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

Abstract

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 L n 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.

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Correspondence to Russell W. Schwab.

Additional information

Communicated by P.-L. Lions

Nestor Guillen was partially supported by NSF grant DMS-0654267 and Russell Schwab was supported by an NSF Postdoctoral Research Fellowship, grant DMS-0903064. This work was carried out during multiple visits of Nestor Guillen to The Center for Nonlinear Analysis at Carnegie Mellon University, for whose support the authors are also grateful. The authors would like to thank Takis Souganidis and Luis Silvestre for comments on the preliminary version of this manuscript, and the authors would like to extend a special thanks to the anonymous referee for helpful observations which led to some significant improvements in this work.

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Guillen, N., Schwab, R.W. Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations. Arch Rational Mech Anal 206, 111–157 (2012). https://doi.org/10.1007/s00205-012-0529-0

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Keywords

  • Fractional Order
  • Viscosity Solution
  • Harnack Inequality
  • Obstacle Problem
  • Regularity Theory