TITLE: Can a Laplace eigenfunction have a level set with a cusp like singularity?
QUESTION [10 upvotes]: Let $\Omega$ be a precompact open subset of ${\mathbb R}^2$ with piecewise 
 smooth boundary. Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either Dirichlet or Neumann conditions on $\partial \Omega$.  Is it possible that some 
level set of $u$ has a `cusp'?  By cusp, I mean a critical point $z$ of $u$ such that the level sets of $u$ topologically foliate a neighborhood of $z$.  
The prototypical example of a cusp is the critical point $(0,0)$ of the function $f(x,y)=x^2+y^3$. A Mathematica generated sketch of the topological foliation is provided here:

S. Y. Cheng observed many years ago that if a critical point lies on the  zero level set of a planar eigenfunction, then near the critical point, the zero level set is akin to that of a harmonic polynomial. It follows  that the
answer to the question is `no' if restricted to critical points that lie in the zero level set.

REPLY [3 votes]: The question has been answered affirmatively by Olti Myrtaj and Kelly Chen as part of a Research Experiences for Undergraduates project. See here