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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).

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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.

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Correspondence to K. Ramachandran.

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The authors declare that they have no competing interests.

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The second author was in part supported by NSF Grant DMS-1362337.

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Gallagher, AK., Lebl, J. & Ramachandran, K. Convexity of level lines of Martin functions and applications. Anal.Math.Phys. 9, 443–452 (2019). https://doi.org/10.1007/s13324-017-0207-3

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