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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?

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

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

Predictably, the reference is to Bernhard Kawohl's book Rearrangements and Convexity of Level Sets in PDE.

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  • $\begingroup$ The book of Kawohl does not seem to contain the exact statement. The statement follows from the log concavity of f which follows from work of Brascamp and Lieb ("Some inequalities for Gaussian measures..."). A short proof is given in Appendix B of a paper by Singer, Wong, Yau, and Yau. numdam.org/article/ASNSP_1985_4_12_2_319_0.pdf $\endgroup$ – Chris Judge Oct 31 '17 at 16:39

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