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Can one construct a harmonic function $f$ defined in unit disk with condition $f(0)\geq1$ such that area of $\{z\in\mathbb{D}: f(z)>0\}$ is small enough?

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  • $\begingroup$ $f$ is real-valued? The question is $\forall\epsilon>0$ $\exists f$ harmonic s.t. area$(\cdots)<\epsilon$? $\endgroup$ Oct 20, 2014 at 6:57
  • $\begingroup$ Small enough for what? $\endgroup$
    – mrf
    Oct 20, 2014 at 7:10
  • $\begingroup$ meaning for $\epsilon>0$ we want to find harmonic function $f$ such that $f(0)\geq 1$ and $area(\{z\in\mathbb{D}: \,f(z)>0\})\leq \epsilon$ $\endgroup$
    – BigM
    Oct 20, 2014 at 14:54

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This question was asked and answered at MathOverflow. The point of this answer is to provide the link and remove this from the list of unanswered questions. (I made this answer community wiki so that I don't gain reputation when I have not earned it and anyone can freely add details.)

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