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.

someeigenfunction to have a level set with the desired singularity. $\endgroup$ – Noam D. Elkies Jul 4 '17 at 22:52