Advances in Differential Equations

Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators

Minoru Murata and Tetsuo Tsuchida

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Abstract

We consider positive solutions of elliptic partial differential equations on non-compact domains of Riemannian manifolds. We establish general theorems which determine Martin compactifications and Martin kernels for a wide class of elliptic equations in skew product form, by thoroughly exploiting parabolic Martin kernels for associated parabolic equations developed in [35] and [25]. As their applications, we explicitly determine the structure of all positive solutions to a Schrödinger equation and the Martin boundary of the product of Riemannian manifolds. For their sharpness, we show that the Martin compactification of ${\mathbb R}^2$ for some Schrödinger equation is so much distorted near infinity that no product structures remain.

Article information

Source
Adv. Differential Equations Volume 22, Number 9/10 (2017), 621-692.

Dates
First available in Project Euclid: 27 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1495850456

Zentralblatt MATH identifier
1376.35015

Subjects
Primary: 31C35: Martin boundary theory [See also 60J50] 35B09: Positive solutions 35C15: Integral representations of solutions 35J08: Green's functions 35K08: Heat kernel 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14]

Citation

Murata, Minoru; Tsuchida, Tetsuo. Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators. Adv. Differential Equations 22 (2017), no. 9/10, 621--692. https://projecteuclid.org/euclid.ade/1495850456


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