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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A Liouville theorem for supersolutions of linear elliptic second-order partial differential equations


Author: Vasilii V. Kurta
Journal: Proc. Amer. Math. Soc. 148 (2020), 611-621
MSC (2010): Primary 35J70, 35B53; Secondary 35J15
DOI: https://doi.org/10.1090/proc/14796
Published electronically: October 18, 2019
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Abstract: We study supersolutions of linear elliptic second-order partial differential equations of the form

$\displaystyle Lu:=\sum \limits _{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j}=0,\tag *{($\ast $)}$    

which are defined and measurable in the whole space $ {\mathbb{R}}^n$, and which belong locally to a Sobolev-type function space associated with the operator $ L$ defined in $ {\mathbb{R}}^n$, $ n\geq 2$. We assume that the coefficients $ a_{ij}(x)$ of the operator $ L$ are measurable, locally bounded and such that $ a_{ij}(x)=a_{ji}(x)$, and that the quadratic form associated with the operator $ L$ is positive-definite. We prove a Liouville theorem for supersolutions of ($ \ast $) defined in $ {{\mathbb{R}}^n}$, in terms of a capacity associated with the operator $ L$. As well, we establish a sharp distance at infinity between any non-constant supersolution of ($ \ast $) in $ {{\mathbb{R}}^n}$ bounded below by a constant and this constant itself, also in terms of the capacity associated with the operator $ L$.

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Additional Information

Vasilii V. Kurta
Affiliation: American Mathematical Society, Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, Michigan 41807-8604
Email: vkurta@umich.edu, vvk@ams.org

DOI: https://doi.org/10.1090/proc/14796
Received by editor(s): March 6, 2019
Published electronically: October 18, 2019
Communicated by: Ariel Barton
Article copyright: © Copyright 2019 American Mathematical Society