# can the first eigenfunction of the Dirichlet Laplacian have any saddle points

Let $\Omega$ be a connected, bounded region of $\mathbb{R}^2$. The Laplacian $\Delta$ has a discrete spectrum of functions satisfying $$\Delta f = \lambda f$$ on $\Omega$ with $f=0$ on the boundary $\partial \Omega$. I am particularly interested in the first eigenfunction $f_1$, i.e. the one with smallest magnitude eigenvalue.

It is known that $f_1$ does not vanish anywhere inside $\Omega$, and so WLOG is positive over the region. Therefore it is superharmonic and so has no local minima inside $\Omega$. Obviously, it has at least one local maximum. Numerical experiments suggests that it "usually" has no other maxima and no saddle points. Is this always true? If not, are there conditions on $\Omega$ that guarantee it?

• I'd imagine that there'd be two local maxima in a dumbbell shaped region. – user940 Dec 13 '13 at 1:41
• @ByronSchmuland: Maybe this follows from a symmetry argument? And maybe the convexity of $\Omega$ guarantees the non-existence of saddle points. – gerw Dec 13 '13 at 8:13
• It seem that we can find some useful information in this paper, however, I have not access to it: journals.cambridge.org/action/… – Tomás Dec 13 '13 at 12:33
• I found the paper here: jjgarmel.webs.ull.es/Articulos/… – Tomás Dec 13 '13 at 12:38
• Thanks @Tomás, from that paper we have that convexity and (discrete) rotational symmetry is sufficient. The symmetry condition is much stronger than needed, though. – user7530 Dec 13 '13 at 14:45

a level set of the first Dirichlet eigenfunction on a bounded convex domain in $\mathbb R^d$ is itself convex [274].