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Let $\Omega \subset \mathbb{R}^n$ be a open bounded domain, and let $g$ be a positive smooth function defined on $\partial \Omega$. Let $v$ be the unique harmonic function in $\Omega$ with boundary data $g$. Does $v$ necessarily need to be positive everywhere, or can it be negative?

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    $\begingroup$ It has to be positive by the maximum principe. Indeed $\inf g\le v\le \sup g$. $\endgroup$ – user99914 Oct 6 '16 at 11:15
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By the maximum principle, $v$ does indeed have to be positive. If it were non-positive, it'd have a strict local minimum not on the boundary, thus it wouldn't be harmonic.

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