Mathematics – Analysis of PDEs
Scientific paper
2010-12-31
Mathematics
Analysis of PDEs
Main theorem improved to treat nondegenerate kernels which leads to new regularity results. Added appendix which provides new
Scientific paper
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 only depends on the $L^\infty$ 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.
Guillen Nestor
Schwab Russell
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