Convexity of level lines of Martin functions and applications

Abstract

Abstract.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).

Level Curves of Minimal Graphs

“…Our next result concerns solutions whose domains are concave. There is a literature (see [3] and references cited there) regarding the propogation of convexity for level curves of solutions to partial differential equations over convex domains.…”