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In the cases I've come across, the lowest eigenfunction of the Laplacian on a given domain has just one maximum. Is this something that has been proven to be true?

I know we can equivalently find eigenfunctions by extremising the Dirichlet energy, $$E[u]=\int_\Omega \lVert \nabla u \rVert^2 dV, $$ where intuitively it would make sense that a single peaked function will have a lower "energy" than one with many peaks.

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I suppose it's for the Dirichlet boundary problem. The answer is negative for general domains. Consider a dumbbell shape formed by two equal balls connected by a thin tube. The lowest eigenfunction will share the symmetries of the domain. Since the tube is thin, it will be near zero within the tube, so there will be two symmetric maxima near the centers of the balls.

For convex domains the maximum is unique. More precisely, the lowest eigenfunction is log-concave (its logarithm is concave). This is mentioned here with a reference to

H.J. Brascamp, E.H. Lieb, Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma, In: Funct. Integr. Appl., Proc. Int. Conf. London 1974, Oxford (1975), 1-14.

Not the easiest paper to find, unfortunately. You don't actually need log-concavity, the convexity of level sets would be enough. For that, a convenient reference is

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics Vol. 1150 (Springer-Verlag, Heidelberg, 1985)

specifically page 103. A more recent development is the paper Convexity in x of the level sets of the first Dirichlet eigenfunction by Chie-Ping Chu which deals with directional convexity: "On a planar domain which is convex in x, the level sets of the first Dirichlet eigenfunction for Laplacian are also convex in x."

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  • $\begingroup$ Thank you for your answer, for both the intuition, and the effort of providing references. $\endgroup$
    – CDCM
    Commented Mar 13, 2018 at 13:38

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