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Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schrödinger equations with exterior conditions
journal contributionposted on 2017-11-29, 16:06 authored by Jozsef Lorinczi, Anup Biswas
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.
- Mathematical Sciences
Published inSIAM Journal on Mathematical Analysis
Pages1543 - 1581
CitationLORINCZI, J. and BISWAS, A., 2019. Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schrödinger equations with exterior conditions. SIAM Journal on Mathematical Analysis, 51(3), pp. 1543-1581.
Publisher© Society for Industrial and Applied Mathematics (SIAM)
- AM (Accepted Manuscript)
Publisher statementThis work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/
NotesThis article was published in the SIAM Journal on Mathematical Analysis [© by SIAM] and the definitive version is available at: https://doi.org/10.1137/18M1171722