Good afternoon.
My question is related to the Maximum principle. In fact:
We consider $\Omega\subset(M,g)$ a bounded domain in a Riemannian manifold and we consider the first Dirichlet eigenfunction $u$ associated to the first Dirichlet eigenvalue $\lambda_1$ of the Laplacian $\Delta$, i,e $$\begin{cases} \Delta u&=\lambda_1u~~~~ in ~~~~\Omega\\ u&=0 ~~~~~~~~over~ \partial\Omega \end{cases}$$ My question is: Does $\frac{\partial u}{\partial\nu}$ is $>0$ or $<0$ over $\partial\Omega$ where $\nu$ is the outward unit normal to $\partial\Omega$ ?
Any explanation will be appreciated.