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Monotonicity of non-Liouville property for positive solutions of skew product elliptic equations

Part of: Representations of solutions Partial differential equations Elliptic equations and systems Parabolic equations and systems Other generalizations

Published online by Cambridge University Press:  29 January 2019

Minoru Murata  and
Tetsuo Tsuchida
Minoru Murata
Affiliation:
5-30-1 Shinyoshida-higashi, Kohoku-ku, Yokohama223-0058Japan (minoru3@math.titech.ac.jp)
Tetsuo Tsuchida
Affiliation:
Department of Mathematics, Meijo University Shiogamaguchi, Tenpaku-ku, Nagoya, 468-8502Japan (tsuchida@meijo-u.ac.jp)
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Abstract

We consider a second-order elliptic operator L in skew product of an ordinary differential operator L1 on an interval (a, b) and an elliptic operator on a domain D2 of a Riemannian manifold such that the associated heat kernel is intrinsically ultracontractive. We give criteria for criticality and subcriticality of L in terms of a positive solution having minimal growth at η (η = a, b) to an associated ordinary differential equation. In the subcritical case, we explicitly determine the Martin compactification and Martin kernel for L on the basis of [24]; in particular, the Martin boundary over η is either one point or a compactification of D2, which depends on whether an associated integral near η diverges or converges. From this structure theorem we show a monotonicity property that the Martin boundary over η does not become smaller as the potential term of L1 becomes larger near η.

Keywords

criticalityMartin boundarymonotonicityskew productpositive solutionintrinsic ultracontractivity

MSC classification

Primary: 31C35: Martin boundary theory
Secondary: 35B09: Positive solutions 35C15: Integral representations of solutions 35J08: Green's functions 35K08: Heat kernel
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1429 - 1449
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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Footnotes

To the memory of Toshimasa Tada

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