An Alexandrov-Bakelman-Pucci estimate for an anisotropic Laplacian with positive drift in unbounded domains

We show an Alexandrov-Bakelman-Pucci (ABP) estimate for a PDE with anisotropic Laplacian in two dimensions in unbounded domains, where the drift vector varies in a segment of the positive quadrant. The unbounded domains are assumed to be bounded below in the x-direction, as well as in the y-directio...

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Bibliographic Details
Published in:Journal of Differential Equations 2021-12, Vol.303, p.183-213
Main Author: Grandits, Peter
Format: Article
Language:English
Subjects:
Alexandrov-Bakelman-Pucci estimate
Anisotropic Laplacian
Positive drift
Singular perturbation
Two-dimensional Brownian motion
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Summary:We show an Alexandrov-Bakelman-Pucci (ABP) estimate for a PDE with anisotropic Laplacian in two dimensions in unbounded domains, where the drift vector varies in a segment of the positive quadrant. The unbounded domains are assumed to be bounded below in the x-direction, as well as in the y-direction. The constant in the upper estimate of the ABP-estimate, which depends in the usual theorems for bounded domains on the diameter of this domain, depends in our case on the small parameter ϵ, appearing in the anisotropic Laplacian. The result is motivated by certain problems of singular perturbation in stochastic control theory, and our methods are probabilistic.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2021.08.039