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$$−\Delta u = f \text{ in } \Omega$$ $$ \frac{\partial u}{\partial n}= g \text{ on } \partial\Omega$$

where $\Omega\subset\mathbb R^n$ is a bounded domain with boundary $\partial\Omega$, $\Delta$ is the Laplace operator, $f$ and $g $ are given smooth functions and $ \frac{\partial u}{\partial n}$ denotes the outer normal derivative of $u$. How to find out necessary and sufficient condition for the above problem to admit a solution?

My try:sorry,i don't know how to proceed.Thank you.

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Hint: Use Green's identity

$$\int_\Omega \Delta u \, dx = \int_{\partial \Omega} \frac{\partial u}{\partial n} \, dS.$$

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  • $\begingroup$ This easily shows necessity, but you should add more details for sufficiency. $\endgroup$ – Alex M. Jan 3 '17 at 13:36
  • $\begingroup$ Sufficiency is much more involved, as it requires proving existence of a solution. I somehow doubt the OP actually wants to fully prove sufficiency. I could be wrong... $\endgroup$ – Jeff Jan 3 '17 at 15:48
  • $\begingroup$ Could you give some details/references for sufficiency. $\endgroup$ – Arun Sep 25 at 15:20

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