3
$\begingroup$

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

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.

1 Answers 1

3

Hint: Use Green's identity

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

  • 0
    This easily shows necessity, but you should add more details for sufficiency. – 2017-01-03
  • 0
    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... – 2017-01-03