Critical points of the first Dirichlet eigenfunction

Problem's Description

Let $\Omega$ be a bounded simply connected and sufficiently smooth plane domain. Let $u$ be the first (positive) Dirichlet eigenfunction in $\Omega$ for the Laplace operator. It is known that the critical points of $u$ are isolated and the number of its maximum points is one more than that of the saddle points. If $\Omega$ is convex, $u$ is log-convex and hence $u$ has only one critical (maximum) point. Problem: find an estimate of the number $N$ of maximum points of $u$ in terms of geometric information about $\Omega$. For instance, a conjecture is that $N$ does not exceed the number of (connected components of local) maximum points of the distance function from the boundary of $\Omega$.

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