SCHWEITZER 2010 Problem 8

Let $D \subset \mathbb {R} ^ {2}$ be a finite Lebesgue measure of a connected open set and $u: D \rightarrow \mathbb {R}$ a harmonic function. Show that it is either a constant $u$ or for almost every $p \in D$ $$f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), f (0) = p$$ has no initial value problem(differentiable everywhere) solution to $f:[0,\infty) \rightarrow D$ .