The Evans-Krylov theorem for nonlocal fully nonlinear equations

By Luis Caffarelli, Luis Silvestre. 2009

190 citations recorded by the source corpus.

We prove a regularity result for solutions of a purely integro-differential Bellman equation. This regularity is enough for the solutions to be understood in the classical sense. If we let the order of the equation approach two, we recover the theorem of Evans and Krylov about the regularity of solutions to concave uniformly elliptic partial differential equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/9038b783fe684312d3c4adcdd2c38e05f34d3652

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

C 1, α Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels (2013, background): “Introduced in [6], the idea is to replace kernels on a small ball around the origin by a multiple of the fractional Laplacian.”
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, unknown): “1): indeed, since we will assume that the nonlinearity f is locally Lipschitz continuous, the regularity theory for viscosity solutions (developed in [13,14]) implies that viscosity and classical solutions coincide in our setting (see [35, Remark 2.”
Nonlocal Diffusion and Applications (2015, background): “Many fundamental topics slipped completely out of these notes: just to name a few, the topological methods and the fine regularity theory in the fractional cases are not presented here, the fully nonlinear or singular/degenerate equations are not taken into account, only very few applications are di”
Localization of nonlocal gradients in various topologies (2015, background): “Various nonlocal operators have also been used implicitly in a number of works such as [5, 6], [8, 9, 25], [13, 14], and [3], for example.”
Nonlocal elliptic equations in bounded domains: a survey (2015, methodology): “We refer to the reader to the following works on: • Existence of solutions: Felsinger-Kassmann-Voigt [25], Hoh-Jacob [31], and Barles-Imbert [3]; • Interior regularity of solutions: Bass-Levin [6], Kassmann [33], CaffarelliSilvestre [13, 14, 15], Barles-Chasseigne-Imbert [5], Kassmann-Mimica [34,…”
Boundary regularity for fully nonlinear integro-differential equations (2014, background): “, with kernels in L0 which are C(2) away from the origin), Caffarelli and Silvestre obtained C interior regularity [14].”
Gradient flows of the entropy for jump processes (2012, unknown): “[6,10] and references therein).”
Nonlocal self-improving properties (2015, unknown): “We finally remark that, in recent times, there has been much attention to the regularity of solutions to nonlocal problems, especially in the basic case of kernels with measurable coefficients; see for instance [1, 3, 5, 7, 8, 14].”
On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, background): “11) holds with infB3/4 u replaced by ‖u‖L∞(Rn), which is proved in [22], [23] and [24] in a much more general setting of fully nonlinear nonlocal equations.”
The Bernstein Technique for Integro-Differential Equations (2020, background): “1 of [19] leads to the local C2s+β regularity for equations (1.”
Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces (2012, methodology): “3We adopt the notion of viscosity solution used in [5, 6, 7].”