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Finite Lebesgue measure of a connected open set and harmonic function

Source: Miklós Schweitzer 2010, P8

September 9, 2020
topologyfunctionharmonic functions

Problem Statement

Let D⊂R2 D \subset \mathbb {R} ^ {2} be a finite Lebesgue measure of a connected open set and u:D→R u: D \rightarrow \mathbb {R} a harmonic function. Show that it is either a constant u u or for almost every p∈D 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,∞)→D f:[0,\infty) \rightarrow D .
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