MathDB

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

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