Convexity of level lines of Martin functions and applications

TL;DR

This paper proves that in convex domains, Martin functions have convex superlevel sets, and in symmetric convex domains, their maxima along slices occur at the center, revealing geometric properties of harmonic functions.

Contribution

It establishes the convexity of superlevel sets of Martin functions in convex domains and characterizes the location of their maxima in symmetric cases.

Findings

Superlevel sets of Martin functions are convex in convex domains.

Maximum of Martin functions along slices occurs at the symmetric center.

Results connect geometric domain properties with harmonic function behavior.

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 $(t,0).$