Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations

By Nestor Guillen, Russell W. Schwab. 2010

64 citations recorded by the source corpus.

In this work we provide an Aleksandrov–Bakelman–Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractional-order setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and, as a consequence, provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the L∞ and Ln norms of the right-hand side, in contrast to previous comparison results which utilize the continuity of the right-hand side for their conclusions. These results appear to be new, even for the linear case of the relevant equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c0edcc24d9262dd825868cc61c2ba64dc38409bc

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Integro-Differential equations with nonlinear directional dependence (2014, methodology): “Via [9, Sections 10, 12], Hölder regularity for a smaller– but different class– of kernels than those treated in [9] follows as a straightforward consequence to the Aleksandrov-Bakelman-Pucci type result obtained in [18].”
Regularity for parabolic integro-differential equations with very irregular kernels (2014, background): “2) is not required to hold at all points are [7], [18], [22], and [24].”
Local Regularity for Parabolic Nonlocal Operators (2012, background): “[15], [12], [8], [14] and recently for fully nonlinear nonlocal parabolic equations in [9].”
Radial symmetry of positive solutions to equations involving the fractional laplacian (2013, methodology): “In [16], Guillen and Schwab showed an ABP estimate (see Theorem 9.”
Boundary regularity for fully nonlinear integro-differential equations (2014, background): “Other important references concerning interior regularity for nonlocal equations in nondivergence form are [24, 23, 17, 2, 21].”
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space (2015, background): “The ABP estimate for bounded domain with nonlocal operator see [21].”