Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local SchrÃ¶dinger equations with exterior conditions
LorincziJozsef
BiswasAnup
2017
We consider Dirichlet exterior value problems related to a class of non-local Schr odinger
operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove
elliptic and parabolic Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain
existence and uniqueness of weak solutions. Next we prove a re ned maximum principle in the
sense of Berestycki-Nirenberg-Varadhan, and a converse. Also, we prove a weak anti-maximum
principle in the sense of Cl ement-Peletier, valid on compact subsets of the domain, and a full
anti-maximum principle by restricting to fractional Schr odinger operators. Furthermore, we show
a maximum principle for narrow domains, and a re ned elliptic ABP-type estimate. Finally, we
obtain Liouville-type theorems for harmonic solutions and for a class of semi-linear equations. Our
approach is probabilistic, making use of the properties of subordinate Brownian motion.