Convexity of level lines of Martin functions and applications

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Abstract

Let \(\Omega \) be an unbounded domain in \(\mathbb {R}\times \mathbb {R}^{d}.\) A positive harmonic function u on \(\Omega \) that vanishes on the boundary of \(\Omega \) is called a Martin function. In this note, we show that, when \(\Omega \) is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition \(\Omega \) has certain symmetry with respect to the t-axis, and \(\partial \Omega \) is sufficiently flat, then the maximum of any Martin function along a slice \(\Omega \cap (\{t\}\times \mathbb {R}^d)\) is attained at (t, 0).

Notes

Acknowledgements

We thank Alexandre Eremenko for useful discussions and suggestions. We are also greatly indebted to the referee for pointing out an error in an earlier version of Theorem 1.2.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsOklahoma State UniversityStillwaterUSA

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