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?