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

Russell Schwab
Carnegie-Mellon University
Mathematical Sciences

Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms (L-n and L-infinity) of the right hand side of the equation has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We will discuss this estimate in
the context of fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])

Presentation (PDF File)

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