Ratio of Eigenfunctions Suppose we have the first two eigenfunctions $\phi_1, \phi_2$ of the Laplace Operator on a bounded domain with smooth boundary with Dirichlet condition. The first eigenfunction is positive in the interior, and hence by Hopf Boundary Lemma, has non-zero gradient on the boundary. Both the first and second eigenfunction are zero on the boundary.
Apparently the ratio $\phi_2/\phi_1$ remains smooth on the closure of the domain. Why is this so?
Reading some papers that use Malgrange's theorem, but I don't understand how we can apply Malgrange to $\phi_2$. I would also like to find a proof that doesn't use such a powerful theorem.
These papers:
http://homepages.math.uic.edu/~yau/35%20publications/An.pdf
 A: This has nothing to do with eigenfunctions; it's really just a basic fact about smooth functions that vanish on the boundary of a smooth domain, when one of them is positive in the interior and has nonvanishing gradient on the boundary. 
Suppose $\Omega$ is a domain with smooth boundary and $\phi_1,\phi_2$ are smooth functions on $\overline\Omega$ that vanish on $\partial \Omega$.  Given any point $p\in \partial \Omega$, there exist a neighborhood $U$ of $p$, a smooth function $u_1\colon U\to\mathbb R$ such that $U\cap \overline \Omega = \{q\in U: u_1(q)\ge 0\}$, and smooth functions $g_1,g_2$ on $\overline\Omega$ such that $\phi_i = u_1g_i$ for $i=1,2$. (See my answer to your previous question for details.) If in addition we know that $\phi_1$ has nonzero gradient on the boundary, it follows that $g_1$ does not vanish on $\partial \Omega\cap U$, and therefore $\phi_2/\phi_1 = g_2/g_1$ in a neighborhood of $p$, which is smooth. This shows that $\phi_2/\phi_1$ is smooth in a neighborhood of each boundary point, and it is smooth on the interior because $\phi_1$ never vanishes there.
