Convexity of level lines of Martin functions and applications

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 $\Omega$ is symmetric, then the maximum of any Martin function along a slice $\Omega \cap (\{t\}\times {\mathbb{R}}^d)$ is attained at $(t,0).$

Statement of Theorem 1.2 revised. 8 pages, 2 figures