A Liouville-type theorem for harmonic functions on exterior domains
Liouville's classical theorem that a function harmonic in \(\mathbb{R}^2\) that is bounded below is a constant does not extend to \(\mathbb{R}^2\) punctured at the origin. Via some interesting preliminaries on convex sets in \(\mathbb{R}^2\), the authors prove that if \(K\) is a non-empty compact convex set in \(\mathbb{R}^2\) and \(f\) is a real harmonic function in \(\mathbb{R}^2\setminus K\) that is bounded below and satisfies the condition that \(f(x,y)= o(\|(x,y)\|)\) as \((x,y)\to \infty\), then \(f\) is constant if \[ \bigl|\nabla f(x,y) \bigr|\cdot\bigl|\nabla f_{x,x}(x,y) \bigr|\leq \bigl|\nabla f_x(x,y) \bigr|^2 \] for every \((x,y)\) in \(\mathbb{R}^2\setminus K\).
scientific article; zbMATH DE number 1441382