Convexity of level lines of Martin functions and applications.

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