$$−\Delta u = f \text{ in } \Omega$$ $$ \frac{\partial u}{\partial n}= g \text{ on } \partial\Omega$$
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.
Hint: Use Green's identity
$$\int_\Omega \Delta u \, dx = \int_{\partial \Omega} \frac{\partial u}{\partial n} \, dS.$$