A criterion for constantness of harmonic functions on exterior domains (Q1399786)

Let \(f: \mathbb{R}^2\to\mathbb{R}\) be a function that is bounded from below and satisfying the Laplace equation: \(f_{xx}+ f_{yy}= 0\). Then Liouville's theorem assures that \(f\) is constant. We can observe that, in general, the result is not true when the harmonic function \(f\) is defined on an exterior domain, that is the complement of a compact set. The main result of this paper is as follows. Let \(K\) be a non-empty convex set of \(\mathbb{R}^2\) and \(f: \mathbb{R}^2\setminus K\to\mathbb{R}\) a harmonic function, bounded from below. Assume that \[ \limsup_{\|(x,y)\|\to+\infty} {f(x,y)\over\|(x, y)\|}= 0. \] Then the function \(f\) is constant, if and only if the relation \[ |\nabla f(x,y)\cdot\nabla f_{xx}(x,y)|\leq \|\nabla f_x(x,y)\|^2, \] holds for every \((x,y)\in \mathbb{R}^2\setminus K\), where the symbol \(\cdot\) denotes the dot product of the gradients \(\nabla f\) and \(\nabla f_{xx}\).