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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.

edited May 30 at 14:43

Willie Wong
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asked May 30 at 14:31

mehdi
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Try it on an interval first? – Otis Chodosh May 30 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 at 14:50

Denis Serre · Answer 1 · 2012-05-30 14:50:43Z UTC

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

answered May 30 at 14:50

Denis Serre
18.7k●1●31●93

Neumann problem in case f=1

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