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Is there a solution to the following problem? $-\Delta u = 1$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$. where $\Omega$ is bounded.

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Try it on an interval first? – Otis Chodosh May 30 '12 at 14:37
Look up "strong maximum principle". For example around page 33 or 34 of Gilbarg and Trudinger, Elliptic Partial Differential Equations of Second Order – Willie Wong May 30 '12 at 14:50

I $\Omega$ is a bounded domain, the answer is NO. Because of $$\int_{\Omega}\Delta v dx=\int_{\partial\Omega}\frac{\partial v}{\partial\nu}ds.$$

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