I have a question about Neumann boundary condition for PDE.

Suppose $\Omega$ is an open bounded set in $R^n$ with a smooth boundary $\partial \Omega $.

Then, a homoegenous Neumann boundary condition is given as

$\frac{\partial u} {\partial n} = 0$ on $\partial \Omega$, where $n$ is the outward normal.

But, is $\frac{\partial u} {\partial n}$ well defined? That is, how do we compute $\frac{\partial u} {\partial n}$ when we do not know whether $u$ is even defined **outside ** $\Omega \cup \partial \Omega$?


It's the one sided limit :

$$ \frac{\partial u}{\partial \vec{n}}(x) = \lim_{\epsilon \to 0^-} \frac{f(x+\epsilon\vec{n}) - f(x)}{\epsilon}$$

For $\epsilon$ small enough (and negative), $f(x+\epsilon\vec{n})$ is in $\Omega$ (because $\partial \Omega$ is smooth enough)

  • $\begingroup$ So, it is in fact an "inward" directional derivative? $\endgroup$ – user74261 Apr 15 '15 at 22:08
  • $\begingroup$ You could say that, but with the opposite sign (as it's positive if what's inside is bigger than what's on the boundary) $\endgroup$ – Tryss Apr 15 '15 at 22:18

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