Aufgaben:Problem 11

Problem

Let \(u\) be a harmonic function on \(\mathbb{R}^3\). Assusme there exists \(C>0\), independent of \(x\), such that \(|u(x)| \leq C(1+|x|)\) on \(\mathbb{R}^3\). Show that then

1. \(\partial_i u\) is constant on \(\mathbb{R}^3\), \(\forall i = 1,2,3\).
2. \(u\) is a linear function, i.e. \(\exists a_0, a_1, a_2, a_3 \in \mathbb{R}\) such that \(u(x) = a_0 + a_1 x_1 + a_2 x_2 + a_3 x_3.\)

Proof Sketch

Proof in 2 dimensions: http://www.math.columbia.edu/~savin/c12d.pdf