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2010 Miklós Schweitzer, 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 $ $$ f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), \quad f (0) = p $$has no initial value problem(differentiable everywhere) solution to $ f:[0,\infty) \rightarrow D $.
function topology harmonic functions algebra calculus combinatorics