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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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

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

First available in Project Euclid: 27 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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.

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