression of the Poisson kernel for a ball [18]; and the explicit…”
Nonlocal problems with Neumann boundary conditions (2014, neutral): “[17, 20, 7, 19]).”
Regularity Results for Nonlocal Equations by Approximation (2009, methodology): “We also recall the interior C estimates from [2] with the hypothesis slightly relaxed.”
Lipschitz regularity of solutions for mixed integro-differential equations (2011, methodology): “The existing theory for second order elliptic equations has been extended to integro-differential equations by Caffarelli and Silvestre in [4].”